Microeconomics BECGE1217 Notes

Microeconomics is a branch of economics that studies the behavior of individual economic decision-makers, such as consumers, households, and firms, in allocating scarce resources to satisfy unlimited human wants. It focuses on how these individual decisions interact to determine the supply and demand for goods and services, as well as the effects of government interventions on these behaviors.

I. Introduction to Microeconomics

A. Why Study Microeconomics?

• Economics is the science of allocating scarce resources to satisfy unlimited human wants.

• Microeconomics studies the economic behavior of individual decision-makers, providing a foundation for understanding the macroeconomy and the role of government.

B. Key Analytical Tools

Microeconomic models rely on three key analytical tools:

1. Constrained optimization: Making the best choice given limitations.

2. Equilibrium analysis: Describing a stable state where no one has an incentive to change their behavior.

3. Comparative statics: Analyzing how an equilibrium changes in response to external shocks.

C. Constrained Optimization

Constrained optimization is an analytical tool for making the best choice, taking into account any possible limitations or restrictions on the choice.

• Objective function: The relationship the decision-maker seeks to maximize or minimize (e.g., sales, utility, profit).

• Constraints: Restrictions or limits imposed on the decision-maker (e.g., budget, time, resources).

• Decision-makers are assumed to be rational, meaning they choose the best available alternative.

• It often involves marginal reasoning, where decisions are made by evaluating the additional benefit versus additional cost of one more unit of an activity.

Example: Advertising Budget Allocation

A brewery manager must allocate a €1 million budget between TV () and radio () ads to maximize beer sales .

</p></blockquote><p>Based on sales data for TV and radio advertising, it is observed that:</p><ul class="tight" data-tight="true"><li><p>Spending the entire budget on TV might not be optimal, even if TV always generates more sales for the same amount spent.</p></li><li><p>The <strong>marginal impact</strong> of advertising (additional sales per additional euro) is decreasing.</p></li><li><p>The optimal solution balances the marginal impact across different channels.</p></li></ul><blockquote><p>The term <strong>marginal</strong> in microeconomics refers to how a dependent variable changes as a result of adding one unit of an independent variable.</p></blockquote><h3>D. Equilibrium Analysis</h3><p>An <strong>equilibrium</strong> is a state that will continue indefinitely as long as factors exogenous to the system remain unchanged.</p><ul class="tight" data-tight="true"><li><p><strong>Equilibrium principle</strong>: Prices adjust until the quantity demanded equals the quantity supplied.</p></li></ul><p>Example: Rental Market for Apartments</p><ul class="tight" data-tight="true"><li><p><strong>Demand curve</strong>: Reflects willingness to pay. Reservation price is the highest price a person is willing to pay. The lower the price, the higher the quantity demanded.</p></li><li><p><strong>Supply curve</strong>: In the short run, the quantity of apartments is often fixed.</p></li><li><p><strong>Market equilibrium</strong>: Determined by the intersection of supply and demand (<span data-latex="p^*" data-type="inline-math"></span>).</p></li><li><p>If <span data-latex="p < p^*" data-type="inline-math"></span>, quantity demanded &gt; quantity supplied <span data-latex="\rightarrow" data-type="inline-math"></span> price rises.</p></li><li><p>If <span data-latex="p > p^*" data-type="inline-math"></span>, quantity demanded &lt; quantity supplied <span data-latex="\rightarrow" data-type="inline-math"></span> price falls.</p></li><li><p>The market efficiently allocates apartments to those with the highest willingness to pay.</p></li></ul><h3>E. Comparative Statics</h3><p><strong>Comparative statics</strong> involves comparing two static equilibria without analyzing the dynamic process of adjustment.</p><ul class="tight" data-tight="true"><li><p>It examines how changes in <strong>exogenous variables</strong> (variables taken as given, e.g., quantity of apartments, income) affect the equilibrium.</p></li></ul><p>Examples:</p><ol class="tight" data-tight="true"><li><p><strong>Construction of new apartments</strong>: Increase in supply <span data-latex="\rightarrow" data-type="inline-math"></span> fall in equilibrium price, rise in equilibrium quantity.</p></li><li><p><strong>Conversion of apartments to condominiums</strong>: Reduction in supply + decrease in demand <span data-latex="\rightarrow" data-type="inline-math"></span> decrease in equilibrium quantity, ambiguous effect on equilibrium price.</p></li></ol><h2>II. Consumer Theory: Budget Constraint</h2><p>Consumer theory aims to understand how consumers make choices about what to buy given their income and prices. The first step in this analysis is understanding the limitations or constraints faced by a consumer.</p><h3>A. Consumption Bundles and Prices</h3><ul class="tight" data-tight="true"><li><p>A consumer chooses between two goods.</p></li><li><p><strong>Consumption bundle</strong> <span data-latex="(x_1, x_2)" data-type="inline-math"></span>: Represents quantities of good 1 and good 2. Sometimes denoted as a vector <span data-latex="X" data-type="inline-math"></span>.</p></li><li><p>Prices of goods are <span data-latex="(p_1, p_2)" data-type="inline-math"></span>.</p></li><li><p>Consumer has an income <span data-latex="m" data-type="inline-math"></span> to spend.</p></li></ul><h3>B. The Budget Set</h3><p>The <strong>budget set</strong> consists of all affordable consumption bundles <span data-latex="(x_1, x_2)" data-type="inline-math"></span> given prices <span data-latex="(p_1, p_2)" data-type="inline-math"></span> and income <span data-latex="m" data-type="inline-math"></span>.</p><p><span data-latex="p1x1 + p2x2 =< m " data-type="inline-math"></span>

C. The Composite Good

• Often, we focus on the demand for one specific good ().

• can represent a composite good: everything else the consumer might want to consume, with its price typically normalized to .

• In this case, the budget constraint becomes: .

D. The Budget Line

The budget line represents bundles that cost exactly .

$</p><p>Rearranging for <span data-latex="x_2" data-type="inline-math"></span>:</p><p><span data-latex="x2 = m/p2 - p1/p2 * x1 " data-type="inline-math"></span>$

• This is a straight line with a vertical intercept of and a slope of .

• The slope of the budget line measures the rate at which the market allows substitution between good 1 and good 2. For an increase of in good 1, consumption of good 2 must decrease by .

• The slope also represents the opportunity cost of consuming good 1 (how many units of good 2 must be forgone for 1 unit of good 1).

E. Changes in the Budget Line

The budget line shifts or rotates when prices or income change:

• Change in income (): A parallel shift of the budget line. Increase in shifts it outwards, decrease shifts it inwards.

• Change in price of one good (e.g., ): Changes the slope of the budget line. An increase in makes the line steeper; a decrease makes it flatter.

• Proportional changes in prices and income: If both prices and income are multiplied by the same factor , the budget line does not change. A perfectly balanced inflation does not alter the budget set.

F. The Numeraire

• The budget line is defined by two prices and income. What matters are relative prices.

• We can normalize one of the prices or income to 1 without changing the budget set.

• The good whose price is set to 1 is called the numeraire. It is the price relative to which other prices and income are measured.

G. Taxes, Subsidies, and Rationing

• Quantity tax ( per unit of good 1): Price of good 1 becomes . $</p></li><li><p><strong>Value tax (ad valorem tax)</strong> (<span data-latex="\tau\%" data-type="inline-math"></span> on sales of good 1): Price of good 1 becomes <span data-latex="(1 + \tau)p_1" data-type="inline-math"></span>. <span data-latex="(1 + ω )p1x1 + p2x2 = m" data-type="inline-math"></span>$

• Quantity subsidy ( per unit of good 1): Price of good 1 becomes .

• Ad valorem subsidy ( for good 1): Price of good 1 becomes .

• Lump-sum tax/subsidy: Changes income directly.

• Rationing constraints: Limits the consumption of a good to a maximum amount (e.g., ), altering the shape of the budget set.

• These interventions can lead to kinks or truncated budget lines. Example: Food Stamp Program.

III. Consumer Theory: Preferences

Preferences define what a consumer considers "best" among available alternatives. They allow consumers to rank different consumption bundles.

A. Definition of Preferences

• Preferences (or tastes) refer to a consumer's ability to compare or rank consumption bundles.

• Given two bundles and consumers can express:

  • Strictly prefers:

  • Indifferent:

  • At least as good as:

• Preference relations are ordinal; they only indicate the order of preference, not the magnitude of preference.

B. Assumptions about Preferences (Axioms of Rational Choice)

To ensure consistent and reasonable choices, preferences are usually assumed to satisfy:

1. Completeness: Any two bundles can be compared. For any and , either , , or .

2. Reflexivity: Any bundle is at least as good as itself ().

3. Transitivity: If and , then . If and , then . These assumptions are crucial for utility maximization.

C. Indifference Curves

• A weakly preferred set for a bundle includes all bundles at least as good as .

• An indifference curve connects all bundles that provide the consumer with the same level of utility (i.e., the consumer is indifferent between them).

• Property: Indifference curves representing distinct levels of preference cannot cross. If they did, it would violate transitivity.

D. Examples of Preferences and Indifference Curves

1. Perfect Substitutes:

   • Consumers are willing to substitute one good for another at a constant rate.

   • Indifference curves are parallel straight lines (e.g., ).

2. Perfect Complements:

   • Goods are always consumed together in fixed proportions.

   • Indifference curves are L-shaped (e.g., ).

3. Bads:

   • A commodity the consumer dislikes. To compensate for a bad, the consumer needs more of the other good.

   • Indifference curves slope upward and to the right.

4. Neutral Goods:

   • A good the consumer doesn't care about; always indifferent to increasing amounts.

   • Indifference curves are horizontal or vertical lines.

5. Satiation (Bliss Point):

   • There is an overall best bundle for the consumer. Consuming beyond this point makes a good a "bad."

   • Indifference curves are typically circular or oval around the satiation point.

6. Discrete Goods:

   • Goods available only in integer amounts (e.g., cars).

   • Indifference "curves" are sets of discrete points.

E. Well-Behaved Preferences (Additional Assumptions)

To ensure that consumer choices can be meaningfully determined, we often impose additional assumptions:

1. Monotonicity (More is Better):

   • If bundle has at least as much of both goods as bundle and more of one, then is strictly preferred to .

   • This implies indifference curves always have a negative slope (ruling out bads and neutral goods).

2. Convexity (Averages are Preferred to Extremes):

   • If two bundles and are on the same indifference curve (), then any weighted average of these bundles (e.g., for ) is at least as good as or .

   • Graphically, the set of bundles weakly preferred to is a convex set.

   • This reflects a preference for balanced consumption. Goods are typically consumed together.

   • Strict convexity: Straight line segment connecting two indifferent bundles is strictly preferred to either extreme bundle. Indifference curves are rounded. Perfect substitutes preferences are convex but not strictly convex (flat spots).

F. The Marginal Rate of Substitution (MRS)

• The MRS is the slope of an indifference curve, measuring the rate at which a consumer is just willing to substitute one good for another while maintaining the same level of utility.

• For a small change in good 1, is the change in good 2 that keeps utility constant.

• . It is negative because of monotonicity.

• It can also be interpreted as the marginal willingness to pay: the amount of good 2 (e.g., euros) a consumer is willing to give up for a marginal amount of good 1.

• Mathematically, for a utility function , the MRS is given by: $</p></li></ul><h2>IV. Consumer Theory: Utility Functions</h2><p>A utility function provides a numerical representation of a consumer's preferences, assigning higher values to more preferred bundles.</p><h3>A. Definition of a Utility Function</h3><ul class="tight" data-tight="true"><li><p>A <strong>utility function</strong> <span data-latex="u(X)" data-type="inline-math"></span> assigns a number to every possible consumption bundle <span data-latex="X" data-type="inline-math"></span> such that more-preferred bundles receive larger numbers than less-preferred bundles. <span data-latex="(x1, x2) > (y1; y2) if and only if u(x1, x2) > u(y1, y2)" data-type="inline-math"></span>$

• This is ordinal utility, meaning only the ranking of bundles matters, not the absolute magnitude of the utility values.

• Monotonic transformation: A function that preserves the order of numbers (). Any monotonic transformation of a utility function represents the same preferences. $</p></li><li><p><strong>Cardinal utility</strong>, which suggests that the magnitude of utility differences is significant, is not typically required to describe choice behavior in microeconomics.</p></li><li><p>Well-behaved preferences (completeness, transitivity, monotonicity, convexity) can be represented by a utility function.</p></li><li><p>A utility function essentially labels indifference curves, with higher indifference curves receiving larger utility values.</p></li></ul><h3>B. Constructing a Utility Function and Indifference Curves</h3><ul class="tight" data-tight="true"><li><p>To draw indifference curves from <span data-latex="u(x_1, x_2)" data-type="inline-math"></span>: plot points <span data-latex="(x_1, x_2)" data-type="inline-math"></span> where <span data-latex="u(x_1, x_2)" data-type="inline-math"></span> equals a constant <span data-latex="k" data-type="inline-math"></span>.</p></li><li><p>Example: <span data-latex="u(x_1, x_2) = x_1 x_2" data-type="inline-math"></span>. Indifference curves are <span data-latex="x_2 = k/x_1" data-type="inline-math"></span>.</p></li><li><p>A monotonic transformation like <span data-latex="v(x_1, x_2) = (x_1 x_2)^2" data-type="inline-math"></span> would represent the same preferences, only changing the labels of the indifference curves.</p></li></ul><h3>C. Examples of Utility Functions</h3><ol class="tight" data-tight="true"><li><p><strong>Perfect Substitutes</strong>: <span data-latex="u(x1, x2) = ax1 + bx2" data-type="inline-math"></span>$

  • Slope of indifference curve (MRS) is constant at .

  • Example: (1 unit of good 1 for 1 unit of good 2).

• Perfect Complements: $</p><ul class="tight" data-tight="true"><li><p>Goods consumed in fixed proportions <span data-latex="b" data-type="inline-math"></span>: <span data-latex="a" data-type="inline-math"></span>.</p></li><li><p>Example: <span data-latex="u(x_1, x_2) = \min(x_1, x_2)" data-type="inline-math"></span> (pairs of shoes).</p></li></ul></li><li><p><strong>Quasilinear Preferences</strong>: <span data-latex="u(x1, x2) = v (x1) + x2" data-type="inline-math"></span>$

  • Linear in good 2, possibly nonlinear in good 1.

  • Indifference curves are vertically shifted versions of each other (e.g., ).

• Cobb-Douglas Utility Function: $</p><ul class="tight" data-tight="true"><li><p>Standard example of well-behaved preferences (convex, monotonic indifference curves).</p></li><li><p>Any monotonic transformation (e.g., taking the logarithm) describes the same preferences. Can be normalized so exponents sum to 1: <span data-latex="x_1^a x_2^{1-a}" data-type="inline-math"></span>.</p></li></ul></li></ol><h3>D. Marginal Utility (MU)</h3><ul class="tight" data-tight="true"><li><p>The <strong>marginal utility</strong> of good 1 (<span data-latex="MU_1" data-type="inline-math"></span>) is the rate of change of the utility function as the quantity of good 1 increases, holding good 2 constant. <span data-latex="$ MU_1 = \frac{\Delta U}{\Delta x_1} = \frac{u(x_1 + \Delta x_1, x_2) - u(x_1, x_2)}{\Delta x_1} " data-type="inline-math">$</p></li><li><p>Marginal utility is affected by monotonic transformations, so its magnitude has no behavioral content. Only the ordering of bundles matters.</p></li></ul><h3>E. Marginal Utility and MRS Relationship</h3><ul class="tight" data-tight="true"><li><p>The MRS is directly related to marginal utilities. For a change <span data-latex="(\Delta x_1, \Delta x_2)" data-type="inline-math"></span> that keeps utility constant (<span data-latex="\Delta U = 0" data-type="inline-math"></span>): <span data-latex="$ MU_1 \Delta x_1 + MU_2 \Delta x_2 = 0 " data-type="inline-math">$</p></li><li><p>This relationship holds for any utility function representing the preferences. The MRS is independent of the utility representation because monotonic transformations scale <span data-latex="MU_1" data-type="inline-math"></span> and <span data-latex="MU_2" data-type="inline-math"></span> by the same factor, which then cancels in the ratio.</p></li></ul><p>Example: Cobb-Douglas MRS</p><p>For <span data-latex="u(x_1, x_2) = x_1^c x_2^d" data-type="inline-math"></span>, the MRS is:</p><p><span data-latex="$ MRS = -\frac{c x_2}{d x_1} " data-type="inline-math">$</p><p>Taking the log transformation <span data-latex="v(x_1, x_2) = c \ln x_1 + d \ln x_2" data-type="inline-math"></span> yields the same MRS, confirming its independence from utility representation.</p><h3>F. Utility in Commuting Decisions</h3><ul class="tight" data-tight="true"><li><p>Utility functions can be estimated to understand consumer behavior in real-world contexts, such as commuting.</p></li><li><p>Example: <span data-latex="U(TW, TT, C) = -0.147 TW - 0.0411 TT - 2.24 C" data-type="inline-math"></span>, where <span data-latex="TW=" data-type="inline-math"></span> walking time, <span data-latex="TT=" data-type="inline-math"></span> travel time, <span data-latex="C=" data-type="inline-math"></span> costs.</p></li><li><p>Coefficients represent the marginal utility of each characteristic.</p></li><li><p>Ratios of coefficients provide MRS (e.g., how much walking time one is willing to substitute for travel time).</p></li></ul><h2>V. Consumer Theory: Optimal Choice</h2><p>The optimal choice for a consumer is the most preferred bundle that is affordable, combining the concepts of the budget set and preferences.</p><h3>A. Optimal Choice Principle</h3><ul class="tight" data-tight="true"><li><p>Consumers choose the best bundle they can afford.</p></li><li><p>This means selecting the bundle within the budget set that lies on the highest possible indifference curve.</p></li><li><p>With well-behaved preferences, the optimal bundle is typically on the budget line (since "more is preferred to less").</p></li></ul><h3>B. Tangency Condition for Optimal Choice</h3><ul class="tight" data-tight="true"><li><p>For well-behaved (smooth, convex, monotonic) preferences, the optimal bundle occurs where an indifference curve is tangent to the budget line.</p></li><li><p>At this point, the slope of the indifference curve (MRS) equals the slope of the budget line (price ratio): <span data-latex="$ MRS = - \frac{p_1}{p_2} " data-type="inline-math">$</p></li><li><p>If indifference curves are not smooth (e.g., kinked for perfect complements) or if the optimum is a corner solution (e.g., consuming zero of one good), tangency may not strictly hold.</p><ul class="tight" data-tight="true"><li><p><strong>Kinky preferences</strong> (e.g., perfect complements): Tangency is not defined at the kink.</p></li><li><p><strong>Boundary optimum</strong>: Optimal consumption involves zero units of one good. Indifference curve is not tangent to the budget line.</p></li></ul></li><li><p>For an <strong>interior optimum</strong> with smooth indifference curves, the tangency condition is a <strong>necessary condition</strong>.</p></li><li><p>With <strong>convex preferences</strong>, any tangency point is also an optimum (a <strong>sufficient condition</strong>). If preferences are strictly convex, there is a unique optimal choice.</p></li></ul><h3>C. Optimality and the MRS</h3><ul class="tight" data-tight="true"><li><p>At an interior optimum, <span data-latex="MRS = -p_1/p_2" data-type="inline-math"></span>. This means the consumer's subjective rate of exchange between goods (MRS) matches the market's objective rate of exchange (<span data-latex="p_1/p_2" data-type="inline-math"></span>).</p></li><li><p>If MRS does not equal the price ratio, the consumer can improve their welfare by trading goods in the market.</p></li></ul><h3>D. Consumer Demand (Marshallian Demand Functions)</h3><ul class="tight" data-tight="true"><li><p>The <strong>demanded bundle</strong> is the optimal choice of goods for given prices <span data-latex="p_1, p_2" data-type="inline-math"></span> and income <span data-latex="m" data-type="inline-math"></span>.</p></li><li><p>A <strong>demand function</strong> expresses the quantities demanded as a function of prices and income: <span data-latex="$ x_1 = x_1(p_1, p_2, m) \quad \text{and} \quad x_2 = x_2(p_1, p_2, m) " data-type="inline-math">$</p></li><li><p>Different preferences lead to different demand functions.</p></li></ul><h3>E. Examples of Demand Functions</h3><ol class="tight" data-tight="true"><li><p><strong>Perfect Substitutes</strong>:</p><ul class="tight" data-tight="true"><li><p>Consumer buys only the cheaper good. If prices are equal, any combination on the budget line is optimal. <span data-latex="$ x_1 = \begin{cases} m/p_1 & \text{if } p_1 < p_2 \\ \text{[0, m/p1]} & \text{if } p_1 = p_2 \\ 0 & \text{if } p_1 > p_2 \end{cases} " data-type="inline-math">$</p></li></ul></li><li><p><strong>Perfect Complements</strong>:</p><ul class="tight" data-tight="true"><li><p>Goods are consumed in fixed proportions. The budget constraint is <span data-latex="p_1 x + p_2 x = m \Rightarrow x = m/(p_1 + p_2)" data-type="inline-math"></span>. Thus, <span data-latex="$ x_1 = x_2 = \frac{m}{p_1 + p_2} " data-type="inline-math">$</p></li></ul></li><li><p><strong>Neutral/Bad Goods</strong>:</p><ul class="tight" data-tight="true"><li><p>Consumer spends all income on the desired good, none on the neutral or bad good. If good 1 is good, good 2 is bad: <span data-latex="$ x_1 = m/p_1, \quad x_2 = 0 " data-type="inline-math">$</p></li></ul></li><li><p><strong>Discrete Goods</strong>:</p><ul class="tight" data-tight="true"><li><p>Optimal choice depends on reservation prices. As price falls, consumer buys more units at specific price thresholds.</p></li></ul></li><li><p><strong>Concave Preferences</strong>:</p><ul class="tight" data-tight="true"><li><p>Optimal choice is usually a corner solution, consuming only one of the goods.</p></li></ul></li></ol><h3>F. Solving the Utility Maximization Problem</h3><p>To find demand functions algebraically:</p><ol class="tight" data-tight="true"><li><p>Set <span data-latex="MRS = -p_1/p_2" data-type="inline-math"></span> (or <span data-latex="\frac{MU_1}{MU_2} = \frac{p_1}{p_2}" data-type="inline-math"></span>).</p></li><li><p>Use the budget constraint: <span data-latex="p_1 x_1 + p_2 x_2 = m" data-type="inline-math"></span>.</p></li><li><p>Solve the system of two equations for <span data-latex="x_1" data-type="inline-math"></span> and <span data-latex="x_2" data-type="inline-math"></span> in terms of <span data-latex="p_1, p_2, m" data-type="inline-math"></span>.</p></li></ol><p>Alternatively, using Lagrange Multipliers:</p><p><span data-latex="$ L = u(x_1, x_2) - \lambda(p_1 x_1 + p_2 x_2 - m) " data-type="inline-math">$</p><p>First-order conditions:</p><p><span data-latex="$ \frac{\partial L}{\partial x_1} = MU_1 - \lambda p_1 = 0 \Rightarrow MU_1 = \lambda p_1 " data-type="inline-math">$</p><p><span data-latex="$ \frac{\partial L}{\partial x_2} = MU_2 - \lambda p_2 = 0 \Rightarrow MU_2 = \lambda p_2 " data-type="inline-math">$</p><p><span data-latex="$ \frac{\partial L}{\partial \lambda} = p_1 x_1 + p_2 x_2 - m = 0 " data-type="inline-math">$</p><ul class="tight" data-tight="true"><li><p>From the first two, <span data-latex="\frac{MU_1}{p_1} = \frac{MU_2}{p_2} = \lambda" data-type="inline-math"></span>. This means the marginal utility per euro spent is equal for all goods.</p></li><li><p><span data-latex="\lambda" data-type="inline-math"></span> (Lagrange multiplier) is the marginal utility of income: how much utility increases from an additional euro of income.</p></li></ul><p>Example: Cobb-Douglas Demand Functions</p><p>For <span data-latex="u(x_1, x_2) = x_1^c x_2^d" data-type="inline-math"></span>, using logarithms for simplicity, demand functions are:</p><p><span data-latex="$ x_1 = \frac{c}{c + d} \frac{m}{p_1} \quad \text{and} \quad x_2 = \frac{d}{c + d} \frac{m}{p_2} " data-type="inline-math">$</p><p>If exponents sum to 1 (<span data-latex="u(x_1, x_2) = x_1^\alpha x_2^{1-\alpha}" data-type="inline-math"></span>):</p><p><span data-latex="$ x_1 = \alpha \frac{m}{p_1} \quad \text{and} \quad x_2 = (1-\alpha) \frac{m}{p_2} " data-type="inline-math">$</p><p>This shows a fixed percentage of income spent on each good.</p><h3>G. Estimating Utility Functions</h3><ul class="tight" data-tight="true"><li><p>In practice, we observe demand behavior and infer the underlying preferences/utility functions.</p></li><li><p>Example: If expenditure shares are constant (as suggested by Cobb-Douglas demand), a Cobb-Douglas utility function can fit observed data well.</p></li></ul><h3>H. Implications of the MRS Condition</h3><ul class="tight" data-tight="true"><li><p>If all consumers face the same prices, are optimizing, and have interior solutions, they must all have the same MRS between any two goods.</p></li><li><p>This implies that individual marginal valuations align with market marginal valuations.</p></li></ul><h3>I. Choosing Taxes: Income Tax vs. Quantity Tax</h3><ul class="tight" data-tight="true"><li><p>A <strong>quantity tax</strong> increases the effective price of a good, shifting the budget line inwards and changing its slope.</p></li><li><p>A revenue-equivalent <strong>income tax</strong> (one that generates the same tax revenue) shifts the budget line inwards parallel to the original budget line.</p></li><li><p>For a single consumer with well-behaved preferences, a revenue-equivalent income tax generally leaves the consumer better off than a quantity tax, because the income tax allows them to choose a bundle on a higher indifference curve.</p></li><li><p>Limitations: This result simplifies for a single consumer, doesn't account for incentives to work, market supply responses, and potential for uniform taxes across heterogenous consumers.</p></li></ul><h3>J. Indirect Utility Function and Roy's Identity</h3><ul class="tight" data-tight="true"><li><p>The <strong>indirect utility function</strong> <span data-latex="v(p_1, p_2, m)" data-type="inline-math"></span> gives the maximum utility attainable for a given set of prices and income. <span data-latex="$ v(p_1, p_2, m) = u(x_1(p_1, p_2, m), x_2(p_1, p_2, m)) " data-type="inline-math">$</p></li><li><p>The partial derivative of indirect utility with respect to income (<span data-latex="\partial v / \partial m" data-type="inline-math"></span>) equals the Lagrange multiplier <span data-latex="\lambda" data-type="inline-math"></span>, confirming <span data-latex="\lambda" data-type="inline-math"></span>'s interpretation as the marginal utility of income.</p></li><li><p><strong>Roy's Identity</strong> connects the indirect utility function to the Marshallian demand functions: <span data-latex="$ x_1(p_1, p_2, m) = - \frac{\partial v / \partial p_1}{\partial v / \partial m} " data-type="inline-math">$ This allows derivation of demand functions from indirect utility.

VI. Consumer Theory: Demand Functions

Demand functions describe how optimal consumption bundles change with variations in prices and income. This section explores various properties and characteristics of these demand functions.

A. Comparative Statics Revisited

Comparative statics analyze how the demanded bundle changes as prices or income vary.

$</p><h3>B. Income Changes and Demand</h3><ul class="tight" data-tight="true"><li><p><strong>Normal good</strong>: Demand increases as income increases (<span data-latex="\partial x_1 / \partial m > 0" data-type="inline-math"></span>).</p></li><li><p><strong>Inferior good</strong>: Demand decreases as income increases (<span data-latex="\partial x_1 / \partial m < 0" data-type="inline-math"></span>). Examples include bus trips or certain inexpensive foods, often dependent on income level.</p></li><li><p><strong>Income offer curve (or income expansion path)</strong>: Illustrates the bundles demanded at different income levels, holding prices constant. If both goods are normal, it has a positive slope.</p></li><li><p><strong>Engel curve</strong>: Plots the demand for one good as a function of income, holding all prices constant. Upward sloping for normal goods, downward sloping for inferior goods.</p></li></ul><p>Examples of Income Offer and Engel Curves:</p><ul class="tight" data-tight="true"><li><p><strong>Perfect Substitutes</strong>: Income offer curve lies along one axis if one good is cheaper, or fills the budget line if prices are equal. Engel curve is linear.</p></li><li><p><strong>Perfect Complements</strong>: Income offer curve is a straight line through the origin. Engel curve is also linear.</p></li><li><p><strong>Cobb-Douglas</strong>: Demand functions are linear in income. Income offer curves are straight lines through the origin. Engel curves are also linear.</p></li><li><p><strong>Luxury good</strong>: Demand increases more than proportionally with income.</p></li><li><p><strong>Necessary good</strong>: Demand increases less than proportionally with income.</p></li><li><p><strong>Homothetic preferences</strong>: If preferences depend only on the ratio of goods. The income offer curve and Engel curves are straight lines through the origin. If income scales by <span data-latex="t" data-type="inline-math"></span>, the demanded bundle also scales by <span data-latex="t" data-type="inline-math"></span>.</p></li><li><p><strong>Quasilinear Preferences</strong>: Demand for good 1 is independent of income (<span data-latex="\partial x_1 / \partial m = 0" data-type="inline-math"></span>). The Engel curve for good 1 is a vertical line. Income changes primarily affect the consumption of good 2 (the linear component).</p></li></ul><h3>C. Price Changes and Demand</h3><ul class="tight" data-tight="true"><li><p><strong>Ordinary good</strong>: Demand increases as its own price decreases (<span data-latex="\Delta x_1 / \Delta p_1 < 0" data-type="inline-math"></span>).</p></li><li><p><strong>Giffen good</strong>: Demand decreases as its own price decreases (<span data-latex="\Delta x_1 / \Delta p_1 > 0" data-type="inline-math"></span>). This is a rare case where the income effect outweighs the substitution effect for an inferior good.</p></li></ul><p>Price Offer Curve and Demand Curve</p><ul class="tight" data-tight="true"><li><p><strong>Price offer curve</strong>: The curve connecting all utility-maximizing bundles as the price of one good changes (e.g., <span data-latex="p_1" data-type="inline-math"></span>), holding other prices and income constant.</p></li><li><p><strong>Demand curve</strong>: Plots the quantity demanded of a good (<span data-latex="x_1" data-type="inline-math"></span>) against its own price (<span data-latex="p_1" data-type="inline-math"></span>), holding <span data-latex="p_2" data-type="inline-math"></span> and <span data-latex="m" data-type="inline-math"></span> fixed.</p></li></ul><p>Examples of Demand Curves:</p><ul class="tight" data-tight="true"><li><p><strong>Perfect Substitutes</strong>: Demand is <span data-latex="m/p_1" data-type="inline-math"></span> if <span data-latex="p_1 < p_2" data-type="inline-math"></span>, 0 if <span data-latex="p_1 > p_2" data-type="inline-math"></span>, and any amount on the budget line if <span data-latex="p_1 = p_2" data-type="inline-math"></span>. The demand curve is stepped.</p></li><li><p><strong>Perfect Complements</strong>: Demand is <span data-latex="m/(p_1 + p_2)" data-type="inline-math"></span>. The demand curve is downward sloping.</p></li><li><p><strong>Discrete Good</strong>: Demand curve is defined by a sequence of reservation prices, forming a step function.</p></li></ul><h3>D. Relation between Goods: Substitutes and Complements</h3><ul class="tight" data-tight="true"><li><p><strong>Substitutes</strong>: If the demand for good 1 increases when the price of good 2 increases (<span data-latex="\partial x_1 / \partial p_2 > 0" data-type="inline-math"></span>).</p></li><li><p><strong>Complements</strong>: If the demand for good 1 decreases when the price of good 2 increases (<span data-latex="\partial x_1 / \partial p_2 < 0" data-type="inline-math"></span>).</p></li></ul><h3>E. The Inverse Demand Function</h3><ul class="tight" data-tight="true"><li><p>The <strong>inverse demand function</strong> expresses price as a function of quantity demanded (e.g., <span data-latex="p_1 = p_1(x_1)" data-type="inline-math"></span>).</p></li><li><p>At the optimal choice, the price ratio equals the absolute value of the MRS: <span data-latex="|MRS| = p_1/p_2" data-type="inline-math"></span>.</p></li><li><p>If good 2 is money (<span data-latex="p_2=1" data-type="inline-math"></span>), then <span data-latex="p_1 = |MRS|" data-type="inline-math"></span>. This means the price of good 1 measures the consumer's marginal willingness to pay for it.</p></li><li><p>A downward-sloping demand curve implies that marginal willingness to pay decreases as consumption of a good increases.</p></li></ul><h2>VII. Consumer Theory: Income and Substitution Effects</h2><p>Changes in price affect consumer demand through two distinct channels: a change in relative prices (substitution effect) and a change in purchasing power (income effect).</p><h3>A. Two Effects of a Price Change</h3><p>When the price of a good changes, there are two primary effects:</p><ol class="tight" data-tight="true"><li><p><strong>Substitution effect</strong>: Change in demand due to a change in the relative price, holding purchasing power constant.</p></li><li><p><strong>Income effect</strong>: Change in demand due to a change in purchasing power, holding relative prices constant.</p></li></ol><h3>B. Decomposing the Price Effect (Slutsky Decomposition)</h3><p>The total change in demand (<span data-latex="\Delta x_1" data-type="inline-math"></span>) due to a price change can be decomposed into substitution and income effects:</p><p><span data-latex="$ \Delta x_1 = \Delta x_1^s + \Delta x_1^n " data-type="inline-math">$</p><p>Methodology:</p><ol class="tight" data-tight="true"><li><p><strong>Step 1 (Substitution Effect)</strong>:</p><ul class="tight" data-tight="true"><li><p>Original bundle <span data-latex="X = (x_1, x_2)" data-type="inline-math"></span> with prices <span data-latex="(p_1, p_2)" data-type="inline-math"></span> and income <span data-latex="m" data-type="inline-math"></span>.</p></li><li><p>New price <span data-latex="p_1'" data-type="inline-math"></span>.</p></li><li><p>Adjust income to <span data-latex="m'" data-type="inline-math"></span> such that the original bundle <span data-latex="X" data-type="inline-math"></span> is just affordable at the new prices <span data-latex="(p_1', p_2)" data-type="inline-math"></span>. <span data-latex="$ m' = p_1' x_1 + p_2 x_2 " data-type="inline-math">\Delta m = m' - m = x_1 (p_1' - p_1)$.</p></li><li><p>Find the new optimal bundle <span data-latex="Y = (x_1(p_1', p_2, m'), x_2(p_1', p_2, m'))" data-type="inline-math"></span>. The movement from <span data-latex="X" data-type="inline-math"></span> to <span data-latex="Y" data-type="inline-math"></span> is the substitution effect.</p></li><li><p>The <strong>substitution effect</strong> (<span data-latex="\Delta x_1^s" data-type="inline-math"></span>) is <span data-latex="x_1(p_1', m') - x_1(p_1, m)" data-type="inline-math"></span>.</p></li><li><p>The substitution effect always moves opposite to the price change: if <span data-latex="p_1" data-type="inline-math"></span> decreases, <span data-latex="x_1" data-type="inline-math"></span> increases due to substitution.</p></li></ul></li><li><p><strong>Step 2 (Income Effect)</strong>:</p><ul class="tight" data-tight="true"><li><p>Original income <span data-latex="m" data-type="inline-math"></span> is restored, but at the new relative prices <span data-latex="(p_1', p_2)" data-type="inline-math"></span>.</p></li><li><p>The movement from bundle <span data-latex="Y" data-type="inline-math"></span> (at <span data-latex="m'" data-type="inline-math"></span>) to the final optimal bundle <span data-latex="Z = (x_1(p_1', p_2, m), x_2(p_1', p_2, m))" data-type="inline-math"></span> is the income effect.</p></li><li><p>The <strong>income effect</strong> (<span data-latex="\Delta x_1^n" data-type="inline-math"></span>) is <span data-latex="x_1(p_1', m) - x_1(p_1', m')" data-type="inline-math"></span>.</p></li><li><p>The income effect can be positive (normal good) or negative (inferior good).</p></li></ul></li></ol><p>Total Change in Demand:</p><p><span data-latex="$ x_1(p_1', m) - x_1(p_1, m) = [x_1(p_1', m') - x_1(p_1, m)] + [x_1(p_1', m) - x_1(p_1', m')] " data-type="inline-math">$</p><p>This equation is the <strong>Slutsky Identity</strong>.</p><h3>C. Implications for Normal and Inferior Goods</h3><ul class="tight" data-tight="true"><li><p><strong>Normal good</strong>: Substitution and income effects work in the same direction. When price increases, both effects lead to a decrease in demand.</p></li><li><p><strong>Inferior good</strong>: Substitution and income effects work in opposite directions. When price increases, the substitution effect decreases demand, but the income effect increases demand (because more income is needed for other goods, leading to more consumption of the inferior good).</p><ul class="tight" data-tight="true"><li><p><strong>Giffen good</strong>: A special type of inferior good where the income effect outweighs the substitution effect, causing demand to rise when the price rises. This implies a positively sloped demand curve. Not all inferior goods are Giffen goods.</p></li></ul></li></ul><h3>D. Examples of Income and Substitution Effects</h3><ul class="tight" data-tight="true"><li><p><strong>Perfect Complements</strong>: Only an income effect. The substitution effect is zero because goods must be consumed in fixed proportions.</p></li><li><p><strong>Perfect Substitutes</strong>: Only a substitution effect if the price difference causes a switch between goods. Income effect can also be present, shifting towards cheaper good across the budget line.</p></li><li><p><strong>Quasilinear Preferences</strong>: Income effect for good 1 is zero. Any change in utility comes from changes in the "money spent on other goods" component.</p></li></ul><h3>E. Slutsky Substitution vs. Hicks Substitution</h3><ul class="tight" data-tight="true"><li><p><strong>Slutsky Substitution Effect</strong>: Holds purchasing power constant by making the original bundle affordable at new prices.</p></li><li><p><strong>Hicks Substitution Effect</strong>: Holds utility constant at the original level, allowing the consumer to reach the original indifference curve with the new prices.</p><ul class="tight" data-tight="true"><li><p>The Hicks substitution effect must always be negative (opposite to the price change).</p></li></ul></li></ul><h3>F. Hicksian Demand Functions</h3><ul class="tight" data-tight="true"><li><p><strong>Marshallian demand functions</strong>: Show demand as a function of prices and income.</p></li><li><p><strong>Hicksian (compensated) demand functions</strong> (<span data-latex="x_i^h(p_1, p_2, \bar{u})" data-type="inline-math"></span>): Show demand as a function of prices and a constant utility level <span data-latex="\bar{u}" data-type="inline-math"></span>.</p></li><li><p>Hicksian demand functions are derived from the <strong>expenditure minimization problem</strong>: <span data-latex="$ \min_{x_1, x_2} p_1 x_1 + p_2 x_2 \quad \text{s.t. } u(x_1, x_2) = \bar{u} " data-type="inline-math">$</p></li><li><p>The solution provides the minimum expenditure needed to achieve a target utility level.</p></li></ul><p>Expenditure Function:</p><ul class="tight" data-tight="true"><li><p>The <strong>expenditure function</strong> <span data-latex="e(p_1, p_2, \bar{u})" data-type="inline-math"></span> gives the minimum cost of achieving a fixed level of utility <span data-latex="\bar{u}" data-type="inline-math"></span> at prices <span data-latex="(p_1, p_2)" data-type="inline-math"></span>. <span data-latex="$ e(p_1, p_2, \bar{u}) = p_1 x_1^h(p_1, p_2, \bar{u}) + p_2 x_2^h(p_1, p_2, \bar{u}) " data-type="inline-math">$</p></li><li><p>Properties of expenditure functions:</p><ul class="tight" data-tight="true"><li><p>Homogeneous of degree 1 in prices.</p></li><li><p>Non-decreasing in prices (<span data-latex="\partial e / \partial p_i \geq 0" data-type="inline-math"></span>).</p></li><li><p>Concave in prices.</p></li></ul></li><li><p><strong>Shephard's Lemma</strong>: Hicksian demand functions can be derived by taking the partial derivative of the expenditure function with respect to the corresponding price: <span data-latex="$ x_1^h(p_1, p_2, \bar{u}) = \frac{\partial e(p_1, p_2, \bar{u})}{\partial p_1} " data-type="inline-math">$</p></li></ul><h3>G. Duality in Consumer Theory (Relationship between Marshallian and Hicksian Demands)</h3><ul class="tight" data-tight="true"><li><p>The utility maximization problem and the expenditure minimization problem are duals of each other.</p></li><li><p><strong>Identities</strong>:</p><ol class="tight" data-tight="true"><li><p><span data-latex="x_1^m(p_1, p_2, m) = x_1^h(p_1, p_2, v(p_1, p_2, m))" data-type="inline-math"></span> (Marshallian demand at income <span data-latex="m" data-type="inline-math"></span> is Hicksian demand at utility <span data-latex="v(m)" data-type="inline-math"></span>).</p></li><li><p><span data-latex="x_1^h(p_1, p_2, \bar{u}) = x_1^m(p_1, p_2, e(p_1, p_2, \bar{u}))" data-type="inline-math"></span> (Hicksian demand at utility <span data-latex="\bar{u}" data-type="inline-math"></span> is Marshallian demand at income <span data-latex="e(\bar{u})" data-type="inline-math"></span>).</p></li></ol></li></ul><h3>H. The Slutsky Equation</h3><p>The Slutsky equation formally decomposes the total change in Marshallian demand due to a price change into the substitution and income effects:</p><p><span data-latex="$ \frac{\partial x_1^m}{\partial p_1} = \frac{\partial x_1^h}{\partial p_1} - \frac{\partial x_1^m}{\partial m} x_1^m " data-type="inline-math">$</p><ul class="tight" data-tight="true"><li><p>The first term, <span data-latex="\frac{\partial x_1^h}{\partial p_1}" data-type="inline-math"></span>, is the <strong>substitution effect</strong> (the slope of the compensated demand curve), which is always negative for ordinary goods. It represents the change in demand from a price change, holding utility constant.</p></li><li><p>The second term, <span data-latex="-\frac{\partial x_1^m}{\partial m} x_1^m" data-type="inline-math"></span>, is the <strong>income effect</strong>. It reflects how changes in <span data-latex="p_1" data-type="inline-math"></span> affect purchasing power, which then impacts demand through the income elasticity.</p></li></ul><h2>VIII. Intertemporal Choice</h2><p>Intertemporal choice analyzes how consumers make decisions about consumption over different time periods, typically involving saving or borrowing.</p><h3>A. The Budget Constraint</h3><ul class="tight" data-tight="true"><li><p>A consumer chooses consumption <span data-latex="(c_1, c_2)" data-type="inline-math"></span> in period 1 and 2.</p></li><li><p>Income in each period is <span data-latex="(m_1, m_2)" data-type="inline-math"></span>, representing the <strong>endowment</strong>.</p></li><li><p>Consumer can borrow or lend money at an interest rate <span data-latex="r" data-type="inline-math"></span>.</p></li><li><p>Prices of consumption in each period are assumed to be 1.</p></li></ul><p>Budget Constraint in Future Value:</p><p>Consumption for a saver if <span data-latex="c_1 < m_1" data-type="inline-math"></span>:</p><p><span data-latex="$ c_2 = m_2 + (1 + r)(m_1 - c_1) " data-type="inline-math">$</p><p>Consumption for a borrower if <span data-latex="c_1 > m_1" data-type="inline-math"></span>:</p><p><span data-latex="$ c_2 = m_2 - (1 + r)(c_1 - m_1) " data-type="inline-math">$</p><p>Both cases yield the same intertemporal budget constraint:</p><p><span data-latex="$ (1 + r)c_1 + c_2 = (1 + r)m_1 + m_2 " data-type="inline-math">$</p><p>Budget Constraint in Present Value:</p><p><span data-latex="$ c_1 + \frac{c_2}{1 + r} = m_1 + \frac{m_2}{1 + r} = \text{PV of income} " data-type="inline-math">$</p><ul class="tight" data-tight="true"><li><p>The present-value form is generally more important as it expresses future values in today's terms.</p></li><li><p>The slope of the intertemporal budget line is <span data-latex="-(1+r)" data-type="inline-math"></span>, representing the opportunity cost of current consumption in terms of future consumption.</p></li></ul><h3>B. Preferences for Consumption Across Time</h3><ul class="tight" data-tight="true"><li><p>Consumers have preferences for consumption in different periods, represented by indifference curves. These are typically convex and downward sloping, reflecting a desire to smooth consumption across periods.</p></li><li><p><strong>Borrowers</strong>: Consume more than current income (<span data-latex="c_1 > m_1" data-type="inline-math"></span>), incurring debt to be paid in period 2.</p></li><li><p><strong>Lenders/Savers</strong>: Consume less than current income (<span data-latex="c_1 < m_1" data-type="inline-math"></span>), lending the surplus to consume more in period 2.</p></li></ul><h3>C. Comparative Statics and Interest Rate Changes</h3><ul class="tight" data-tight="true"><li><p>An increase in the interest rate <span data-latex="r" data-type="inline-math"></span> makes the budget line steeper, pivoting around the endowment point <span data-latex="(m_1, m_2)" data-type="inline-math"></span>.</p><ul class="tight" data-tight="true"><li><p>For a <strong>lender</strong>: An increase in <span data-latex="r" data-type="inline-math"></span> makes them better off and they will remain a lender (possibly increasing <span data-latex="c_2" data-type="inline-math"></span> due to higher returns on savings).</p></li><li><p>For a <strong>borrower</strong>: An increase in <span data-latex="r" data-type="inline-math"></span> makes them worse off if they remain a borrower, as their debt becomes more expensive. Using revealed preferences, if they remain a borrower, they're definitely on a lower indifference curve. They might switch to being a lender if the income effect of higher interest rates encourages saving.</p></li></ul></li></ul><h3>D. Present Value Calculations</h3><ul class="tight" data-tight="true"><li><p>The <strong>present value</strong> (PV) measures everything in today's currency. The PV of €1 to be received next period is <span data-latex="\frac{1}{1+r}" data-type="inline-math"></span>.</p></li><li><p>A consumption plan is affordable if the present value of consumption equals the present value of income.</p></li><li><p>Consumers prefer income patterns with a higher present value if they can freely borrow and lend.</p></li><li><p>The concept extends to multiple periods: <span data-latex="$ c_1 + \frac{c_2}{1+r} + \frac{c_3}{(1+r)^2} = m_1 + \frac{m_2}{1+r} + \frac{m_3}{(1+r)^2} " data-type="inline-math">tp_t = \frac{1}{(1+r)^{t-1}}$.</p></li><li><p><strong>Net Present Value (NPV)</strong>: Compares revenues and costs of an investment over time. For an income stream <span data-latex="(M_1, M_2)" data-type="inline-math"></span> and payments <span data-latex="(P_1, P_2)" data-type="inline-math"></span>: <span data-latex="$ NPV = (M_1 - P_1) + \frac{M_2 - P_2}{1+r} " data-type="inline-math">$ An investment should be undertaken if NPV is positive.

IX. Choice Under Uncertainty

This section explores how individuals make decisions when outcomes are uncertain, incorporating the concepts of probability, expected value, and utility.

A. Describing Risky Outcomes

• An investment or event with uncertain outcome is a lottery.

• A probability distribution describes all possible payoffs of a lottery and their probabilities (summing to 1).

• Expected Value (EV): The weighted average of all possible outcomes, where weights are their probabilities. \pi_iiO_i$ is its value)

• Riskiness of a lottery: Measured by variance or standard deviation (square root of variance). Higher variance/standard deviation implies higher risk.

• A fair bet has an expected value of zero.

B. Expected Utility Theory

• Individuals typically do not maximize expected value; they maximize expected utility.

• Most humans are risk-averse; they will not take gambles even with small positive expected values.

• An expected utility function (or von Neumann-Morgenstern utility function) takes the form: v(c_i)i$)</p></li><li><p>Expected utility functions are unique up to a positive affine transformation (<span data-latex="f(u) = au + b" data-type="inline-math"></span>, with <span data-latex="a > 0" data-type="inline-math"></span>).</p></li><li><p>This framework assumes <strong>independence</strong> between outcomes in different states of nature.</p></li></ul><h3>C. Risk Aversion, Loving, and Neutrality</h3><ul class="tight" data-tight="true"><li><p>A consumer is <strong>risk-averse</strong> if the utility of expected wealth is greater than the expected utility of wealth. <span data-latex="$ u(EV) > \sum \pi_i u(O_i) " data-type="inline-math">$ The utility function of a risk-averse consumer is concave. Risk-averse individuals prefer a certain outcome to a gamble with the same expected value.

• A consumer is risk-loving if they prefer a gamble to its expected value. $ The utility function of a risk-loving consumer is convex.

• A consumer is risk-neutral if they are indifferent between a gamble and its expected value. $ The utility function of a risk-neutral consumer is linear. They only care about the expected value of wealth.

D. Risk Premium

• The risk premium (RP) is the minimum difference between the expected value of a lottery and the payoff of a sure thing that would make a decision-maker indifferent between the lottery and the sure thing. </p></li><li><p>For risk-averse individuals, RP &gt; 0. The higher the variance of the lottery (for a given expected value), the larger the risk premium.</p></li></ul><h3>E. The Demand for Insurance</h3><ul class="tight" data-tight="true"><li><p>Insurance offers a way to change the probability distribution of wealth, smoothing consumption across states of nature.</p></li><li><p>A <strong>contingent consumption plan</strong> specifies what will be consumed in each different state of nature.</p></li><li><p>The decision to buy insurance involves trading off consumption in a good state (paying premium) for consumption in a bad state (receiving payout).</p></li><li><p>The slope of the budget line facing someone buying insurance is determined by the insurance premium rate.</p></li><li><p><strong>Fair insurance</strong>: The premium rate <span data-latex="\gamma" data-type="inline-math"></span> equals the probability of the loss <span data-latex="\pi" data-type="inline-math"></span> (<span data-latex="\gamma = \pi" data-type="inline-math"></span>), meaning the insurance company expects zero profit.</p></li><li><p>If offered fair insurance, a risk-averse consumer will always choose to <strong>fully insure</strong>, equalizing consumption across all states of nature (<span data-latex="c_1 = c_2" data-type="inline-math"></span>). This is because their marginal utility of income would be equal across states.</p></li></ul><h2>X. Consumer's Surplus</h2><p>Consumer's surplus is a crucial metric for measuring the benefit consumers receive from consuming a good, beyond what they pay for it. It is particularly clear for discrete goods and with quasilinear utility.</p><h3>A. Demand for a Discrete Good</h3><ul class="tight" data-tight="true"><li><p>For a discrete good, consumers decide whether to buy 0, 1, 2, ... units.</p></li><li><p>A <strong>reservation price</strong> (<span data-latex="r_n" data-type="inline-math"></span>) is the maximum price a consumer is willing to pay for the <span data-latex="n" data-type="inline-math"></span>-th unit of a good. At this price, the consumer is indifferent between consuming <span data-latex="n-1" data-type="inline-math"></span> units and <span data-latex="n" data-type="inline-math"></span> units. <span data-latex=" u(n-1, m - (n-1)p) = u(n, m - np) " data-type="inline-math">$</p></li><li><p>The demand curve for a discrete good is a step function, where each step corresponds to a reservation price.</p></li></ul><p>Quasilinear Utility and Reservation Prices:</p><p>If <span data-latex="u(x_1, x_2) = v(x_1) + x_2" data-type="inline-math"></span> and <span data-latex="v(0)=0" data-type="inline-math"></span> (where <span data-latex="x_1" data-type="inline-math"></span> is discrete, <span data-latex="x_2" data-type="inline-math"></span> is money):</p><ul class="tight" data-tight="true"><li><p><span data-latex="r_1 = v(1)" data-type="inline-math"></span>. (Price making consumer indifferent between 0 and 1 unit).</p></li><li><p><span data-latex="r_n = v(n) - v(n-1)" data-type="inline-math"></span>. (Price making consumer indifferent between <span data-latex="n-1" data-type="inline-math"></span> and <span data-latex="n" data-type="inline-math"></span> units).</p></li><li><p>The assumption of convex preferences implies <span data-latex="r_1 > r_2 > r_3 > \ldots" data-type="inline-math"></span></p></li></ul><h3>B. Constructing Utility from Demand for Discrete Goods</h3><ul class="tight" data-tight="true"><li><p>If <span data-latex="v(0)=0" data-type="inline-math"></span>, the utility from consuming <span data-latex="n" data-type="inline-math"></span> units, <span data-latex="v(n)" data-type="inline-math"></span>, is the sum of the first <span data-latex="n" data-type="inline-math"></span> reservation prices. This area under the demand curve is the <strong>gross consumer's surplus</strong>.</p></li><li><p><strong>Consumer's surplus (CS)</strong> or <strong>net consumer's surplus</strong> is the total willingness to pay minus the actual expenditure: <span data-latex="$ CS = v(n) - pn " data-type="inline-math">n$ units of the good.

C. Other Interpretations of Consumer's Surplus

• The consumer's surplus is how much money one would need to give to a consumer to fully compensate them for losing access to a discrete good. R = v(n) - pn$, directly equal to the consumer's surplus.</p></li></ul><h3>D. Approximating a Continuous Demand</h3><ul class="tight" data-tight="true"><li><p>For continuous goods, the total utility from consuming <span data-latex="x" data-type="inline-math"></span> units is the area under the inverse demand curve (from 0 to <span data-latex="x" data-type="inline-math"></span>). <span data-latex="$ v(x) = \int_0^x p(t) dt " data-type="inline-math">$</p></li><li><p>Consumer's surplus for a continuous good is the area under the demand curve and above the price line.</p></li><li><p>Using the area under the demand curve to measure utility is precisely correct only when the utility function is quasilinear, as it implies no income effects for that good.</p></li><li><p>If income effects are small, the change in consumer's surplus is a reasonable approximation of the change in utility.</p></li></ul><h3>E. Interpreting the Change in Consumer's Surplus</h3><ul class="tight" data-tight="true"><li><p>When the price of a good increases, consumer's surplus decreases. This loss can be divided into:</p><ul class="tight" data-tight="true"><li><p>Loss from paying more for units still consumed.</p></li><li><p>Loss from reduced consumption due to the higher price.</p></li></ul></li></ul><h3>F. Compensating Variation (CV) and Equivalent Variation (EV)</h3><p>These are monetary measures of welfare change, especially useful when utility is not quasilinear.</p><ul class="tight" data-tight="true"><li><p><strong>Compensating Variation (CV)</strong>: The amount of money that would need to be given to (or taken from) the consumer <em>after</em> a price change to restore them to their initial utility level. <span data-latex="$ u_f(m + CV) = u_i(m) " data-type="inline-math">u_iu_f$ is utility at new prices).

• Equivalent Variation (EV): The amount of money that would have to be taken from (or given to) the consumer before a price change to leave them as well off as they would be after the price change. $</p></li><li><p>In general, CV and EV are not equal.</p></li><li><p>For <strong>quasilinear utility</strong>, CV = EV, and both are equal to the change in consumer's surplus. This is due to the lack of income effects.</p></li></ul><h2>XI. Market Demand and Elasticity</h2><p>This section moves from individual demand to market demand, introducing the crucial concept of elasticity to measure responsiveness to changes in prices and income.</p><h3>A. From Individual to Market Demand</h3><ul class="tight" data-tight="true"><li><p><strong>Individual demand function</strong>: <span data-latex="x_{1i}(p_1, p_2, m_i)" data-type="inline-math"></span> for consumer <span data-latex="i" data-type="inline-math"></span>.</p></li><li><p><strong>Market demand</strong>: The sum of individual demands across all consumers: <span data-latex="$ X^1(p_1, p_2, m_1, \ldots, m_n) = \sum_{i=1}^n x_{1i}(p_1, p_2, m_i) " data-type="inline-math">$</p></li><li><p>Market demand depends on prices and the distribution of income.</p></li><li><p>If we assume a <strong>representative consumer</strong>, market demand can be written as <span data-latex="X^1(p_1, p_2, M)" data-type="inline-math"></span>, where <span data-latex="M" data-type="inline-math"></span> is total income.</p></li><li><p><strong>Inverse market demand function</strong> <span data-latex="P(X)" data-type="inline-math"></span>: Measures the market price for <span data-latex="X" data-type="inline-math"></span> units demanded.</p></li></ul><h3>B. Price Elasticity of Demand (<span data-latex="\epsilon" data-type="inline-math"></span>)</h3><ul class="tight" data-tight="true"><li><p>The <strong>price elasticity of demand</strong> is the percentage change in quantity demanded divided by the percentage change in price. <span data-latex="$ \epsilon = \frac{\% \Delta q}{\% \Delta p} = \frac{\Delta q / q}{\Delta p / p} = \frac{dq}{dp} \frac{p}{q} " data-type="inline-math">$</p></li><li><p>Elasticity is typically negative (due to the law of demand for ordinary goods), but the absolute value is often used.</p><ul class="tight" data-tight="true"><li><p><span data-latex="|\epsilon| > 1" data-type="inline-math"></span>: <strong>Elastic</strong> (quantity demanded is very responsive to price changes).</p></li><li><p><span data-latex="|\epsilon| = 1" data-type="inline-math"></span>: <strong>Unit elastic</strong>.</p></li><li><p><span data-latex="|\epsilon| < 1" data-type="inline-math"></span>: <strong>Inelastic</strong> (quantity demanded is not very responsive to price changes).</p></li></ul></li><li><p>For a linear demand curve <span data-latex="q = a - bp" data-type="inline-math"></span>, elasticity is <span data-latex="\epsilon = -b \frac{p}{q}" data-type="inline-math"></span>. It varies along the curve, being more elastic at higher prices and less elastic at lower prices.</p></li></ul><h3>C. Elasticity and Revenue</h3><p>Total revenue <span data-latex="R(p) = p \cdot q(p)" data-type="inline-math"></span>.</p><ul class="tight" data-tight="true"><li><p>How revenue changes with price: <span data-latex="R'(p) = q(p) (1 + \epsilon)" data-type="inline-math"></span>. (using <span data-latex="R'(p) = p q'(p) + q(p)" data-type="inline-math"></span> and rearranging)</p><ul class="tight" data-tight="true"><li><p>If demand is **elastic** (<span data-latex="|\epsilon| > 1" data-type="inline-math"></span>), an increase in price leads to a decrease in total revenue. (A price decrease increases revenue).</p></li><li><p>If demand is **inelastic** (<span data-latex="|\epsilon| < 1" data-type="inline-math"></span>), an increase in price leads to an increase in total revenue. (A price decrease decreases revenue).</p></li><li><p>If demand is **unit elastic** (<span data-latex="|\epsilon| = 1" data-type="inline-math"></span>), a change in price does not affect total revenue.</p></li></ul></li><li><p>A demand curve with constant elasticity of -1 (e.g., <span data-latex="q = \bar{R}/p" data-type="inline-math"></span>) has constant revenue.</p></li><li><p>A general constant elasticity demand function is <span data-latex="q = Ap^\epsilon" data-type="inline-math"></span>.</p></li></ul><h3>D. Elasticity and Marginal Revenue</h3><p><strong>Marginal revenue (MR)</strong> is the change in total revenue from producing one more unit of output.</p><p><span data-latex="$ MR = \frac{dR(q)}{dq} = p(q) + q \frac{dp(q)}{dq} " data-type="inline-math">$</p><p>Expressed in terms of elasticity:</p><p><span data-latex="$ MR = p(q) \left(1 + \frac{1}{\epsilon_p}\right) = p(q) \left(1 - \frac{1}{|\epsilon_p|}\right) " data-type="inline-math">$</p><ul class="tight" data-tight="true"><li><p>When <span data-latex="|\epsilon_p| = 1" data-type="inline-math"></span>, <span data-latex="MR = 0" data-type="inline-math"></span>.</p></li><li><p>When <span data-latex="|\epsilon_p| > 1" data-type="inline-math"></span> (elastic demand), <span data-latex="MR > 0" data-type="inline-math"></span>.</p></li><li><p>When <span data-latex="|\epsilon_p| < 1" data-type="inline-math"></span> (inelastic demand), <span data-latex="MR < 0" data-type="inline-math"></span>.</p></li><li><p>A profit-maximizing firm will never choose a price on the inelastic part of the demand curve, because doing so implies <span data-latex="MR < 0" data-type="inline-math"></span>, meaning decreasing production would actually increase revenue while also decreasing costs.</p></li></ul><p>Marginal Revenue Curves:</p><ul class="tight" data-tight="true"><li><p>For a linear inverse demand curve <span data-latex="p(q) = a - bq" data-type="inline-math"></span>, the MR curve is <span data-latex="MR(q) = a - 2bq" data-type="inline-math"></span>. It has the same vertical intercept as demand but is twice as steep.</p></li><li><p>For constant elasticity demand, MR is a fixed proportion of price.</p></li></ul><h3>E. Income Elasticity (<span data-latex="\epsilon_m" data-type="inline-math"></span>)</h3><ul class="tight" data-tight="true"><li><p>The <strong>income elasticity of demand</strong> is the percentage change in quantity demanded divided by the percentage change in income. <span data-latex="$ \epsilon_m = \frac{\% \Delta q}{\% \Delta m} = \frac{\Delta q / q}{\Delta m / m} " data-type="inline-math">$</p></li><li><p>Interpretation:</p><ul class="tight" data-tight="true"><li><p><span data-latex="\epsilon_m < 0" data-type="inline-math"></span>: Inferior good.</p></li><li><p><span data-latex="\epsilon_m > 0" data-type="inline-math"></span>: Normal good.</p></li><li><p><span data-latex="\epsilon_m > 1" data-type="inline-math"></span>: Luxury good.</p></li></ul></li><li><p>The weighted average of income elasticities for all goods (where weights are expenditure shares <span data-latex="s_i = p_i x_i / m" data-type="inline-math"></span>) sum to 1: <span data-latex="$ s_1 \epsilon_{m1} + s_2 \epsilon_{m2} = 1 " data-type="inline-math">$ This implies that if one good is inferior, at least one other good must be a luxury good (or a strong normal good) to balance it out.

F. The Laffer Curve (Application to Tax Revenue)

• The Laffer curve illustrates the relationship between tax rates and total tax revenue, suggesting that tax revenue might initially increase with the tax rate but eventually decrease after a certain point.

• Example: In a simple labor market model, if labor is taxed at rate , then the effective wage received by workers is . Tax revenue is , where is labor supply.

• Tax revenue decreases when the elasticity of labor supply is greater than . This happens when a high tax rate significantly discourages labor supply, outweighing the revenue gain from the higher rate.

XII. Production Theory: Technology

Production theory analyzes how firms transform inputs into outputs, focusing on the technological constraints they face. This is the first step in understanding firm behavior.

A. Definition of Technology

• A technology is a process that converts inputs into outputs.

• Inputs (factors of production): Land, labor, capital, raw materials.

• A production set is the set of all technologically feasible combinations of inputs and outputs .

• A production function states the maximum amount of output possible from an input bundle.

• A production plan is feasible if .

• An isoquant is the set of all input bundles that yield a specific, constant output level .

B. Examples of Technology (Production Functions)

1. Fixed Proportions (Leontief): $</p><ul class="tight" data-tight="true"><li><p>No substitution possibilities between inputs; they are complements and used in fixed ratios.</p></li><li><p>Isoquants are L-shaped.</p></li></ul></li><li><p><strong>Perfect Substitutes (Linear)</strong>: <span data-latex="y= a1x1 + a2x2 +... + anxn" data-type="inline-math"></span>$

   • Inputs can be substituted at a constant rate.

   • Isoquants are straight lines.

Cobb-Douglas:

• Represents a balance between substitution and diminishing returns.

• Isoquants are smooth, convex curves.

C. Properties of Technology

• Monotonicity (Free Disposal): If you increase the amount of at least one input, you produce at least as much output as before. (Implicitly means discarding extra inputs is free).

• Convexity: If there are two ways to produce units of output, then a weighted average of these input bundles will produce at least units of output. This implies that diversified input bundles are preferred, and isoquants are convex.

D. The Marginal Product (MP)

• The marginal product of input () is the rate of change of output as the level of input changes, holding all other input levels fixed. $</p></li><li><p><strong>Diminishing Marginal Product</strong>: The marginal product of an input decreases as the level of that input increases, holding other inputs constant. <span data-latex="$ \frac{\partial MP_i}{\partial x_i} < 0 " data-type="inline-math">$</p></li></ul><h3>E. The Technical Rate of Substitution (TRS)</h3><ul class="tight" data-tight="true"><li><p>The <strong>Technical Rate of Substitution (TRS)</strong> measures the trade-off between two inputs, indicating the rate at which a firm must substitute one input for another to keep output constant.</p></li><li><p>It is the slope of an isoquant. <span data-latex="$ TRS = \frac{dx_2}{dx_1} = - \frac{MP_1}{MP_2} = - \frac{\partial y / \partial x_1}{\partial y / \partial x_2} " data-type="inline-math">$</p></li><li><p><strong>Diminishing TRS</strong> (due to strict convexity of isoquants): The absolute value of the isoquant's slope decreases as <span data-latex="x_1" data-type="inline-math"></span> increases.</p></li></ul><h3>F. The Long Run and the Short Run</h3><ul class="tight" data-tight="true"><li><p><strong>Long run</strong>: All factors of production can be varied.</p></li><li><p><strong>Short run</strong>: At least one factor of production is fixed (e.g., fixed land, plant size, number of machines).</p></li><li><p>Short-run production means some inputs are fixed, affecting how output changes with variable inputs.</p></li></ul><h3>G. Returns to Scale</h3><p><strong>Returns to scale</strong> describe how output changes when all inputs are increased proportionally by a factor <span data-latex="k > 1" data-type="inline-math"></span>.</p><ul class="tight" data-tight="true"><li><p><strong>Constant Returns to Scale (CRS)</strong>: Output increases by the same proportion as inputs. <span data-latex="$ f(kx_1, kx_2, \ldots) = k f(x_1, x_2, \ldots) " data-type="inline-math">$</p></li><li><p><strong>Increasing Returns to Scale (IRS)</strong>: Output increases by more than the proportion of inputs. <span data-latex="$ f(kx_1, kx_2, \ldots) > k f(x_1, x_2, \ldots) " data-type="inline-math">$</p></li><li><p><strong>Decreasing Returns to Scale (DRS)</strong>: Output increases by less than the proportion of inputs. <span data-latex="$ f(kx_1, kx_2, \ldots) < k f(x_1, x_2, \ldots) " data-type="inline-math">$</p></li><li><p>A production function is <strong>homogeneous of degree </strong><span data-latex="n" data-type="inline-math"></span> if <span data-latex="f(kx_1, \ldots) = k^n f(x_1, \ldots)" data-type="inline-math"></span>.</p><ul class="tight" data-tight="true"><li><p><span data-latex="n=1 \rightarrow" data-type="inline-math"></span> CRS</p></li><li><p><span data-latex="n>1 \rightarrow" data-type="inline-math"></span> IRS</p></li><li><p><span data-latex="n<1 \rightarrow" data-type="inline-math"></span> DRS</p></li></ul></li><li><p>Note: A technology can exhibit IRS or CRS even if all of its marginal products are diminishing. Diminishing marginal product applies when only one input increases, while returns to scale applies when all inputs increase proportionally.</p></li></ul><h3>H. Examples of Returns to Scale</h3><ul class="tight" data-tight="true"><li><p><strong>Perfect Substitutes</strong> production function (<span data-latex="y = a_1 x_1 + \ldots" data-type="inline-math"></span>): Exhibits CRS.</p></li><li><p><strong>Perfect Complements</strong> production function (<span data-latex="y = \min(a_1 x_1, \ldots)" data-type="inline-math"></span>): Exhibits CRS.</p></li><li><p><strong>Cobb-Douglas</strong> production function (<span data-latex="y = x_1^a x_2^b \ldots" data-type="inline-math"></span>):</p><ul class="tight" data-tight="true"><li><p>CRS if <span data-latex="a+b+\ldots = 1" data-type="inline-math"></span>.</p></li><li><p>IRS if <span data-latex="a+b+\ldots > 1" data-type="inline-math"></span>.</p></li><li><p>DRS if <span data-latex="a+b+\ldots < 1" data-type="inline-math"></span>.</p></li></ul></li></ul><h2>XIII. Profit Maximization</h2><p>Firms aim to maximize profits in competitive markets. This involves choosing optimal input and output levels based on prices, technology, and whether they are operating in the short or long run.</p><h3>A. Profits</h3><ul class="tight" data-tight="true"><li><p>A firm produces outputs (<span data-latex="y_i" data-type="inline-math"></span>) and uses inputs (<span data-latex="x_j" data-type="inline-math"></span>).</p></li><li><p>Output prices are <span data-latex="(p_i)" data-type="inline-math"></span> and input prices are <span data-latex="(w_j)" data-type="inline-math"></span>.</p></li><li><p><strong>Profits (</strong><span data-latex="\pi" data-type="inline-math"></span><strong>)</strong>: Total Revenue - Total Cost. <span data-latex="$ \pi = \sum p_i y_i - \sum w_j x_j " data-type="inline-math">$</p></li><li><p>All factors must be valued at their market rental prices, reflecting <strong>opportunity costs</strong>.</p></li><li><p><strong>Fixed factor</strong>: An input used in a fixed amount (at least in the short run).</p></li><li><p><strong>Variable factor</strong>: An input that can be used in different amounts.</p></li><li><p><strong>Short run</strong>: Some factors are fixed. Firms can make negative profits but must cover variable costs to continue operating.</p></li><li><p><strong>Long run</strong>: All factors are variable. Firms must make non-negative profits (or zero economic profit) to stay in business.</p></li><li><p><strong>Quasi-fixed factors</strong>: Used in a fixed amount if output is positive, but not if output is zero.</p></li></ul><h3>B. Short-Run Profit Maximization</h3><p>Assume input <span data-latex="x_2" data-type="inline-math"></span> is fixed at <span data-latex="\tilde{x}_2" data-type="inline-math"></span>. The firm chooses <span data-latex="x_1" data-type="inline-math"></span> to maximize profits:</p><p><span data-latex="$ \max_{x_1} p f(x_1, \tilde{x}_2) - w_1 x_1 - w_2 \tilde{x}_2 " data-type="inline-math">$</p><ul class="tight" data-tight="true"><li><p><strong>First-order condition</strong>: <span data-latex="p \cdot MP_1(x_1^*, \tilde{x}_2) = w_1" data-type="inline-math"></span>. The value of the marginal product of a factor equals its price.</p></li><li><p>If <span data-latex="p \cdot MP_1 > w_1" data-type="inline-math"></span>, use more of factor 1. If <span data-latex="p \cdot MP_1 < w_1" data-type="inline-math"></span>, use less.</p></li><li><p>An <strong>isoprofit line</strong> shows all combinations of inputs and outputs that yield a constant profit level. For a given output <span data-latex="y" data-type="inline-math"></span>: <span data-latex="$ y = \frac{\pi}{p} + \frac{w_2}{p} \tilde{x}_2 + \frac{w_1}{p} x_1 " data-type="inline-math">$</p></li><li><p>Profit maximization occurs where the slope of the production function (MP_1) equals the slope of the isoprofit line (<span data-latex="w_1/p" data-type="inline-math"></span>).</p></li><li><p><strong>Comparative statics</strong>:</p><ul class="tight" data-tight="true"><li><p>Increase in <span data-latex="w_1" data-type="inline-math"></span> (input price): Decreases optimal <span data-latex="x_1" data-type="inline-math"></span>. Short-run factor demand curves are downward sloping.</p></li><li><p>Decrease in <span data-latex="p" data-type="inline-math"></span> (output price): Decreases optimal <span data-latex="x_1" data-type="inline-math"></span>. Short-run supply function is upward sloping.</p></li><li><p>Increase in <span data-latex="w_2" data-type="inline-math"></span> (fixed input price): No effect on optimal <span data-latex="x_1" data-type="inline-math"></span> or <span data-latex="y" data-type="inline-math"></span>; only profits change.</p></li></ul></li></ul><h3>C. Long-Run Profit Maximization</h3><p>All inputs (<span data-latex="x_1, x_2" data-type="inline-math"></span>) are variable. The firm maximizes profits by choosing both <span data-latex="x_1" data-type="inline-math"></span> and <span data-latex="x_2" data-type="inline-math"></span>:</p><p><span data-latex="$ \max_{x_1, x_2} p f(x_1, x_2) - w_1 x_1 - w_2 x_2 " data-type="inline-math">$</p><ul class="tight" data-tight="true"><li><p><strong>First-order conditions</strong>: <span data-latex="$ p \cdot MP_1(x_1^*, x_2^*) = w_1 " data-type="inline-math">$ The value of the marginal product of each factor equals its price.

• These conditions define the factor demand curves (optimal as functions of ).

• Inverse factor demand curves: Show what factor prices must be for a given quantity of inputs demanded.

Example: Cobb-Douglas Production Function

• The long-run factor demands and supply function can be derived by solving the first-order conditions.

• If there are constant returns to scale (), the supply function is not well-defined, and firms in competitive markets make zero profits in the long run.

D. Profit Maximization and Returns to Scale

• In perfectly competitive markets, if a firm has constant returns to scale (CRS) and makes positive economic profits in the long run, it contradicts the assumption of profit maximization. Why? Because the firm could seemingly expand indefinitely to earn infinite profits.

• This paradox is resolved by considering:

  • Limitations to efficient scaling, invalidating CRS at very large scales.

  • Market dominance, invalidating competitive assumptions.

  • Entry of other firms, driving down prices and profits to zero.

• Conclusion: In long-run competitive equilibrium, firms with CRS earn zero economic profits.

XIV. Cost Minimization

Firms choose an optimal combination of inputs to produce a given level of output at the lowest possible cost. This leads to the concept of cost functions and their properties.

A. The Long-Run Cost Minimization Problem

The firm seeks to minimize the cost of producing output given factor prices :

$</p><ul class="tight" data-tight="true"><li><p>The minimum cost achieved is the <strong>cost function</strong> <span data-latex="c(w_1, w_2, y)" data-type="inline-math"></span>.</p></li><li><p>An <strong>isocost line</strong> shows all combinations of inputs that yield the same total cost <span data-latex="C" data-type="inline-math"></span>: <span data-latex="$ x_2 = \frac{C}{w_2} - \frac{w_1}{w_2} x_1 " data-type="inline-math">-w_1/w_2$.</p></li><li><p>The cost minimization problem involves finding the point on the isoquant (output <span data-latex="y" data-type="inline-math"></span>) that touches the lowest possible isocost line.</p></li><li><p><strong>Optimal solution</strong> (interior, smooth isoquant): The Technical Rate of Substitution (TRS) must equal the factor price ratio: <span data-latex="$ TRS(x_1^*, x_2^*) = - \frac{MP_1(x_1^*, x_2^*)}{MP_2(x_1^*, x_2^*)} = - \frac{w_1}{w_2} " data-type="inline-math">\frac{MP_1}{w_1} = \frac{MP_2}{w_2}$ (marginal product per dollar spent is equal across inputs).

The optimal choices of inputs and are called conditional factor demands (or derived factor demands).

B. Comparative Statics of Cost Minimization

• Increase in an input price (e.g., ): Rotates the isocost line, leading to a decrease in the cost-minimizing quantity of that input (assuming normal inputs and initial positive use). This is reflected in the downward slope of conditional factor demand curves.

• An expansion path connects cost-minimizing input combinations as output varies, holding factor prices constant.

• Normal input: Conditional demand increases with output.

• Inferior input: Conditional demand decreases with output (rare).

C. The Cost Function (Long-Run Total Cost)

• The total cost function describes the minimum total cost of producing output at given input prices. This is a long-run concept since all inputs are adjusted.

• It is obtained by substituting the conditional factor demands into the total cost equation: $</p></li></ul><h3>D. Duality and Shephard's Lemma</h3><ul class="tight" data-tight="true"><li><p>The cost function is <strong>dual</strong> to the production function; one can be derived from the other.</p></li><li><p><strong>Shephard's Lemma</strong>: Partial differentiation of the cost function with respect to an input price yields the conditional factor demand for that input. <span data-latex="$ x_1^*(y, w_1, w_2) = \frac{\partial c(y, w_1, w_2)}{\partial w_1} " data-type="inline-math">$</p></li></ul><h3>E. Returns to Scale and the Cost Function</h3><ul class="tight" data-tight="true"><li><p><strong>Unit cost function</strong>: <span data-latex="c(w_1, w_2, 1)" data-type="inline-math"></span> is the cost of producing one unit of output.</p></li><li><p>Relationship between returns to scale and cost function behavior:</p><ul class="tight" data-tight="true"><li><p><strong>Constant Returns to Scale (CRS)</strong>: Cost function is linear in output: <span data-latex="c(w_1, w_2, y) = c(w_1, w_2, 1) \cdot y" data-type="inline-math"></span>. Average Cost is constant.</p></li><li><p><strong>Increasing Returns to Scale (IRS)</strong>: Costs increase less than linearly in output: <span data-latex="c(w_1, w_2, y) < c(w_1, w_2, 1) \cdot y" data-type="inline-math"></span>. Average Cost is declining.</p></li><li><p><strong>Decreasing Returns to Scale (DRS)</strong>: Costs increase more than linearly in output: <span data-latex="c(w_1, w_2, y) > c(w_1, w_2, 1) \cdot y" data-type="inline-math"></span>. Average Cost is rising.</p></li></ul></li></ul><h3>F. Long-Run and Short-Run Costs</h3><ul class="tight" data-tight="true"><li><p><strong>Short-run cost function</strong> (<span data-latex="c_s(y, \tilde{x}_2)" data-type="inline-math"></span>): Minimum cost to produce <span data-latex="y" data-type="inline-math"></span> output when some factors (e.g., <span data-latex="x_2" data-type="inline-math"></span>) are fixed at <span data-latex="\tilde{x}_2" data-type="inline-math"></span>. <span data-latex="$ c_s(y, \tilde{x}_2) = \min_{x_1} w_1 x_1 + w_2 \tilde{x}_2 \quad \text{s.t. } f(x_1, \tilde{x}_2) = y " data-type="inline-math">$</p></li><li><p><strong>Long-run cost function</strong> (<span data-latex="c(y)" data-type="inline-math"></span>): Minimum cost when all factors are variable. <span data-latex="$ c(y) = \min_{x_1, x_2} w_1 x_1 + w_2 x_2 \quad \text{s.t. } f(x_1, x_2) = y " data-type="inline-math">$</p></li><li><p>The long-run cost function is the lower envelope of all possible short-run cost functions.</p><ul class="tight" data-tight="true"><li><p>For any output level <span data-latex="y" data-type="inline-math"></span>, the long-run cost <span data-latex="c(y)" data-type="inline-math"></span> will always be less than or equal to any short-run cost <span data-latex="c_s(y, \tilde{x}_2)" data-type="inline-math"></span>.</p></li><li><p>At the optimal fixed factor level <span data-latex="k^*(y)" data-type="inline-math"></span> for a given output <span data-latex="y" data-type="inline-math"></span>, <span data-latex="c(y) = c_s(y, k^*(y))" data-type="inline-math"></span>.</p></li></ul></li></ul><h2>XV. Cost Curves</h2><p>Cost curves illustrate how costs vary with output, providing crucial insights into a firm's production decisions and market supply.</p><h3>A. Average Costs</h3><p>Total costs <span data-latex="c(y)" data-type="inline-math"></span> are sum of variable costs <span data-latex="c_v(y)" data-type="inline-math"></span> and fixed costs <span data-latex="F" data-type="inline-math"></span> (<span data-latex="c(y) = c_v(y) + F" data-type="inline-math"></span>).</p><ul class="tight" data-tight="true"><li><p><strong>Average Cost (AC)</strong>: Total cost per unit of output. <span data-latex="$ AC(y) = \frac{c(y)}{y} = \frac{c_v(y)}{y} + \frac{F}{y} = AVC(y) + AFC(y) " data-type="inline-math">$</p></li><li><p><strong>Average Variable Cost (AVC)</strong>: Variable cost per unit of output.</p></li><li><p><strong>Average Fixed Cost (AFC)</strong>: Fixed cost per unit of output. AFC declines continuously as output increases.</p></li><li><p>The AC curve is typically U-shaped, a result of the combination of a decreasing AFC and an increasing AVC.</p></li></ul><h3>B. Marginal Costs (MC)</h3><ul class="tight" data-tight="true"><li><p><strong>Marginal Cost (MC)</strong>: Change in total cost (or variable cost) for a one-unit change in output. <span data-latex="$ MC(y) = \frac{\Delta c(y)}{\Delta y} = \frac{\Delta c_v(y)}{\Delta y} " data-type="inline-math">MC(y) = \frac{dc(y)}{dy} = \frac{dc_v(y)}{dy}.</p></li><li><p>Relationship between MC and AC/AVC:</p><ul class="tight" data-tight="true"><li><p>If MC &lt; AVC (or AC), then AVC (or AC) is falling.</p></li><li><p>If MC &gt; AVC (or AC), then AVC (or AC) is rising.</p></li><li><p>The MC curve must intersect the AVC (and AC) curve at its minimum point.</p></li></ul></li><li><p>The area beneath the MC curve up to output <span data-latex="y" data-type="inline-math"></span> equals the total variable cost <span data-latex="c_v(y)" data-type="inline-math"></span>.</p></li></ul><h3>C. Marginal Costs and Variable Costs for Multiple Plants</h3><ul class="tight" data-tight="true"><li><p>If a firm operates multiple plants, it minimizes total cost by allocating production such that the marginal cost of each plant is equal. <span data-latex=" MC_1(y_1) = MC_2(y_2) " data-type="inline-math">$</p></li><li><p>The firm's overall marginal cost curve is the horizontal sum of the individual plants' marginal cost curves.</p></li></ul><h3>D. Long-Run and Short-Run Cost Curves</h3><ul class="tight" data-tight="true"><li><p>In the long run, all factors are variable, so a firm can produce 0 output at 0 cost.</p></li><li><p>The long-run average cost (LRAC) curve is the lower envelope of all possible short-run average cost (SRAC) curves. <span data-latex="$ LRAC(y) \leq SRAC(y, k) " data-type="inline-math">k$ is a short-run fixed factor).

• At the optimal plant size for a given output , .

• This implies that at the common tangency point, the long-run marginal cost (LRMC) equals the short-run marginal cost (SRMC) for the optimal plant size. $</p></li></ul><h2>XVI. Market Structures: Perfect Competition</h2><p>Perfect competition serves as a benchmark market structure where many firms produce identical products and individual firms have no market power.</p><h3>A. Market Environments</h3><ul class="tight" data-tight="true"><li><p>Firms face <strong>technological constraints</strong> (feasible input-output combinations) and <strong>market constraints</strong> (how much can be sold at a given price).</p></li><li><p>The <strong>demand curve facing a firm</strong> shows the relationship between its price and the quantity it sells.</p></li><li><p><strong>Market environment</strong> describes how firms interact in their pricing and output decisions.</p></li></ul><h3>B. Perfect Competition Defined</h3><ul class="tight" data-tight="true"><li><p>A market structure where each firm assumes the market price is independent of its own output level.</p></li><li><p>Characteristics:</p><ul class="tight" data-tight="true"><li><p>Many firms.</p></li><li><p>Identical product (homogeneous goods).</p></li><li><p>Each firm is a small part of the market (price taker).</p></li><li><p>Free entry and exit in the long run.</p></li><li><p>Perfect information.</p></li></ul></li><li><p>The <strong>market demand curve</strong> is downward sloping. The <strong>demand curve facing a competitive firm</strong> is perfectly horizontal at the market price.</p></li></ul><h3>C. The Supply Decision of a Competitive Firm</h3><p>A competitive firm maximizes profits by choosing output <span data-latex="y" data-type="inline-math"></span>:</p><p><span data-latex="$ \max_y \ py - c(y) \quad \text{s.t. } y \geq 0 " data-type="inline-math">$</p><ul class="tight" data-tight="true"><li><p><strong>First-order condition</strong>: <span data-latex="p - c'(y^*) = 0 \Rightarrow p = MC(y^*)" data-type="inline-math"></span>. Price equals marginal cost.</p></li><li><p><strong>Second-order condition</strong>: <span data-latex="-c''(y^*) \leq 0 \Rightarrow MC'(y^*) \geq 0" data-type="inline-math"></span>. Marginal cost must be increasing.</p></li><li><p>The <strong>supply curve of a competitive firm</strong> is its marginal cost curve above the average variable cost (AVC).</p></li><li><p><strong>Shutdown condition</strong>: A firm shuts down in the short run if total revenue is less than variable costs (<span data-latex="py < c_v(y)" data-type="inline-math"></span>), or equivalently, if market price <span data-latex="p < AVC(y)" data-type="inline-math"></span>. Fixed costs are sunk, so only variable costs matter for short-run decision.</p></li></ul><h3>D. Profits and Producer's Surplus</h3><ul class="tight" data-tight="true"><li><p><strong>Profits</strong> (<span data-latex="\pi" data-type="inline-math"></span>) = Total Revenue (TR) - Total Cost (TC) = <span data-latex="py - c_v(y) - F" data-type="inline-math"></span>.</p></li><li><p><strong>Producer's surplus (PS)</strong>: Total Revenue minus Total Variable Cost = <span data-latex="py - c_v(y)" data-type="inline-math"></span>.</p></li><li><p>PS is the area above the supply curve and below the price.</p></li><li><p>Change in PS = Change in profits (since fixed costs are constant).</p></li></ul><h3>E. The Long-Run Supply Curve of a Firm</h3><ul class="tight" data-tight="true"><li><p>In the long run, all costs are variable. The firm can enter or exit the market.</p></li><li><p>Long-run profit maximization condition: <span data-latex="p = LRMC(y)" data-type="inline-math"></span>.</p></li><li><p><strong>Entry/Exit condition</strong>: In the long run, firms enter if <span data-latex="p > LRAC(y)" data-type="inline-math"></span> (positive profits) and exit if <span data-latex="p < LRAC(y)" data-type="inline-math"></span> (negative profits).</p></li><li><p>In long-run competitive equilibrium, <span data-latex="p = LRMC = LRAC_{min}" data-type="inline-math"></span>. Firms earn zero economic profit.</p></li><li><p>The long-run supply curve is more elastic than the short-run supply curve because firms have more flexibility to adjust all factors of production.</p></li><li><p>If the long-run technology exhibits constant returns to scale, the LRAC is constant, and the long-run industry supply curve is a horizontal line at <span data-latex="p = LRAC_{min}" data-type="inline-math"></span>.</p></li></ul><h2>XVII. General Equilibrium</h2><p>General equilibrium analysis examines how prices and quantities are determined simultaneously across multiple interconnected markets, considering feedback effects that partial equilibrium models ignore.</p><h3>A. Partial vs. General Equilibrium</h3><ul class="tight" data-tight="true"><li><p><strong>Partial equilibrium analysis</strong>: Studies price and output determination in a single market, assuming prices in other markets are fixed.</p></li><li><p><strong>General equilibrium analysis</strong>: Studies price and output determination in multiple markets simultaneously, accounting for interdependencies. (e.g., changes in the coffee market affect the tea market, and vice versa).</p></li></ul><h3>B. The Edgeworth Box</h3><ul class="tight" data-tight="true"><li><p>A tool to analyze exchange economy with two goods and two consumers.</p></li><li><p>Dimensions of the box represent total available quantities of goods 1 and 2 (<span data-latex="(\omega_A^1 + \omega_B^1)" data-type="inline-math"></span> and <span data-latex="(\omega_A^2 + \omega_B^2)" data-type="inline-math"></span>).</p></li><li><p>Consumer A's consumption is measured from the bottom-left origin (<span data-latex="O_A" data-type="inline-math"></span>). Consumer B's is from the top-right origin (<span data-latex="O_B" data-type="inline-math"></span>).</p></li><li><p>Indifference curves for both consumers can be depicted within the box.</p></li><li><p>An <strong>allocation</strong> is a distribution of goods between consumers <span data-latex="(X_A, X_B)" data-type="inline-math"></span>.</p></li><li><p>A <strong>feasible allocation</strong> means total consumption equals total endowment: <span data-latex="$ x_A^1 + x_B^1 = \omega_A^1 + \omega_B^1 " data-type="inline-math">$</p></li></ul><h3>C. Trade and Gains from Exchange</h3><ul class="tight" data-tight="true"><li><p>Starting from an initial endowment (<span data-latex="W" data-type="inline-math"></span>), mutually advantageous trade occurs if both consumers can reach a higher indifference curve.</p></li><li><p>The set of all allocations where both A and B are better off than at their initial endowment is the <strong>lens-shaped area</strong> between their indifference curves passing through <span data-latex="W" data-type="inline-math"></span>.</p></li></ul><h3>D. Pareto Efficient Allocations</h3><ul class="tight" data-tight="true"><li><p>A <strong>Pareto improving allocation</strong> makes at least one person better off without making anyone worse off.</p></li><li><p>A <strong>Pareto efficient allocation</strong> is where no one can be made better off without making someone else worse off. This happens when all gains from trade are exhausted.</p></li><li><p><strong>Condition for Pareto efficiency</strong>: The indifference curves of the two agents must be tangent at any interior Pareto efficient allocation. This implies: <span data-latex="$ MRS_A = MRS_B " data-type="inline-math">$</p></li><li><p>The set of all Pareto efficient points in the Edgeworth box is called the <strong>Pareto Set</strong> or <strong>Contract Curve</strong>.</p></li></ul><h3>E. Market Trade and Equilibrium</h3><ul class="tight" data-tight="true"><li><p>In a competitive market, each consumer is a price-taker, maximizing utility given prices <span data-latex="(p_1, p_2)" data-type="inline-math"></span> and their endowment.</p></li><li><p><strong>Gross demand</strong>: Total amount of a good wanted at given prices.</p></li><li><p><strong>Net/Excess demand</strong>: Gross demand minus initial endowment (<span data-latex="e_A^1 = x_A^1 - \omega_A^1" data-type="inline-math"></span>).</p></li><li><p>A <strong>market equilibrium (competitive equilibrium, Walrasian equilibrium)</strong> is a set of prices <span data-latex="(p_1^*, p_2^*)" data-type="inline-math"></span> where:</p><ul class="tight" data-tight="true"><li><p>Each consumer chooses the most preferred affordable bundle.</p></li><li><p>Demand equals supply for every good (aggregate excess demand is zero).</p></li></ul></li><li><p>At equilibrium, each agent's indifference curve is tangent to their budget line (slope <span data-latex="-p_1/p_2" data-type="inline-math"></span>), meaning <span data-latex="MRS = p_1/p_2" data-type="inline-math"></span> for both. Consequently, <span data-latex="MRS_A = MRS_B" data-type="inline-math"></span> at equilibrium.</p></li></ul><h3>F. Walras' Law</h3><ul class="tight" data-tight="true"><li><p><strong>Walras' Law</strong> states that the value of aggregate excess demand is identically zero for any positive prices <span data-latex="(p_1, p_2)" data-type="inline-math"></span>, whether equilibrium or not. <span data-latex="$ p_1 z_1(p_1, p_2) + p_2 z_2(p_1, p_2) = 0 " data-type="inline-math">z_ii$).</p></li><li><p>Proof relies on consumers spending all their budget (budget constraint always holds).</p></li><li><p>Implication: If <span data-latex="k-1" data-type="inline-math"></span> markets are in equilibrium, the <span data-latex="k" data-type="inline-math"></span>-th market must also be in equilibrium. This means only <span data-latex="k-1" data-type="inline-math"></span> independent prices need to be determined; one price can be set as a numeraire.</p></li></ul><h3>G. First Fundamental Theorem of Welfare Economics</h3><ul class="tight" data-tight="true"><li><p>An equilibrium achieved by a set of competitive markets will be <strong>Pareto efficient</strong>.</p></li><li><p>Proof: Assumes competitive equilibrium is not Pareto efficient, which leads to a contradiction (implies a bundle preferred by both agents is also more expensive for both, which is impossible if they already maximized utility within their budget).</p></li><li><p>This implies that the "invisible hand" of the market leads to an efficient allocation.</p></li><li><p>Does not guarantee fairness; initial endowments can lead to skewed, but still efficient, outcomes.</p></li></ul><h3>H. Second Fundamental Theorem of Welfare Economics</h3><ul class="tight" data-tight="true"><li><p>If all agents have convex indifference curves, then any Pareto efficient allocation can be achieved as a market equilibrium for an appropriate assignment of endowments.</p></li><li><p>This implies that a society can achieve any desired Pareto efficient distribution (e.g., more equitable) by redistributing initial endowments (e.g., through lump-sum taxes/transfers) and then allowing markets to operate freely.</p></li><li><p>Prices play two roles: <strong>allocative</strong> (signaling relative scarcity) and <strong>distributive</strong> (determining purchasing power). The second welfare theorem suggests these roles can be separated.</p></li></ul><h2>XVIII. Monopoly</h2><p>A monopoly is a market structure where a single firm dominates the market, giving it significant control over price and output. This section explores the profit-maximization behavior of monopolists and monopsonists, as well as the unique issues arising in vertically integrated monopolies.</p><h3>A. Monopoly in the Output Market</h3><ul class="tight" data-tight="true"><li><p>A <strong>monopoly</strong> is an industry structure with only one firm.</p></li><li><p>Unlike competitive firms, a monopolist is a price-setter, facing a downward-sloping market demand curve <span data-latex="p(y)" data-type="inline-math"></span>.</p></li><li><p><strong>Profit-maximization problem</strong>: <span data-latex="$ \max_y \ r(y) - c(y) " data-type="inline-math">r(y) = p(y)yc(y)$ is total cost.

• First-order condition: . Marginal Revenue equals Marginal Cost.

• Marginal Revenue (MR) for a monopolist: p'(y) < 0MR < p(y)$.</p></li><li><p>MR in terms of price elasticity of demand (<span data-latex="\epsilon" data-type="inline-math"></span>): <span data-latex="$ MR = p(y) \left(1 + \frac{1}{\epsilon}\right) = p(y) \left(1 - \frac{1}{|\epsilon|}\right) " data-type="inline-math">$</p></li><li><p>A monopolist will always choose to operate on the <strong>elastic part of the demand curve</strong> (<span data-latex="|\epsilon| > 1" data-type="inline-math"></span>), where <span data-latex="MR > 0" data-type="inline-math"></span>. If demand is inelastic, <span data-latex="MR < 0" data-type="inline-math"></span>, meaning reducing output would increase revenue and decrease costs, which would increase profits.</p></li><li><p>For a linear demand curve <span data-latex="p(y) = a - by" data-type="inline-math"></span>, <span data-latex="MR(y) = a - 2by" data-type="inline-math"></span>. The MR curve is twice as steep as the demand curve.</p></li><li><p><strong>Markup pricing</strong>: A monopolist's price is a markup over marginal cost: <span data-latex="$ p(y) = \frac{MC(y)}{1 - 1/|\epsilon(y)|} " data-type="inline-math">$1 - 1/|\epsilon(y)| < 1" data-type="inline-math"></span> (for elastic demand), <span data-latex="p(y) > MC(y)" data-type="inline-math"></span>.</p></li></ul><h3>B. Monopoly and Factor Demand</h3><ul class="tight" data-tight="true"><li><p>The monopolist's decision for input <span data-latex="x" data-type="inline-math"></span> to produce output <span data-latex="y = f(x)" data-type="inline-math"></span> leads to: <span data-latex="$ \frac{dR(x)}{dx} = (p(y) + p'(y)y) f'(x) = MR_y MP_x " data-type="inline-math">$ This is the Marginal Revenue Product (MRP) of the input.

• In perfect competition, .

• In monopoly, . Since , the monopolist's MRP is less than the value of the marginal product ().

• This implies a monopolist demands less of an input than a competitive firm would, all else equal.

C. Monopsony

• A monopsony is a market where there is a single buyer of a factor of production.

• A monopsonist faces an upward-sloping factor supply curve . To hire more , it must offer a higher price.

• Profit-maximization problem: $</p></li><li><p><strong>First-order condition</strong>: <span data-latex="p f'(x) = w(x) + w'(x)x" data-type="inline-math"></span>. The value of the marginal product of the factor equals its marginal expenditure (ME).</p></li><li><p>The marginal expenditure on a factor (<span data-latex="ME_x" data-type="inline-math"></span>) is greater than the factor price (<span data-latex="w(x)" data-type="inline-math"></span>) for a monopsonist. <span data-latex="$ ME_x = w(x) \left(1 + \frac{1}{\eta}\right) " data-type="inline-math">\eta$ is the supply elasticity of the factor).

• A monopsonist hires less of the factor and pays a lower price () compared to a competitive factor market.

• This results in a Pareto inefficient outcome (too little of the factor hired).

• Minimum wage effect: In a monopsonized labor market, a minimum wage set between the monopsony wage () and the competitive wage () can increase employment and wages simultaneously, potentially moving towards a more efficient outcome.

D. Upstream and Downstream Monopolies (Double Marginalization)

• This occurs when an upstream monopolist sells an input to a downstream monopolist, who then sells the final good.

• Both firms try to extract monopoly profit, leading to a "double markup."

• Example: Upstream firm produces at cost , sells at price . Downstream firm uses as input for , faces demand .

  • Downstream profit maximization: .

  • Upstream profit maximization: Based on residual demand for input from downstream, chooses where .

• The overall output in this scenario is lower, and the final price is higher, than if the two firms merged into a single integrated monopolist. The integrated monopolist would only apply one markup over the true marginal cost of production.

• This double marginalization is inefficient from a social perspective and even from the perspective of maximizing total monopoly profits.