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Microeconomic theory

Andreu Mas-Colell, Michael D. Whinston, Jerry R. Green

Chapter 17

The Positive Theory of Equilibrium - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that for a pure exchange economy with $J=1$ and $Y_1=-\mathbf{R}^L$., " $y_1^* \leq 0$, $p \cdot y_1^*=0$, and $p \geq 0$ " if and only if " $y_1^* \in Y_1$ and $p \cdot y_1^* \geq p \cdot y_1$ for all $y_1 \in Y_1$ "

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Problem 1

Argue that the replica procedure described at the beginning of Section 17.1 does effectively include the case where the numbers of consumers of different types are not the same (assume, for simplicity, that the proportions of the different types are rational numbers). [ Hin : Redefine the size of the original economy.]

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Problem 1

Suppose that the system of excess demand functions $z(p)$ satisfies the gross substitute property. Consider the tâtonnement price dynamics
$$\frac{d p_{\ell}}{d t}=z_\gamma(p) \quad \text { for every } \ell$$
For any price vector $p$ let $\psi(p)=\operatorname{Max}\left\{z_1(p) / p_1, \ldots, z_L(p) / p_L\right\}$.
(a) Argue that if $p(t)$ is a solution for the above tatonnement dynamics (i.e., $d p_1(t) / d t=z_2(p(t))$ for every $t$ and $t$ ) and $z(p(0)) \neq 0$ then $\psi(p(t))$ should be decreasing through time. [Hint: If $z_r(p(t)) / p_r(t)=\psi(p(t))$ then $p_r(t) / p_r(t)$ cannot decrease at $t$ for any $\ell^{\prime}$. Hence, $z,(p(t))$ cannot increase, whereas $p$, surely increases.]
(b) Argue that $p(t)$ converges to an equilibrium price as $t \rightarrow \infty$. [Hint: Recall that for the dynamics $\left({ }^*\right)$ Walras* law implies that $\Sigma, p_r^2(t)=$ constant.]

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Problem 1

Verify that the correspondence $f(\cdot)$ introduced in the proof of Proposition 17.C.1 is convex-valued.

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Problem 1

Suppose that in an exchange economy (and with the normalization $p_L=1$ ) we are given equilibrium prices $p\left(\dot{\omega}_1\right)$ as a differentiable function defined as an open domain of the endowments of the first $L-1$ goods of the first consumer, $\omega_1=\left(\omega_{11}, \ldots, \omega_{L-1,1}\right)$. All the remaining endowments are kept fixed. Suppose that the demand function of the first consumer is strictly normal in the sense that $D_{m_1} x_1\left(p_1 w_1\right) \gg 0$ through the relevant domain of $\left(p, w_1\right)$. Show then that for any $\dot{\hat{\omega}}_1$ and $\bar{p}=p\left(\hat{\omega}_1\right)$, we have rank $D_{\partial_1} z_1\left(\bar{p} ; \hat{\omega}_1\right)=L-1$ and rank $D p\left(\hat{\omega}_1\right)=L-1$, where $t_1\left(p ; \hat{\omega}_1\right)$ is the excess demand function of the first consumer for the first $L-1$ goods.

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Problem 1

Show that expression (17.F.2) gives rise to a negative semidefinite matrix of price effects, $D:(p)$, if initial endowments are proportional among themselves or if consumptions are proportional among themselves.

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Problem 1

Consider an exchange economy with two commodities and two consumers. Both consumers have homothetic preferences of the constant elasticity variety. Moreover, the elasticity of substitution is the same for both consumers and is small (i.e., goods are close to perfect complements). Specifically,
$$
u_1\left(x_{11}, x_{21}\right)=\left(2 x_{11}^p+x_{21}^p\right)^{1 / p} \quad \text { and } \quad u_2\left(x_{12}, x_{22}\right)=\left(x_{12}^p+2 x_{22}\right)^{1 / p} .
$$
and $\rho=-4$. The endowments are $\omega_1=(1,0)$ and $\omega_2=(0,1)$.
Compute the excess demand function of this economy and verify that there are multiple equilibria.

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Problem 1

Derive expressions (17.E.1) and (17.E.2).

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05:56

Problem 2

Prove property (v) of Proposition 17.B.2. The proof of Proposition 17.B. 2 in the text contains a hint. Recall also the following technical fact: any bounded sequence in $\mathbf{R}^L$ has a convergent subsequence.

Jimmy Yao
Jimmy Yao
Numerade Educator

Problem 2

Consider for a one-input, one-output problem the production function $q=v^2$, where $v$ is the amount of input. Show that the corresponding production set $Y$ is additive but that the smallest cone containing it, $Y^*$, is not closed. Discuss in what sense the nonconvexity in $Y$ is large. Argue that, whatever the number of consumers, there is no useful sense in which an equilibrium (nearly) exists.

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Problem 2

There is an output good and a numeraire. The price of the output good is $p$. The data of our problem are given by two functions: The consumption side of the economy provides an excess demand function $z(p)$ for the output good, and the production side an increasing inverse output supply function $p(z)$. Both functions are differentiable. In addition, their graphs cross at ( 1,1 ), which is the equilibrium we will concentrate on in this exercise.
Given this setting we can define two one-variable dynamics:
(i) In Walras price dynamics we assume that at $p$ the price increases or decreases according to the sign of the difference between excess demand and (direct) supply at $p$.
(ii) In Marshall quantity dynamics we assume that at $z$ production increases or decreases according to the sign of the difference between the demand price (i.e., the inverse excess demand) and the supply price (i.e., $p(z)$ ) at $z$.
(a) Write the above formally and interpret economically.
(b) Suppose that the technology is nearly of the constant returns type. Show then that around the equilibrium $(1,1)$ the system is always Walrasian stable but that Marshallian stability depends on the slope of the excess demand function (in what way?).
(c) Write general price and quantity dynamics where prices move à la Walras and quantities à la Marshall. Draw a $(p, z)$ phase diagram and argue that in the typical case dynamic trajectories will spiral around the equilibrium.
(d) Go back to the technology specification of (b). Show that the system in (c) is locally stable if and only if the equilibrium is Marshallian stable.
(e) Consider the simplest price and quantity dynamics in the limit case where there are constant returns and excess demand is also a constant function. Draw the phase diagram. Suppose now that the quantity dynamics is modified by making the quantity responses depend not only on price and cost but also on the "expectation of sales," that is, on the excess demand. Will this have a stabilizing or a destabilizing effect?

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Problem 2

The setting is as in Exercise 17.G. 1 or as in Proposition 17.G.2. Suppose that $z\left(\bar{p}_{;} \dot{\omega}_1\right)=0$. Show that there are economies with $D_p \xi\left(\hat{p}_{;} \dot{\omega}_1\right)$ an $(L-1) \times(L-1)$ negative definite matrix but where $\partial p_1\left(\omega_1\right) / \partial \omega_{11}>0$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 2

Complete the requested verification of Example 17.F.1.

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Problem 2

Derive expression (17.E.3).

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Problem 2

Apply the implicit function theorem to show that if $f(v)=0$ is a system of $M$ equations in $N$ unknowns and if at $\bar{v}$ we have $f(\bar{v})=0$ and rank $D f(\bar{v})=M$, then in a neighborhood of $\tilde{v}$ the solution set of $f(\cdot)=0$ can be parameterized by means of $N-M$ parameters.

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Problem 2

Show that a convex-valued correspondence $z(\cdot)$ defined on $\mathbf{R}_{++}^{\mathrm{L}}$ and satisfying the conditions (i) to (v) listed below (parallel to the corresponding conditions in Proposition 17.C.1) admits a solution; that is, there is a $p$ with $0 \in z(p)$.
(i) $z(\cdot)$ is upper-hemicontinuous.
(ii) $z(\cdot)$ is homogeneous of degree zero.
(iii) For every $p$ and $z \in z(p)$ we have $p^* z=0$ (Walras' law).
(iv) There is $s \in \mathbf{R}$ such that $z_f>-s$ for any $z \in z(p)$ and $p$.
(v) If $p^* \rightarrow p \neq 0, z^* \in z\left(p^*\right)$ and $p_l=0$ for some $l$, then $\operatorname{Max}\left\{z_1^*, \ldots, z_l^*\right\} \rightarrow \infty$.

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Problem 3

Carry out explicitly the computations for Proposition 17.D.4.

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Problem 3

There are three commodities the first is a high-quality good, the second is a low-quality good, and the third is labor. The first and second goods can be produced from labor according to the production functions $f_1(v)=\operatorname{Min}\{v, 1\}$ and $f_2(v)=\operatorname{Min}\left\{v^f, 1\right\}$ for $0<\beta<1$. The economy has one unit of labor in the aggregate. Labor has no utility value. There are two equally sized classes of agents, with a very large number of each. "Rich" and "poor" have identical endowments, but the rich own all the shares in the firms of the economy. The rich spend all their wealth on the high-quality good; the poor must buy either one quality or the other-they cannot buy both. The utility function of the poor is $u\left(x_1, x_2\right)=x_1+\frac{1}{2} x_2$, defined for ( $x_1, x_2$ ) not both positive.
(a) Which standard hypothesis of the general model does this economy fail to satisfy?
(b) Show that there can be no equilibria other than one in which both qualities of product are produced.
(c) Show that an equilibrium exists.

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Problem 3

Suppose that $2(\cdot)$ is an aggregate excess demand function satisfying conditions (i) to (v) of Proposition 17.8.2. Let $p^* \rightarrow p$ with some, but not all, of the components of $p$ being zero.
(a) Show that as $n$ becomes large, the maximal excess demand is always obtained for some commodity whose price goes to zero.
(b) Argue (if possible by example) that a commodity whose price goes to zero may actually remain in excess supply for all $n$. [Hint: Relative prices matter.]

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Problem 3

For $L=3$ draw an example similar to Figure 17.H. 2 but in which there is a single equilibrium that, moreover, is locally totally unstable. Could you make it a saddle?

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Problem 3

Provide explicit utility functions rationalizing at a given price vector $p$ the individual excess demands $z_i(p)$ and matrices of price effects $D z_i(p)$ constructed in the proof of Proposition 17.E.2.

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Problem 3

The setting is as in Exercise 17.F.16, except that now we have two functions $g(p) \in \mathbf{R}^N$ and $g(p) \in \mathbf{R}^N$. Each of these functions satisfies the conditions of Exercise 17.F.16 (in particular the SDS property). In addition, we assume that $g(\cdot)$ is an upward shift of $g(\cdot)$; that is, $g(p) \geq g(p)$ for every $p \in[0, r]^N$. Prove that if $\left(p^{\text {min }}, p^{\text {max }}\right)$ and $\left(p^{\text {min }}, p^{\text {max }}\right)$ are the minimal and maximal equilibrium price vectors (see Exercise 17.F.16) for $g(\cdot)$ and $g(\cdot)$, respectively, then $p^{\text {mis }} \geq p^{\text {mit }}$ and $p^{\text {max }} \geq p^{\text {max }}$. [You can assume that $g(\cdot)$ and $g(\cdot)$ are continuous; if this makes things simpler, assume also that both functions have a unique solution.] Represent graphically for the case $N=1$.

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Problem 3

Consider an exchange economy in which every consumer $i$ has continuous, strongly monotone, strictly convex preferences, and $\omega_4>0$. The peculiarity of the equilibrium problem to be considered is that the consumer will now pay a type of tax on his gross consumption; moreover, this tax can differ across commodities and consumers. We will also assume that total tax receipts are rebated equally across consumers and in a lump-sum fashion. Specifically, for every $i$ there is a vector of given tax rates $t_i=\left(t_{i,} \ldots, t_L\right) \geq 0$ and for every price vector $p \gg 0$ the budget set of consumer $i$ is
$$B_i\left(p, w_i\right)=\left\{x_i \in R^i: \sum_i\left(1+t_{i, i}\right) p_i x_{i, i} \leq w_i\right\} .$$
An equilibrium with taxes is then a price vector $p \gg 0$ and an allocation ( $x_i^*, \ldots, x_i^*$ ) with $\sum x_i^*=\sum{ }_i \omega_i$ such that every $i$ maximizes preferences in $B_i\left(p_i p^* \omega_i+(1 / l)\left(\sum_l t_l p_l x_l\right)\right)$.
(a) Mustrate the notion of an equilibrium with taxes in an Edgeworth box. Verify that an equilibrium with taxes need not be a Pareto optimum.
(b) Apply Proposition 17.C.1 to show that an equilibrium with taxes exists.
(c) As formulated here, the taxes are on gross consumptions. If they were imposed instead on net consumptions, that is, on amounts purchased or sold, then (assuming the same rate for buying or selling) the budget set would be
$$
B_i\left(p, T_i\right)=\left\{x_i \in \mathbf{R}_{+}^L: p \cdot\left(x_i-\omega_i\right)+\sum_l t\left|p_{l i}\left(x_{l i}-\omega_{l i}\right)\right| \leq T_i\right\},
$$
where the $T_i$ are the lump-sum rebates. In what way does this budget set differ from that described previously for the case of taxes on gross consumptions? Represent graphically. Notice the kinks.
(d) Write down a budget set for the situation similar to (e) except that the tax rates for amounts bought or sold may be different.
(c) (More advanced) How would you approach the existence issue for the modification described in (c)?

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Problem 3

There are four goods and two consumers. The endowments of the consumers are $\omega_1=\left(\omega_{11}, \omega_{21}, 0,0\right)$ and $\omega_2=\left(\omega_{12}, \omega_{22}, 0,0\right)$. Consumer 1 spends all his wealth on good 3 while consumer 2 does the same on good 4 . Specify some values of $\omega_1$ and $\omega_2$ for which the corresponding excess demand of this economy does not satisfy the weak axiom of revealed preference.

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Problem 4

Consider a two-commodity, two-consumer exchange economy satisfying the appropriate differentiability conditions on utility and demand functions. There is a total endowment vector $\omega \gg 0$. Show that for almost every $\omega_1 \ll \omega$ the economy defined by the initial endowments $\omega_1$ and $\omega_2=\tilde{\omega}-\omega_1$ has a finite number of equilibria. This differs from the situation in Proposition 17.D. 2 in that total endowments are kept fixed. [Hint: You should use the properties of the Slutsky matrix.]

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Problem 4

Suppose that there are $J$ firms whose production sets $Y_1, \ldots, Y_j \subset \mathbf{R}^L$ are closed, strictly convex, and bounded above. Suppose also that a strictly positive consumption bundle is producible using the initial endowments and the economy's aggregate production set $Y=\sum_i Y_j$ (i.e., there is an $\bar{x} \gg 0$ such that $\bar{x} \in\left\{\Sigma, \omega_i\right\}+Y$ ). Show that the production inclusive aggregate excess demand function $\bar{z}(p)$ in (17.B.3) satisfies properties (i) to (v) of Proposition 17.B.2.

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Problem 4

Suppose that there are $L$ goods but that for every consumer there is a good such that at any price the consumer spends all his wealth on that good (perhaps goods are distinguished by their location). Show that the aggregate excess demand will satisfy the (weak) gross substitute property.

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Problem 4

Consider a pure exchange economy. The only novelty is that a progressive tax system is instituted according to the following rule: individual wealth is no longer $p \cdot \omega_i$; instead, anyone with wealth above the mean of the population must contribute half of the excess over the mean into a fund, and those below the mean receive a contribution from the fund in proportion to their deficiency below the mean.
(a) For a two-consumer society with endowments $\omega_1=(1,2)$ and $\omega_2=(2,1)$, write the after-tax wealths of the two consumers as a function of prices.
(b) If the consumer preferences are continuous, strictly convex, and strongly monotone, will the excess demand functions satisfy the conditions required for existence in Proposition 17.C.I given that wealth is being redistributed in this way?
condition (ii) not included in the list?

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Problem 4

Consider the two-commodity case. Give an example of a function $z(p)$ defined on $P_{\varepsilon}=\left\{\left(p_1, p_2\right) \geqslant 0: \varepsilon<\left(p_1 / p_2\right)<(1 / \varepsilon)\right\}$, and with values in $\mathbf{R}^2$, that is continuous, is homogeneous of degree zero, satisfies Walras' law, and cannot be generated from a rational preference relation. Represent graphically the offer curve associated with this function. Note that it goes through the initial endowment point and compare with the construction used in Figure 17.E.2.

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Problem 5

Consider a two-commodity, two-consumer exchange economy satisfying the appropriate differentiability conditions on utility and demand functions. Set the equilibrium problem as an equation system in the consumption variables $x_1 \in \mathbf{R}_{+}^2$ and $x_2 \in \mathbf{R}_{+}^2$, the price variables $p \in \mathbf{R}^2$, and the reciprocals of the marginal utilities of wealth $\lambda_1 \in \mathbf{R}$ and $\lambda_2 \in \mathbf{R}$ (neglect the possibility of boundary equilibria). The parameters of the system are the initial endowments $\left(\omega_1, \omega_2\right) \in \mathbf{R}_{+}^4$. Prove without further aggregation that (after deleting one equation and one unknown) the system satisfies the full rank condition of the transversality theorem.

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Problem 5

Consider a two-commodity, two-consumer exchange economy satisfying the appropriate differentiability conditions on utility and demand functions. Set the equilibrium problem as an equation system in the consumption variables $x_1 \in \mathbf{R}_{+}^2$ and $x_2 \in \mathbf{R}_{+}^2$, the price variables $p \in \mathbf{R}_4^2$, and the reciprocals of the marginal utilities of wealth $\lambda_1 \in \mathbf{R}$ and $\lambda_2 \in \mathbf{R}$ (neglect the possibility of boundary equilibria). The parameters of the system are the initial endowments $\left(\omega_1, \omega_2\right) \in \mathbf{R}_4^4$. Prove without further aggregation that (after deleting one equation and one unknown) the system satisfies the full rank condition of the transversality theorem.

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Problem 5

Suppose that there are $J$ firms. Each firm produces a single output under conditions of constant returns. The unit cost function of firm $j$ is $c_j(p)$, which we assume to be differentiable. The consumption side of the economy is expressed by an aggregate excess demand function $z(p)$. Write down an equation system similar to (17.B.4)-(17.B.5) for the equilibria of this economy.

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Problem 5

Show that the choices represented in Figure 17.E. 3 cannot be generated from consumers with endowment vectors bounded above by $(1,1)$ and nonnegative consumption.

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Problem 5

Complete the missing steps of Example 17.F.2.

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Problem 6

The setup is identical to Exercise 17.D. 5 except that an externality is allowed: The (differentiable) utility function of consumer 1 may depend on the consumption of consumer 2 ; that is, it has the form $u_1\left(x_1, x_2\right)$ where $x_i$ is consumer $i$ 's consumption bundle [but we still have $\left.u_2\left(x_2\right)\right]$. Equilibrium is defined as usual, with the proviso that consumer 1 takes consumer 2's consumption as given. Show that, generically on initial endowments $\left(\omega_1, \omega_2\right) \in \mathbf{R}_{+}^4$, the number of equilibria is finite.

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Problem 6

Suppose that there is a single production set $Y$ and that $Y$ is a closed, convex cone satisfying free disposal. Consider the following exchange equilibrium problem. Given prices $p=\left(p_1 \ldots \ldots p_2\right)$, every consumer $i$ chooses a vector $v_i \in \mathbf{R}^L$ so as to maximize $\succsim_i$ on the set $\left\{x_i \in X_i: p \cdot v_i \leq p+\omega_i\right.$, and $x_i=v_i+y$ for some $\left.y \in Y\right\}$. The price vector $p$ and the choices $v^*=\left(v_1^*, \ldots, v_i^*\right)$ are in equilibrium if $\sum_i v_i^*=\sum, \omega_i$. Show that, under the standard assumptions on preferences and consumption sets, the price vector and the individual consumptions constitute a Walrasian equilibrium for the economy with production. Interpret.

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Problem 6

Consider a two-consumption-good, two-factor model with constant returns and no joint production. In fact, suppose that the production functions for the two consumption goods are Cobb-Douglas. Consumers have holdings of factors and have preferences only for the two consumption goods. The economy is a closed economy (at equilibrium, consumption must equal production). Suppose that the two goods are normal and gross substitutes in the demand function of the consumers. Define an induced exchange economy for factors of production by assuming that at any vector of factor prices the two goods are priced at average cost and the final demand for them is met. Show that the resulting aggregate excess demand for factors of production has the gross substitute property and, consequently, that there is a unique equilibrium for the overall economy.

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Problem 6

Show that the excess demand function $z_i(p)=e^d-p_i p$. defined for $\|p\|=1$, is proportionally one-to-one in the sense used in the general proof of Proposition 17.E. 3 (at the end of Section 17.E).

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Problem 7

Show directly that the excess demand function $z_i(p)=e^1-p_i p$ used in the general proof of Proposition 17.E. 3 satisfies the strong axiom of revealed preference.

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Problem 7

The setup is identical to Exercise 17.D. 5 except that an externality is allowed: The (differentiable) utility function of consumer 1 may depend on the consumption of consumer 2 ; that is, it has the form $u_1\left(x_1, x_2\right)$ where $x_i$ is consumer $i$ 's consumption bundle [but we still have $u_2\left(x_2\right)$ ]. Equilibrium is defined as usual, with the proviso that consumer 1 takes consumer 2's consumption as given. Show that, generically on initial endowments $\left(\omega_1, \omega_2\right) \in \mathbf{R}_{+}^4$, the number of equilibria is finite.

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01:00

Problem 7

Prove expression (17.F.3) for $L=2$.

Raj Bala
Raj Bala
Numerade Educator

Problem 8

Show that expression (17.F.3) implies that the set of solutions to $z(p)=0$ is convex.

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Problem 8

Suppose the agents of an overall exchange economy are distributed across $N$ islands with no communication among them. Each island economy has three equilibria.
(a) Argue that the number of equilibria in the overall economy is $3^N$.
(b) Suppose now that the islands' economies are identical and that there is a possibility of communication across the islands: free and costless transportation of commodities. Show that then the number of equilibria is 3 .
17.D.8^ Show by explicit computation that the index of the equilibrium of a one-consumer Cobb-Douglas pure exchange economy is +1 .

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01:20

Problem 9

Consider an economy with a single constant returns production set $Y$. Preferences are continuous, strictly convex, and strongly monotone. Suppose that the feasible consumptions ( $x_1, \ldots, x_t$ ) are associated with a Walrasian equilibrium. Assume, moreover, that no trade is required to attain these consumptions if $Y$ is freely available to all consumers; that is $x_i-\omega_i \in Y$ for all $i$. Show then that those are the only possible equilibrium consumptions.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 10

Show that expression (17.F.3) implies that $D z(p)$ is negative semidefinite at an equilibrium $p$.

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Problem 11

Show that if $z(p)=0$, $\operatorname{rank} D z(p)=L-1$, and $D z(p)$ is negative semidefinite, then, for any $\ell$, the $(L-1) \times(L-1)$ matrix obtained from $D z(p)$ by deleting the $\ell$ th row and column has a determinant of sign $(-1)^{L-1}$.

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Problem 12

Show that if $z(p)=0$ and $D z(p)$ has the gross substitute sign pattern, then the $(L-1) \times(L-1)$ matrix obtained from $D z(p)$ by deleting the $\ell$ th row and column has a negative dominant diagonal (see Section M.D of the Mathematical Appendix for this concept) and is therefore negative definite.

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00:20

Problem 13

Provide the missing computation for Example 17.F.3.

Amy Jiang
Amy Jiang
Numerade Educator

Problem 14

Consider a firm that produces good 1 out of goods $\ell=2, \ldots, L$ by means of a production function $f\left(v_2, \ldots, v_2\right)$. Assume that $f(\cdot)$ is concave, increasing, and twice continuously differentiable. We say that $l^{\prime}$ and $l^{\prime}$ are complements at the input combination $v=\left(v_2, \ldots, v_L\right)$ if $\partial^2 f(v) / \partial v_f \partial v_f>0$.
(a) Verify that for the Cobb-Douglas production function $f\left(v_2, \ldots, v_L\right)=v_2^* \times \cdots \times v_L^*$. $x_2+\cdots+x_L \leq 1$, any two inputs are complements at any $v$.
(b) Suppose that $f(\cdot)$ is of the constant returns type. Show that at any $v$ and for any $\ell$ there is an $l^{\prime}$ that is a complement to $l$ at $v$.
(c) Suppose now that $f(\cdot)$ is strietly concave and that any two inputs are complements at any $v$. Let $v_f\left(p_1, \ldots, p_L\right)$ be the input demand functions. Show that, for any $\ell, \partial v_{\ell} / \partial p_1>0$, $\partial v_l / \partial p_l<0$, and $\partial v_l / \partial p_c<0$ for $l^{\prime} \neq l$.
(d) Discuss the implications of (a) to (c) for uniqueness theorems that rely on the gross substitute property.

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Problem 15

Consider a one-consumer economy with production and strictly convex preferences. There is a system of ad valorem taxes $t=\left(t_1, \ldots, t_L\right)$ creating a wedge between consumer and producer prices; that is, $p_l=\left(1+t_l\right) q_l$ where $p_l$ and $q_l$ are, respectively, the consumer and producer price for good $\ell$. Tax receipts are turned back in lump-sum fashion. Write the definition of (distorted) equilibrium. Show that the equilibrium is unique if the production sector is of the Leontief type (a single primary factor, no joint production, constant returns) and all goods are normal in consumption. Can you argue by example the nondispensability of the last normality condition? If this is simpler, you can limit your discussion to the case of two commodities (one input and one output).

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Problem 16

Suppose that $g(p)=\left(g_1(p), \ldots, g_N(p)\right)$ is defined in the domain $[0, r]^N$ and that $g(0, \ldots, 0)>0(0, \ldots, 0), g(r, \ldots, r) \ll(0, \ldots, 0)$. Note that we do not assume Walras' law, homogeneity of degree zero, or, for that matter, continuity. The function $g(\cdot)$ could, for example, be the system of excess demands corresponding to a subgroup of markets with the prices of commodities outside the group kept fixed.
(a) We say that $g(\cdot)$ satisfies the strong gross substitute property (SGS) if for some $\alpha>0$ every coordinate of the function $\alpha g(p)+p$ is strictly increasing in $p$ and $(\alpha g(p)+p) \in[0, r]^\mu$ for every $p \in[0, r]^N$. Show that if $g(p)$ has the SGS property then it also has the GS property.
(b) Show by example that the GS property does not imply the SGS property. Establish, however, that if $g(\cdot)$ is continuously differentiable and the GS property is satisfied then the SGS property holds.

From now on we assume that $g(\cdot)$ satisfies the SGS property.
(c) Show that there is an equilibrium, that is, a $p$ with $g(p)=0$. Illustrate graphically for the case $N=1$.
(d) Give an example for $N=2$ where the equilibrium is not unique.
(e) Suppose that $g(p)=g\left(p^{\prime}\right)=0$. Show that there must be an equilibrium $p^*$ such that $p^{\prime \prime} \geq p$ and $p^{\prime \prime} \geq p^{\prime}$. Similarly, there is an equilibrium $p^{\prime \prime}$ such that $p^{\prime \prime} \leq p$ and $p^{\prime \prime} \leq p^{\prime}$.
(f) Argue (you can assume continuity here) that the equilibrium set satisfies a strong and very special property, namely, that it has a maximal and a minimal equilibrium. That is, there are $p^{\text {max }}$ and $p^{\text {min }}$ such that $g\left(p^{\text {max }}\right)=g\left(p^{\text {min }}\right)=0$ and $p^{\text {min }} \leq p \leq p^{\text {max }}$ whenever $g(p)=0$.
(g) Assume now that $g(\cdot)$ is also differentiable. Suppose that we know that at equilibrium, that is, whenever $g(p)=0$, the matrix $D g(p)$ has a negative dominant diagonal; that is, $D g(p) v \propto 0$ for a $v \gg 0$. Argue (perhaps nonrigorously) that the equilibrium must then be unique.
(h) Suppose that $g(\cdot)$ is the usual excess demand system for the first $N$ goods of an economy with $N+1$ goods in which the last price has been fixed to equal 1 and the overall $(N+1)$-good excess demand system satisfies the gross substitute property. Apply (g) to show that the equilibrium is unique.

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Problem 17

Grandmont (1992)] Suppose that $L=2$ and you have a continuum of consumers. All consumers have the same initial endowments; they are not rational, however. Given a budget set, they choose at random from consumption bundles on the budget line using a uniform distribution among the nonnegative consumptions. Let $z(p)$ be the average excess demand ( $=$ expected value of a single consumer's choice). Show that $z(\cdot)$ can be generated from preference maximization of a Cobb-Douglas utility function (thus the economy admits a positive representative consumer in the sense of Section 4.D).

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