Chapter 10Advanced Microeconomics

Introduction

Chapter 6 introduced consumer theory through utility maximization and the Lagrangian. This chapter strips away the crutch of specific functional forms and builds the theory from axiomatic foundations. We ask: when can preferences be represented by a utility function? What properties must demand functions satisfy? And under what conditions does a system of competitive markets allocate resources efficiently?

The shift in method is from computation to proof. Part II solved optimization problems. Part III proves theorems — establishing which results are robust and which depend on special assumptions.

By the end of this chapter, you will be able to:
  1. State the preference axioms and the conditions for utility representation
  2. Define WARP and SARP and test revealed preference consistency
  3. Derive the expenditure function and Hicksian demand using duality
  4. State and verify properties of the Slutsky matrix
  5. Define Walrasian equilibrium and prove the First Welfare Theorem
  6. State the Second Welfare Theorem and explain its policy implications

Prerequisites: Chapters 6–7. Mathematical prerequisites: real analysis basics (open/closed sets, continuity, fixed-point theorems), convex analysis, matrix algebra. See Appendix A.

Named literature: Mas-Colell, Whinston & Green (MWG); Debreu Theory of Value; Arrow & Debreu (1954); Varian Microeconomic Analysis.

10.1 Choice Theory: Axioms and Utility Representation

Preference Axioms

Preference relation. A binary relation $\succsim$ on a consumption set $X \subseteq \mathbb{R}^n_+$. We define: $x \succ y$ (strict preference) if $x \succsim y$ and not $y \succsim x$; and $x \sim y$ (indifference) if $x \succsim y$ and $y \succsim x$.

The standard axioms:

Axiom 1 (Completeness). For all $x, y \in X$: $x \succsim y$ or $y \succsim x$ (or both).
Axiom 2 (Transitivity). For all $x, y, z \in X$: if $x \succsim y$ and $y \succsim z$, then $x \succsim z$.
Axiom 3 (Continuity). For all $y \in X$, the sets $\{x : x \succsim y\}$ and $\{x : y \succsim x\}$ are closed. Equivalently, the strict preference sets $\{x : x \succ y\}$ and $\{x : y \succ x\}$ are open.
Theorem (Debreu). If $\succsim$ is complete, transitive, and continuous, then there exists a continuous utility function $u: X \to \mathbb{R}$ such that $x \succsim y \iff u(x) \geq u(y)$.

Proof sketch. Fix a ray $\{te : t \geq 0\}$ where $e = (1,1,\ldots,1)$. For each $x$, by completeness and continuity, there exists a unique $t(x) \geq 0$ such that $x \sim t(x)e$. Set $u(x) = t(x)$. Transitivity ensures the representation is consistent; continuity ensures $u$ is continuous.

The utility function is ordinal — any monotonic transformation $v = g(u)$ with $g' > 0$ represents the same preferences. Cardinal properties (magnitudes of utility differences) are meaningless.

Additional Properties

Monotonicity (more is better). If $x \geq y$ (component-wise) and $x \neq y$, then $x \succ y$.
Convexity. If $x \succsim y$, then $\lambda x + (1-\lambda)y \succsim y$ for all $\lambda \in [0,1]$. Convexity means indifference curves are convex to the origin — the consumer prefers mixtures.
Strict convexity. If $x \succsim y$, $x \neq y$, and $\lambda \in (0,1)$, then $\lambda x + (1-\lambda)y \succ y$. Strict convexity guarantees unique optimal bundles.
Example 10.1a — Checking Preference Axioms

Consider lexicographic preferences on $\mathbb{R}^2_+$: $x \succ y$ if $x_1 > y_1$, or $x_1 = y_1$ and $x_2 > y_2$.

Completeness: Satisfied — for any $x, y$, either $x_1 > y_1$, $y_1 > x_1$, or $x_1 = y_1$ and we compare $x_2, y_2$.

Transitivity: Satisfied — if $x \succ y$ and $y \succ z$, then $x \succ z$ (follows from transitivity of $>$ on $\mathbb{R}$).

Continuity: Fails. Consider $y = (1, 1)$. The set $\{x : x \succ y\}$ includes $(1, 1.5)$ but not $(0.999, 100)$. The "at least as good" set is not closed — there is a jump at $x_1 = 1$.

Consequence: No continuous utility function represents lexicographic preferences. This shows that continuity is essential for Debreu's utility representation theorem.

10.2 Revealed Preference

Instead of assuming preferences, we can infer them from observed choices.

Weak Axiom of Revealed Preference (WARP). If bundle $x$ is chosen when bundle $y$ is affordable (i.e., $p \cdot x \geq p \cdot y$ where $p$ is the price vector), then $y$ is never chosen when $x$ is affordable.

Formally: if $x$ is revealed preferred to $y$ ($xRy$: $x$ chosen at prices where $y$ was affordable), then $y$ is not revealed preferred to $x$.

Strong Axiom of Revealed Preference (SARP). The revealed preference relation has no cycles: there is no sequence $x^1 R x^2 R \cdots R x^k R x^1$.

SARP is necessary and sufficient for observed choices to be consistent with utility maximization (Afriat's theorem). WARP is necessary but not sufficient in general (though it is sufficient with two goods).

Example 10.1 — WARP Check

A consumer's choices at two price-income situations:

SituationPrices $(p_1, p_2)$Chosen bundle $(x_1, x_2)$Expenditure
A(1, 2)(4, 2)8
B(2, 1)(2, 4)8

Check WARP: At prices A, could the consumer afford bundle B? \$1(2) + 2(4) = 10 > 8$. No. At prices B, could the consumer afford bundle A? \$1(4) + 1(2) = 10 > 8$. No. WARP is satisfied — the data are consistent with utility maximization.

Interactive: Revealed Preference Checker

Enter price vectors and chosen bundles for up to 6 observations. The checker will test WARP and SARP automatically.

Obs.$p_1$$p_2$$x_1$$x_2$Expenditure
1 8.0
2 8.0
3 6.0
4 —
5 —
6 —
Click "Check WARP & SARP" to analyze the data.

Interactive 10.1. Enter price-bundle observations and test for revealed preference consistency. WARP checks direct pairwise reversals; SARP checks for cycles of any length. Violations are highlighted with explanations.

10.3 Duality: Expenditure Function and Hicksian Demand

Chapter 6 solved the primal problem: maximize utility subject to a budget. The dual problem minimizes expenditure to achieve a target utility level.

The Expenditure Minimization Problem

Expenditure function. $e(p, \bar{u}) = \min_{x \geq 0} p \cdot x$ subject to $u(x) \geq \bar{u}$. It gives the minimum cost of achieving utility level $\bar{u}$ at prices $p$. The expenditure function is homogeneous of degree 1 in prices and concave in prices.
$$e(p, \bar{u}) = \min_{x \geq 0} \; p \cdot x \quad \text{subject to} \quad u(x) \geq \bar{u}$$ (Eq. 10.1)
Hicksian (compensated) demand. The demand function $h(p, \bar{u})$ that solves the expenditure minimization problem. It shows how consumption responds to price changes holding utility constant (compensating the consumer for the price change). Unlike Marshallian demand, Hicksian demand isolates the pure substitution effect.

The solution is the Hicksian (compensated) demand $h(p, \bar{u})$:

Shephard's lemma. Hicksian demand can be recovered directly from the expenditure function by differentiation: $h_i(p, \bar{u}) = \partial e(p, \bar{u}) / \partial p_i$. This is the dual analogue of Roy's identity.
$$h_i(p, \bar{u}) = \frac{\partial e(p, \bar{u})}{\partial p_i} \quad \text{(Shephard's lemma)}$$ (Eq. 10.2)

Properties of the Expenditure Function

  1. Homogeneous of degree 1 in $p$: $e(tp, \bar{u}) = te(p, \bar{u})$
  2. Non-decreasing in $p$: Higher prices mean more expenditure to reach $\bar{u}$
  3. Concave in $p$: $e(\lambda p + (1-\lambda)p', \bar{u}) \geq \lambda e(p, \bar{u}) + (1-\lambda)e(p', \bar{u})$
  4. Non-decreasing in $\bar{u}$: Higher target utility means more expenditure

Connecting Primal and Dual

The indirect utility function $V(p, m)$ gives the maximum utility achievable at prices $p$ with income $m$:

$$V(p, m) = \max_{x} \; u(x) \quad \text{s.t.} \quad p \cdot x \leq m$$

The key duality relationships:

$$e(p, V(p, m)) = m$$ (Eq. 10.3)
$$V(p, e(p, \bar{u})) = \bar{u}$$ (Eq. 10.4)
$$h(p, \bar{u}) = x(p, e(p, \bar{u}))$$ (Eq. 10.5)
Roy's identity. Marshallian demand can be recovered from the indirect utility function: $x_i(p, m) = -(\partial V / \partial p_i) / (\partial V / \partial m)$. A price increase reduces welfare in proportion to the quantity consumed, scaled by the marginal utility of income.

Roy's identity provides a shortcut for deriving Marshallian demand from the indirect utility function:

$$x_i(p, m) = -\frac{\partial V / \partial p_i}{\partial V / \partial m}$$ (Eq. 10.6)

Intuition for Roy's identity: A small increase in $p_i$ has two effects on welfare (measured by $V$): (1) it directly reduces utility by making good $i$ more expensive (the numerator $\partial V/\partial p_i < 0$), and (2) the magnitude of this effect is proportional to how much of good $i$ the consumer buys ($x_i$) times the marginal utility of income ($\partial V/\partial m$). Dividing (1) by the marginal utility of income gives the quantity of good $i$.

Example 10.2 — CES Duality

CES utility: $u(x_1, x_2) = (x_1^\rho + x_2^\rho)^{1/\rho}$, $\rho < 1$, $\rho \neq 0$.

The expenditure function is: $e(p, \bar{u}) = \bar{u} \cdot (p_1^r + p_2^r)^{1/r}$ where $r = \rho/(\rho - 1)$.

Hicksian demand (Shephard's lemma): $h_i = \bar{u} \cdot p_i^{r-1} / (p_1^r + p_2^r)^{(r-1)/r}$.

As $\rho \to 0$ (elasticity of substitution $\sigma = 1/(1-\rho) \to 1$), this converges to the Cobb-Douglas case.

Interactive: Duality Explorer

Cobb-Douglas utility $u = x_1^{0.5} x_2^{0.5}$ with income $m = 10$. Slide $p_1$ to see how all three representations — budget-line tangency, Marshallian demand, and expenditure function — encode the same information.

\$1.50 \$1.00 \$1.00
At $p_1 = 2.00$: Marshallian: $x_1^* = 2.50$, $x_2^* = 2.50$  |  $V(p, m) = 2.50$  |  $e(p, \bar{u}) = 10.00$

Interactive 10.2. Three views of the same consumer. Left: indifference curve tangent to budget line (primal). Center: Marshallian demand for good 1 as a function of $p_1$. Right: expenditure function $e(p_1, p_2, \bar{u})$ needed to achieve the current utility level. All three encode the same preferences.

10.4 The Slutsky Matrix

Slutsky matrix. The $n \times n$ matrix $S$ with entries $S_{ij} = \partial h_i / \partial p_j$, measuring substitution effects between goods. If demand is generated by utility maximization, $S$ must be symmetric and negative semidefinite. These are testable restrictions on observed demand.

The Slutsky equation from Chapter 6 (Eq. 6.7) generalizes to a matrix. Define the Slutsky (substitution) matrix with entries:

$$S_{ij} = \frac{\partial h_i}{\partial p_j} = \frac{\partial x_i}{\partial p_j} + x_j \frac{\partial x_i}{\partial m}$$ (Eq. 10.7)

Properties of the Slutsky Matrix

If demand is generated by utility maximization, the Slutsky matrix must be:

  1. Symmetric: $S_{ij} = S_{ji}$ (cross-substitution effects are equal)
  2. Negative semidefinite: $v'Sv \leq 0$ for all vectors $v$ (own-substitution effects are non-positive: $S_{ii} \leq 0$)
  3. $S \cdot p = 0$: Compensated demand is homogeneous of degree zero in prices

These are testable restrictions — if observed demand violates them, it cannot have been generated by a rational consumer maximizing a well-behaved utility function.

Integrability. Conversely, if a demand system satisfies: (a) Walras' law ($p \cdot x(p,m) = m$), (b) homogeneity of degree zero, (c) Slutsky symmetry and negative semidefiniteness — then there exists a utility function that generates it. This is the integrability theorem.
Example 10.3 — Slutsky Symmetry for Cobb-Douglas

Cobb-Douglas demand: $x_1 = am/p_1$, $x_2 = (1-a)m/p_2$.

$S_{12} = \partial x_1/\partial p_2 + x_2 \cdot \partial x_1/\partial m = 0 + [(1-a)m/p_2] \cdot [a/p_1] = a(1-a)m/(p_1 p_2)$

$S_{21} = \partial x_2/\partial p_1 + x_1 \cdot \partial x_2/\partial m = 0 + [am/p_1] \cdot [(1-a)/p_2] = a(1-a)m/(p_1 p_2)$

$S_{12} = S_{21}$ ✓

Interactive: Slutsky Decomposition (Advanced)

Adjust the price of good 1 to see how Marshallian demand, Hicksian (compensated) demand, and the income effect respond. Uses Cobb-Douglas utility $u(x_1,x_2)=x_1^a x_2^{1-a}$ with $a=0.6$, $p_2=3$, $m=120$.

1 (cheaper)4 (more expensive)

Figure 10.2. Left: Slutsky decomposition in commodity space. The original bundle (blue), compensated bundle (orange, on original indifference curve at new prices), and new bundle (green). The substitution effect moves from blue to orange; the income effect moves from orange to green. Right: Slutsky matrix entries $S_{11}$ and $S_{12}$ as $p_1$ varies, confirming negative semidefiniteness ($S_{11} \leq 0$) and symmetry.

10.5 General Equilibrium: Walrasian Equilibrium

Exchange Economy

Exchange economy. An economy with $I$ consumers and $L$ goods but no production. Each consumer has an initial endowment $\omega_i$ and preferences $\succsim_i$. Trade occurs at market prices; the question is whether a set of prices exists that clears all markets simultaneously.

Consider an economy with $I$ consumers and $L$ goods. Consumer $i$ has endowment $\omega_i \in \mathbb{R}^L_+$ and preferences $\succsim_i$.

At prices $p$, consumer $i$'s wealth is $m_i = p \cdot \omega_i$. She demands $x_i(p, m_i)$.

Walrasian (competitive) equilibrium. A price vector $p^*$ and allocation $(x_1^*, \ldots, x_I^*)$ such that: (1) Each consumer maximizes utility: $x_i^*$ solves $\max u_i(x_i)$ s.t. $p^* \cdot x_i \leq p^* \cdot \omega_i$; (2) Markets clear: $\sum_i x_i^* = \sum_i \omega_i$.

Aggregate excess demand:

$$z(p) = \sum_i x_i(p, p \cdot \omega_i) - \sum_i \omega_i$$ (Eq. 10.8)

Equilibrium requires $z(p^*) = 0$.

Walras' law. For any price vector $p$: $p \cdot z(p) = 0$. The total value of excess demand is always zero. This follows from budget exhaustion: $p \cdot x_i = p \cdot \omega_i$ for each consumer.

Implications: (1) If $L - 1$ markets clear, the $L$th clears automatically. (2) Only relative prices matter — we can normalize one price to 1 (the numeraire).

Existence

Theorem (Arrow-Debreu, 1954). Under standard conditions (continuous, strictly convex, locally nonsatiated preferences; positive aggregate endowment of each good), a Walrasian equilibrium exists.

Proof strategy (sketch). Normalize prices to the unit simplex $\Delta$. Define a price-adjustment map $f: \Delta \to \Delta$ that raises the price of goods in excess demand. By Brouwer's fixed-point theorem, $f$ has a fixed point $p^*$. At the fixed point, $z(p^*) = 0$ — all markets clear.

The Edgeworth Box

Edgeworth box. A diagram for a 2-consumer, 2-good exchange economy. The box dimensions equal total endowments. Consumer 1's origin is at the bottom-left, consumer 2's at the top-right. Every point in the box is a feasible allocation; the contract curve connects all Pareto-efficient points (tangencies of indifference curves).

For a 2-consumer, 2-good economy, the Edgeworth box provides a complete visualization. The box dimensions equal total endowments. Consumer 1's origin is at bottom-left, consumer 2's at top-right. Every point in the box is a feasible allocation.

Interactive: Edgeworth Box

Two consumers with Cobb-Douglas preferences. Drag the endowment point to explore how the Walrasian equilibrium, contract curve, and core change.

15 (total x = 10)9
14 (total y = 8)7
Endowment: C1 = (6, 2), C2 = (4, 6)  |  Equilibrium: $p_x/p_y = 1.00$, C1 gets (5.0, 5.0)

Figure 10.1 (Interactive). The Edgeworth box. The orange dot is the endowment. The green dot is the Walrasian equilibrium. The red curve is the contract curve (all Pareto-efficient allocations). The shaded core region shows allocations both consumers prefer to the endowment. The budget line passes through the endowment with slope $-p_x/p_y$.

Example 10.4 — Symmetric Exchange Economy

Consumer 1: $u_1 = x_1^{1/2}y_1^{1/2}$, endowment $(4, 0)$. Consumer 2: $u_2 = x_2^{1/2}y_2^{1/2}$, endowment $(0, 4)$.

Market clearing gives $p_x = p_y$, and the equilibrium allocation is $x_1^* = y_1^* = 2$, $x_2^* = y_2^* = 2$.

Each consumer trades half their endowment for the other good, ending up with equal amounts of both goods.

10.6 The First Welfare Theorem

First Welfare Theorem. If preferences are locally nonsatiated, then every Walrasian equilibrium allocation is Pareto optimal.
Pareto optimal (efficient). An allocation $x^*$ is Pareto optimal if there is no other feasible allocation $x'$ such that $u_i(x'_i) \geq u_i(x_i^*)$ for all $i$ and $u_j(x'_j) > u_j(x_j^*)$ for some $j$.

Proof. We proceed by contradiction. Suppose the Walrasian equilibrium allocation $x^*$ at prices $p^*$ is not Pareto optimal. Then there exists a feasible allocation $x'$ with everyone at least as well off and someone strictly better off.

Step 1. For consumer $j$ who is strictly better off: since $x_j^*$ was utility-maximizing and $x_j'$ is strictly preferred, $x_j'$ must have been unaffordable: $p^* \cdot x_j' > p^* \cdot \omega_j$.

Step 2. For every consumer $i$: by local nonsatiation, $p^* \cdot x_i' \geq p^* \cdot \omega_i$.

Step 3. Summing: $\sum_i p^* \cdot x_i' > \sum_i p^* \cdot \omega_i$.

Step 4. But feasibility requires $\sum_i x_i' = \sum_i \omega_i$, giving $\sum_i p^* \cdot x_i' = \sum_i p^* \cdot \omega_i$. Contradiction. $\square$

The proof uses only local nonsatiation and budget exhaustion. It does not require convexity, differentiability, or any specific functional form. This generality is what makes the theorem powerful.

Interpretation. The First Welfare Theorem is the formal statement of Adam Smith's "invisible hand." Competitive markets produce an allocation that no rearrangement can improve upon without making someone worse off. But the assumptions (complete markets, price-taking, no externalities, no public goods, full information) define exactly when the invisible hand fails.

Example 10.6 — First Welfare Theorem in a 2-Consumer Economy

Consumer 1: $u_1 = x_1^{1/2}y_1^{1/2}$, endowment $(4, 0)$. Consumer 2: $u_2 = x_2^{1/2}y_2^{1/2}$, endowment $(0, 4)$.

From Example 10.4, the equilibrium is $x_1^* = y_1^* = x_2^* = y_2^* = 2$ at $p_x = p_y$.

Check Pareto optimality: At the equilibrium, $MRS_1 = y_1/x_1 = 1$ and $MRS_2 = y_2/x_2 = 1$. Since $MRS_1 = MRS_2 = p_x/p_y$, the indifference curves are tangent — the allocation is on the contract curve.

Verify no Pareto improvement: Any reallocation giving Consumer 1 more of good $x$ (say $x_1 = 3$) requires $x_2 = 1$. Then $u_1 = \sqrt{3 \cdot y_1}$ and $u_2 = \sqrt{1 \cdot y_2}$ with $y_1 + y_2 = 4$. For Consumer 1 to gain ($u_1 > \sqrt{4} = 2$), we need \$1y_1 > 4$, so $y_1 > 4/3$, leaving $y_2 < 8/3$, giving $u_2 = \sqrt{8/3} < 2 = u_2^*$. Consumer 2 is worse off. No Pareto improvement exists.

Interactive: First Welfare Theorem Visualization

The Walrasian equilibrium lies on the contract curve (Pareto efficient). Toggle "Pareto improvements?" to verify: at the equilibrium, the lens-shaped region where both consumers can gain is empty. At the endowment, it is not.

At the Walrasian equilibrium: No Pareto improvements exist — the lens-shaped region is empty. This IS the First Welfare Theorem.

Interactive 10.3. Toggle between viewing the equilibrium (where no Pareto improvements exist) and the endowment (where the shaded lens shows mutually beneficial trades). The equilibrium's position on the contract curve proves efficiency visually.

10.7 The Second Welfare Theorem

Second Welfare Theorem. Under convexity assumptions (convex preferences, convex production sets), any Pareto optimal allocation can be achieved as a Walrasian equilibrium — after appropriate redistribution of endowments (lump-sum transfers of wealth).

Interpretation. The Second Welfare Theorem says efficiency and equity are separable problems. Society can choose any Pareto-efficient distribution through two steps:

  1. Redistribute endowments using lump-sum transfers
  2. Let markets operate from the new endowments

The markets will then produce a competitive equilibrium that is both efficient (by the First Welfare Theorem) and achieves the desired distribution.

Why it matters for policy. Don't distort markets to achieve equity (that sacrifices efficiency). Instead, use lump-sum transfers to redistribute, then let markets work. The right-wing implication: let markets operate freely. The left-wing implication: redistribute as much as you want. Both can be achieved simultaneously — in theory.

Why it fails in practice. Lump-sum transfers require information about individuals' types that the government does not have. Real-world redistribution uses distortionary taxes (income, capital gains, wealth) that change incentives and create deadweight loss. This information problem is the subject of mechanism design (Chapter 11) and optimal taxation (Chapter 16).

Core Equivalence

In large economies, the set of core allocations (allocations that no coalition can improve upon) shrinks to the set of Walrasian equilibrium allocations. This is the core equivalence theorem — competitive equilibrium is the unique outcome that survives competition among all possible coalitions.

Maya's Enterprise

We model Maya's lemonade market as a 2-consumer, 2-good Edgeworth box exchange economy.

Setup: Maya and Alex. Two goods: lemonade ($L$) and cookies ($C$). Maya starts with 45 lemonade and 0 cookies. Alex starts with 0 lemonade and 40 cookies.

Preferences: $u_M = L_M^{0.5}C_M^{0.5}$, $u_A = L_A^{0.3}C_A^{0.7}$.

Market clearing gives $p_L/p_C = 8/15 \approx 0.533$.

Equilibrium: Maya: $(L_M, C_M) = (22.5, 12)$. Alex: $(L_A, C_A) = (22.5, 28)$.

By the First Welfare Theorem, this allocation is Pareto optimal.

The Historical Lens

Arrow-Debreu (1954): The Existence Proof. Kenneth Arrow and Gerard Debreu proved that a competitive equilibrium exists under weak assumptions (convex preferences, no externalities). Using Kakutani's fixed-point theorem, they showed that a set of prices exists clearing all markets simultaneously — formalizing Adam Smith's "invisible hand" two centuries after The Wealth of Nations.

The mathematical achievement was remarkable: reducing the problem to showing that a certain correspondence (excess demand as a function of prices) satisfies the conditions for a fixed point. The result required only local nonsatiation and convexity — not differentiability or specific functional forms.

Debreu's Theory of Value (1959) distilled this framework into a rigorous axiomatic system, earning him the 1983 Nobel Prize. Arrow had already received the Nobel in 1972 for his broader contributions to general equilibrium and social choice. Their existence proof remains the mathematical foundation for welfare economics and the two welfare theorems proved in this chapter.

Summary

Key Equations

LabelEquationDescription
Eq. 10.1$e(p, \bar{u}) = \min p \cdot x$ s.t. $u(x) \geq \bar{u}$Expenditure minimization
Eq. 10.2$h_i = \partial e / \partial p_i$Shephard's lemma
Eq. 10.3–10.4$e(p, V(p,m)) = m$; $V(p, e(p,\bar{u})) = \bar{u}$Duality identities
Eq. 10.5$h(p, \bar{u}) = x(p, e(p, \bar{u}))$Hicksian = Marshallian at compensated income
Eq. 10.6$x_i = -(\partial V/\partial p_i)/(\partial V/\partial m)$Roy's identity
Eq. 10.7$S_{ij} = \partial h_i/\partial p_j = \partial x_i/\partial p_j + x_j \partial x_i/\partial m$Slutsky matrix entry
Eq. 10.8$z(p) = \sum_i x_i(p) - \sum_i \omega_i$Aggregate excess demand

Exercises

Practice

  1. Preferences are defined by $x \succsim y \iff x_1 + x_2 \geq y_1 + y_2$. Verify completeness, transitivity, and continuity. Write down a utility function that represents these preferences.
  2. A consumer makes the following choices: at prices (2, 1) she buys (3, 4); at prices (1, 3) she buys (5, 1). Check WARP.
  3. For Cobb-Douglas utility $u = x_1^{1/3}x_2^{2/3}$: (a) derive the expenditure function, (b) verify Shephard's lemma, (c) verify Roy's identity.
  4. In a 2-consumer, 2-good exchange economy: $u_1 = x_1 y_1$, $\omega_1 = (6, 2)$; $u_2 = x_2 y_2$, $\omega_2 = (2, 6)$. Find the Walrasian equilibrium prices and allocation.

Apply

  1. A consumer's observed demand function is $x_1 = m/(p_1 + p_2)$ and $x_2 = m/(p_1 + p_2)$. (a) Check Walras' law. (b) Check homogeneity of degree zero. (c) Compute the Slutsky matrix and check symmetry and negative semidefiniteness. (d) Can this demand be generated by utility maximization?
  2. Explain why the First Welfare Theorem does not apply to an economy with externalities (connect to Chapter 4). Identify the specific assumption that fails.
  3. The Second Welfare Theorem says any efficient allocation can be achieved via competitive markets with lump-sum transfers. Explain why, in practice, governments use distortionary taxes instead. What information problem makes lump-sum transfers infeasible?
  4. Using the Edgeworth box, illustrate: (a) an allocation in the core but not a competitive equilibrium, (b) a Pareto improvement from the endowment point, (c) why the endowment point itself is generally not Pareto efficient.

Challenge

  1. Prove that if the expenditure function $e(p, \bar{u})$ is concave in $p$, then the Slutsky matrix is negative semidefinite. (Hint: the Hessian of a concave function is negative semidefinite, and $\partial^2 e/\partial p_i \partial p_j = \partial h_i/\partial p_j = S_{ij}$.)
  2. Prove the First Welfare Theorem for the 2-consumer, 2-good case with locally nonsatiated preferences. Then identify where the proof breaks down if one consumer has satiated preferences (a bliss point).
  3. In an Edgeworth box economy with Leontief preferences ($u = \min(x, 2y)$) for both consumers, does a Walrasian equilibrium exist? If so, find it. If not, explain which existence condition fails.
  4. State Afriat's theorem precisely. Using a dataset of 4 observations (price vectors and chosen bundles), construct an example where WARP is satisfied but SARP is violated.
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