Chapter 6, Clearing markets Video Solutions, Competitive Equilibrium: Theory and Applications

Problem 1

Prove that $\Delta_w \Phi(p)=\Phi(p)-\sum_{i \in I} w_i$.

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Problem 4

Extend the results of Section 6.1 to a constant returns to scale production economy. To an Arrow-Debreu production economy.

Lucas Finney

Numerade Educator

01:46

Problem 5

Find all of the fixed points of $f:[0,1] \rightarrow[0,1]$ for
(a) $f: x \mapsto x^2$;
(b) $f: x \mapsto \sqrt{x}$;
(c) $f: x \mapsto 1 /(1+x)$.

Ankit Gupta

Numerade Educator

03:11

For $\emptyset \neq S \subset[0,1]$, provide an example where $f: S \rightarrow S$ fails to have a fixed point because:
(a) $S$ is compact and convex, but $f$ is not continuous;
(b) $f$ is continnons and $S$ is compact, but $S$ is not convex;
(c) $f$ is continuous and $S$ is convex, but $S$ is not compact.

Chris Trentman

Numerade Educator

01:20

Problem 7

Using a net trade diagram, illustrate the free disposal equilibrinm for the economy portrayed in Figure 6.3.

Kaylee Mcclellan

Numerade Educator

08:43

Problem 8

Let $V=\{(0,0),(1,0),(1,1),(0,1)\} \subset \mathbf{R}^2$ and $S=\operatorname{co} V$. Show that the midpoint of the square $S$ is expressible in more than one way as an affine combination of the vertices of $S$. Which points in $S$, if any, are uniquely expressible as an affine combination of its vertices?

Ahmad Reda

Numerade Educator

00:50

Problem 9

For appropriate choices of the parameters, can any $p \in \Delta$ be attained as a solution to the model described in Table 6.1?

Ankit Gupta

Numerade Educator

03:27

Problem 10

Illustrate the sets $M_j$ in a drawing of the simplex for the example economy of Section 6.2 .4 with $\mu_1=\mu_2=\mu_3=1 / 3$, and verify that they satisfy the requirements of the KKM Theorem. Experiment with some alternative values for the $\mu_t$.

Nick Johnson

Numerade Educator

01:51

Problem 11

Complete the proof of Theorem 6.25.

Joseph Liao

Numerade Educator

01:02

Problem 12

For $V=\{(0,0),(1,0),(0,1)\} \subset \mathbf{R}^2$ and $S=\operatorname{co} V$, give examples where the conclusion of Theorem 6.24 fails to hold because
(a) the sets $M_j$ are not closed;
(b) the sets $M_j$ do not cover $S$;
(c) the condition co $\left\{v^j \in V \mid j \in T\right\} \subset \cup_{j \in T} M_j$ for all nonempty $T \subset\{1, \ldots, n\}$ is not satisfied.

Raj Bala

Numerade Educator

01:18

Problem 13

How many distinct permutations are there of the form
$$
\left(\begin{array}{cccc}
1 & 2 & \ldots & n-1 \\
\psi(1) & \psi(2) & \ldots & \psi(n-1)
\end{array}\right) ?
$$

Aymara Gallardo

Numerade Educator

00:54

Problem 14

Let
$$
b=\left(\begin{array}{l}
5 \\
4 \\
3 \\
4
\end{array}\right)
$$
be a base vertex in the 4-dimensional unit simplex with grid denominator $D=16$. Describe the vertices of the subsimplex generated by this base vertex and the permutation
$$
\psi=\left(\begin{array}{lll}
1 & 2 & 3 \\
2 & 1 & 3
\end{array}\right) .
$$

Sriparna Bhattacharjee

Numerade Educator

04:51

Problem 15

Sperner's Leinma has all of the makings of an excellent parlor game. Give a friend a copy of Figure 6.5 and tell the friend that she is free to attach labels 1,2 , or 3 to any of the vertices subject only to the restrictions that:
$\bullet$ the three main vertices corresponding to $e^{\mathrm{I}}, \hat{e}^2$, and $e^3$ must have labels 1, 2, and 3 respectively; and
$\bullet$ the vertices along the main facets must have labels (a) 2 or 3 if opposite the main vertex labeled 1 ; (b) 1 or 3 if opposite the main vertex labeled 2; and (c) 1 or 2 if opposite the main vertex labeled 3.
Wager that, however she chooses the labels inside the main triangle, there will always be an odd number of small triangles which are completely labeled. Yon can't lose!

Lucía Guerrero

Numerade Educator

Problem 16

Consider the Kuhn simplicial subdivision for the three-dimensional unit simplex with $D=2$. There are ten vertices of the form
$$
\left(\pi_1 / 2, \pi_2 / 2, \pi_3 / 2, \pi_4 / 2\right) \quad\left(\sum_{i=1}^4 \pi_i=2\right)
$$
to which are attached the following labels:
$$
\begin{array}{ccc}
\text { Vertex } & \text { Coordinates } & \text { Label } \\
\hline \text { A } & (2,0,0,0) & 1 \\
\text { B } & (0,2,0,0) & 2 \\
\text { C } & (0,0,2,0) & 3 \\
\text { D } & (0,0,0,2) & 4 \\
\text { E } & (1,1,0,0) & 2 \\
\text { F } & (0,1,1,0) & 2 \\
\text { G } & (0,0,1,1) & 4 \\
\text { H } & (1,0,1,0) & 3 \\
\text { I } & (0,1,0,1) & 4 \\
\text { J } & (1,0,0,1) & 1
\end{array}
$$
(a) Illustrate this simplex as a pyramid in $\mathbf{R}^3$ with vertices $A, B, C$ as the base and $D$ as the top. Indicate on the diagram the positions of the remaining vertices.
(b) Verify that this labeling satisfies the labeling rule of Sperner's Lemma.
(c) Starting with the base vertex
$$
A=\left(\begin{array}{l}
2 \\
0 \\
0 \\
0
\end{array}\right)
$$
and each possible permutation $\psi:\{1,2,3\} \rightarrow\{1,2,3\}$, find all of the subsimplices contained in the main simplex which have $A$ as a base.
(d) Repeat the procedure of part (c) for each of the remaining vertices $B, C, D, E, F, G, H, I$, and $J$ (as in part (c), you need only determine the subsimplices contained in the main simplex).
(e) List the subsimplices determined in parts (c) and (d), and illustrate the simplicial subdivision in your diagram.
(f) Starting with the base vertex
$$
\left(\begin{array}{l}
2 \\
0 \\
0 \\
0
\end{array}\right)
$$
compute the path to a completely labeled subsimplex. Illustrate the path in your diagram.

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00:44

Problem 17

Suppose that in Figure 6.9 you start at some almost completely labeled subsimplex not on the boundary. Is it possible to cycle?

James Kiss

Numerade Educator

01:51

Problem 18

Compute the matrices corresponding to the path to an approximate equilibrium illustrated in Figure 6.11. Compute the excess demands for each of the three commodities at each of the three vertices of the completely labeled subsimplex, letting the endowment parameter $b=10$.

Anand Jangid

Numerade Educator

Problem 19

Verify the claim in the proof of the KKM Theorem that the labeling satisfies the requirements of Sperner's Lemma.

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Problem 20

Use the Brouwer Theorem to prove the KKM Theorem. Yon may use without proof the following fact about "partitions of nnity": if $\left\{G_i \subset \Delta \mid i=1, \ldots, n\right\}$ is an opening covering of the simplex $\Delta$, then there exists a collection of functions $f_i: \Delta \rightarrow[0,1]$ such that (a) $f_i(x)>0$ if $x \in G_i$ and $f_i(x)=0$ if $x \notin G_i ;$ and (b) $\sum_{i=1}^n f_i(x)=1$ for all $x \in \Delta$.

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01:27

Problem 21

Suppose that a pure exchange economy with two types of commodity has the market excess demand function for the first commodity
(Figure Cant Copy)
(a) Apply Merrill's algorithm to this economy with $D=100$ and a starting estimate $\pi^*=(48,52)$. Illustrate your results in a diagram similar to Figure 6.14 .
(b) Repeat part (a) with the initial estimate $\pi^*=(39,61)$, and illustrate your results in a diagram.
(c) Is there any way to use the Merrill algorithm to reach the third equilibrium point?

Nick Johnson

Numerade Educator

Problem 22

Letting $D=9$ and taking $\pi=(3,3,3)$ as a starting estimate, use a diagram like that of Figure 6.5 to show the labels assigned by Merrill's algorithm to the vertices on the artificial layer. Verify that the 2-dimensional simplex on the artificial layer defined by the starting 3-dimensional simplex is the only 2-dimensional simplex on the artificial layer which is completely labeled. Why is this important to the successful functioning of Merrill's algorithm?

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01:18

Problem 23

Letting $D=8$ and taking $\pi^*=(3,2,3)$ as the starting estimate, apply Merrill's algorithm to the simplex illustrated in Figure 6.8. Give a rough sketch of the Merrill sandwich for this case. Is there any way that you can get the algorithm to find the other two completely labeled subsimplices in Figure 6.8?

Audrey Fong

Numerade Educator

Problem 24

Practical applications of the Scarf algorithm usually involve production. Assuming constant returns and the absence of intermediate production, it is possible to solve for equilibrium in such models by iterating through a low dimensional simplex even though the number of produced commodities is very large. This exercise gives a flavor of such applications.
Consider an economy with four commodities and a single consumer. Commodities 1 and 2 are produced subject to constant returns to scale with a Cobb-Douglas production function using commodities 3 and 4 as inputs. Average cost of production for commodities 1 and 2 is given by $p_3^\beta p_4^{1-\beta}$ and $p_3^{1-\beta} p_4^\beta$ respectively where $0<\beta<1$. The consumer has initial endowment $w=(0,0, b, b)$, $b>0$, of commodities 3 and 4 , which she is willing to supply perfectly inelastically. Adopting the normalization $p_3+p_4=1$, we conclude that consumer wealth equals $b$. The consumer's preferences are described by a Cobb-Douglas utility function $x_1^\alpha x_2^{1-\alpha}, 0<\alpha<1$, for commodities 1 and 2 . (Commodities 3 and 4 are not desired by the consumer, which explains why her supply of these input commodities is perfectly inelastic.)
(a) Show that, in solving for equilibrium, it suffices to clear only one of the markets for the "primary factors" (i.e., either commodity 3 or commodity 4) rather than three out of four markets as would be the case in pure exchange. Find the Walrasian equilibrium prices for commodities 3 and 4 as functions of the parameters $\alpha$ and $\beta$.
(b) Letting $\alpha=\beta=1 / 4$, solve for the equilibrium prices for all four commodities.
(c) Focusing solely on the prices of the primary factors, illustrate the Merrill algorithm for this economy using a grid denominator $D=8$ and initial estimate $\left(\pi_3, \pi_4\right)=(7,1)$. Show how the path through the simplex can be represented by a series of matrices with the movement from matrix to matrix governed by the Kuhn replacement method.
(d) If the consumer's utility is an arbitrary quasi-concave function of all four commodities, does it still suffice to clear the markets for commodities 3 and 4 ? If not, what more needs to be assumed for this procedure to suffice?

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03:11

Problem 25

Let $K \subset[0,1]$ and $\Psi: K \rightarrow\left(2^K\right)_0^F$. Give an example where $\Psi$ has no fixed point because:
(a) $\Psi$ satisfies the conditions of Theorem $6.34, K$ is compact, but $K$ is not convex;
(b) $\Psi$ satisfies the conditions of Theorem $6.34, K$ is convex, but $K$ is not compact;
(c) $K=[0,1]$ and $\Psi$ is uhc, but $\Psi$ is not convex-valued;
(d) $K=[0,1]$ and $\Psi$ is convex-valued, but $\Psi$ is not uhe.

Chris Trentman

Numerade Educator

Competitive Equilibrium: Theory and Applications

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