Chapter 7, Walras meets Nash Video Solutions, Competitive Equilibrium: Theory and Applications

Problem 1

Replacing $s_i$ by $x_i$ and $s_{-i}$ by $e$, show that the definition of $\Gamma\left(P_i\right)$ in Section 7.1.1 is equivalent to the definition given in Chapter 5 ,
$$
\Gamma\left(P_i\right):=\left\{\left(e, x_i, x_i^{\prime}\right) \in E \times X_i \times X_i \mid x_i^{\prime} \in P_i\left(e, x_i\right)\right\} .
$$

Sajin Shajee

Numerade Educator

05:24

Problem 2

Consider a two-person game in which player 1 has strategy set $S_1=$ $\{U, D\}$ and player 2 strategy set $S_2=\{L, R\}$. Assume that payoffs are given by the following payoff matrix
$\begin{array}{ccc} & L & R \\ U & (5,5) & (-3,8) \\ D & (8,-3) & (0,0)\end{array}$
with the interpretation $\left(u_1(D, L), u_2(D, L)\right)=(8,-3)$ for the lower left hand entry and so on. Determine $B_1(s)$ and $B_2(s)$ for $s=(U, L)$ and for $s=(D, R)$. Using the fixed point characterization, show that one of these strategy vectors is a Nash equilibrium and one is not.

Manasvee Singh

Numerade Educator

Problem 3

Replacing $s$ by $\left(s_i, s_{-i}\right)$ and using the fact that $K_i$ does not actually depend on $s_i$, show that $B_i(s)$ given in Section 7.1.3 does not depend on $s_i$. Conclude, therefore, that the definition of $B_i(s)$ is not circular.

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Problem 4

Let $\left\{S_i, P_i \mid i \in I\right\}$ be an ordinary game (not an abstract economy!) in which each of the strategy sets $S_i$ is nonempty, compact, and convex. Assume that preferences of each player satisfy the assnmptions listed in Theorem 7.5. Using Theorem 7.5 as a model, state and prove a theorem establishing existence of a Nash equilibrium for this game.

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04:42

Problem 5

Under the conditions of Lemma 7.7 prove that if the fixed point $s^*=$ $\left(p^*, x^*, y^*\right)$ satisfies $x_i^* \in \operatorname{cl} P_i\left(x^*\right)$ for all $i \in I$, then $p^* \cdot Z\left(x^*, y^*\right)=0$. (Hint: Look at the proof of Theorem 6.9, the strong form of Walras' Law.)

Yiyang Wang

Numerade Educator

Problem 6

Fixing $p \in \Delta$ and assuming that the conditions of Theorem 7.8 are satisfied, verify each of the following assertions about consumer $i \in I$ :
(a) $\left\{x_i \in L \mid p \cdot x_i \leq \omega_i(p)\right\}$ is closed;
(b) $\beta_i(p):=X_i \cap\left\{x_i \in L \mid p \cdot x_i \leq \omega_i(p)\right\}$ is compact;
(c) $\left\{x_i \in L \mid p \cdot x_i \leq \omega_i(p)\right\}$ is convex;
(d) $\beta_i(p):=X_i \cap\left\{x_i \in L \mid p \cdot x_i \leq \omega_i(p)\right\}$ is convex;
(e) $\beta_i(p)$ is nonempty.

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01:10

Problem 7

Show that if $\succ_i$ has open graph in $X_i \times X_i$ and $\succ_i$ is independent of $p$, then $\succ_i$ has open graph in $\Delta \times X_i \times X_i$.

Manik Pulyani

Numerade Educator

01:59

Problem 8

Show that if $\phi_i(s):=\phi_i(p, x, y)$ is independent of $x_{-i}$ and $y$ and if $\phi_i$ is uhc in $p$, then $\phi_i$ is uhc in $s$.

Wen Zheng

Numerade Educator

Problem 9

Assuming that $Y_k$ is convex, prove that the upper contour set $\left\{y_k^{\prime} \in Y_k \mid p \cdot y_k^{\prime} \geq p \cdot y_k\right\}$ is convex.

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Problem 10

Prove that for fixed $x$ and $y$, the upper contour set $\left\{p^{\prime} \in \Delta \mid p^{\prime} \cdot Z(x, y) \geq p \cdot Z(x, y)\right\}$ of the Auctioneer is convex.

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01:33

Problem 11

Justify the assertion in Section 7.2 .1 that $p \cdot Z(x, y)$ is continuous in $p, x$, and $y$.

Linh Vu

Numerade Educator

Problem 12

Consider a two-person exchange economy. Consumer 1 has utility function $u_1\left(x_1\right)=\sqrt{x_{11}}+\sqrt{x_{12}}$ and endowment $w_1=(2,0)$. Consumer 2 has utility function $u_2\left(x_2\right)=\min \left\{2 x_{21}, x_{22}\right\}$ and endowment $w_2=(2,4)$. Consumption sets $X_1=X_2=[0,10]^2$. Normalize prices so that $p_1+p_2=1$.
(a) Verify that consumer 1's budget correspondence is not continuous and his demand correspondence is not uhc at $p=(0,1)$.
(b) Does a Walrasian equilibrium exist for this economy?
(c) Does your answer to (b) change if $u_2\left(x_2\right)=x_{22}$ ?
(d) Does your answer to (b) change if $u_2\left(x_2\right)=x_{22}$ and $X_1=X_2=$ $[0,4]^2 ?$

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Problem 13

Complete the proof of Theorem 7.11 .

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Problem 14

In the first version of the model presented in Section 7.3.2 with $\alpha=\beta$ and strictly positive production, justify the assertion that the equilibrium is Pareto optimal. Show explicitly that if $n k / 2 \alpha \leq 1$, then an equilibrium with no trade and no production does exist. (For greater specificity, take $n=2, k=1$, and $\alpha=\beta=1$ if you wish.)

Victor Salazar

Numerade Educator

02:16

Problem 15

Using the Lagrangian
$$
L=u_1\left(x_{11}, \bar{x}_2\right)+\sum_{i=2}^n \lambda_i\left(\bar{u}_i-u_i\left(x_{i 1}, \bar{x}_2\right)\right)+\gamma\left(6 n-\sum_{i=1}^n x_{i 1}-n \bar{x}_2\right),
$$
derive the Samuelson condition for the optimal provision of a pure public good: $\sum_{i \in I} \operatorname{MRS}_i\left(x_i\right)=n$.

Akash M

Numerade Educator

01:21

Problem 16

Suppose that $S_1=(0,1)$ and $S_2=(2,3)$.
(a) Letting $x=3$ and using a diagram like Figure 7.5, illustrate the Shapley-Folkman Theorem.
(b) Again letting $x=3$, find closed sets of the type $\tilde{S}_i$ described in the proof of the Shapley-Folkman Theorem.

Sriram Soundarrajan

Numerade Educator

01:46

Problem 17

Suppose that $S_i=\{0,1\}$ for $i \in I=\{1, \ldots, 100\}$.
(a) Describe the sets $\sum_{i \in I} S_i$ and co $\sum_{i \in I} S_i$.
(b) Illustrate the Shapley-Folkman Theorem for $x=42.3$.

Darren Mckaig

Numerade Educator

04:51

Problem 18

Verify that the set $C(x)$ described in the proof of the ShapleyFolkman Theorem is nonempty, compact, and convex as claimed. Also verify that $z^*$ and $z^{* *}$ belong to $P(x)$ as claimed.

Chris Trentman

Numerade Educator

Problem 19

Consider a production economy with three commodities: the first two are not produced while the third is produced using commodities one and two as inputs. Production is performed by an arbitrary number of identical firms each with cost function
$$
c\left(p_1, p_2, y_3\right)=\sqrt{p_1 \overline{p_2}}\left(1+y_3^2\right)
$$
where $p_1$ and $p_2$ are the prices of commodities one and two and $y_3$ the output of commodity three. Assume that consumers have strictly positive initial endowments of commodities one and two and that utility functions $u_i\left(x_i\right)$ are strictly monotonic and continuous.
(a) Compute the supply correspondence for a representative firm and demonstrate that for some prices it fails to be convex-valued.
(b) Describe the effect of "convexifying" this economy on the cost function of each of the firms.
(c) Assuming that an equilibrium exists for the convexified economy, apply the Shapley-Folkman Theorem to prove the existence of an approximate equilibrium for this economy: i.e., a price vector $p$ such that all markets clear, every consumer receives a commodity bundle which maximizes utility subject to his or her budget constraint, and the production of all but three firms is both feasible and profit maximizing.
(d) Use the Shapley-Folkman Theorem to prove that the approximate equilibrium can allow all but one firm a profit maximizing, feasible production vector.

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Problem 20

Show that the Walrasian Auctioneer satisfies the assumptions of Theorem 7.6.

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01:43

Problem 21

Illustrate with an example in $\mathbf{R}^2$ that it is possible to have $x_i \notin P_i(s)$ (irreflexivity) but $x_i \in \operatorname{co} P_i(s)$.

Kelsey Van Horn

Numerade Educator

00:44

Problem 22

Using equation (7.8), verify that the strict upper contour sets in Figure 7.8 are correct in depicting that $(2 / 3,1 / 3) \notin P_i(1 / 3,2 / 3)$ and $(1 / 3,2 / 3) \notin P_i(2 / 3,1 / 3)$. Demonstrate by example that these preferences are in fact intransitive.

Eric Mockensturm

Numerade Educator