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# Econ经济作业代写Economics代考|Exchange economy: positive theory

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## Exchange Edgeworth box: prices and equilibria|Econ经济作业代写Economics代考

It is possible to add price information into Edgeworth boxes. If household $A$ buys a bundle $\left(x_{1}^{A}, x_{2}^{A}\right)$ with the same worth as his endowment, we have
$$p_{1} x_{1}^{A}+p_{2} x_{2}^{A}=p_{1} \omega_{1}^{A}+p_{2} \omega_{2}^{A}$$
Starting from an endowment point, positive prices $p_{1}$ and $p_{2}$ lead to negatively sloped budget lines for both individuals. In figure XIX.1, two price lines with prices $p_{1}^{l}<p_{1}^{h}$ are depicted. The indifference curves indicate which bundles the households prefer.

EXERCISE XIX.1. Why do the two price lines in figure XIX.1 cross at the endowment point w?

Of course, we would like to know whether these prices are compatible in the sense of allowing both agents to demand the preferred bundle. If that is the case, the prices and the bundles at these prices constitute a Walras equilibrium.

## Definition of an exchange economy|ECON经济作业代写ECONOMICS代考

DEFINITION XIX.1 (EXCHANGE ECONOMY). An exchange economy is a tuple
$$\mathcal{E}=\left(N, G,\left(\omega^{i}\right){i \in N},\left(\precsim^{i}\right){i \in N}\right)$$
consisting of

• the set of agents $N={1,2, \ldots, n}$,
• the finite set of goods $G={1, \ldots, \ell}$,
and for every agent $i \in N$
• an endowment $\omega^{i}=\left(\omega_{1}^{i}, \ldots, \omega_{\ell}^{i}\right) \in \mathbb{R}_{+}^{\ell}$, and
• a preference relation $\precsim$.
Thus, every agent has property rights on endowments. The total endowment of an exchange economy is given by $\omega=\sum_{i \in N} \omega^{i}$. A household’s consumption possibilities are described by the budget. We refer the reader to chapter VI.

## Excess Demand and Market Clearance|Econ经济作业代写Economics代考

Let us begin with the first case and consider player 1. The truthful announcement of his type leads to the provision of the public good and $z_{1}=0$. His utility is
$$u_{1}(1,0, \ldots)=t_{1}-\frac{C}{n}$$
Player 1 has no incentive to overstate his willingness to pay (that will not change anything). Indeed, a utility change occurs only if player 1 understates his willingness to pay so that the public good is not provided. Then, player 1 has to pay $\sum_{j \in N \backslash{1}} m_{j}-\frac{n-1}{n} C$ (third case). Player 1 is harmed by this understatement:
\begin{aligned} u_{1}(1,0, \ldots) &=t_{1}-\frac{C}{n} \ & \geq\left(C-\sum_{j \in N \backslash{1}} m_{j}\right)-\frac{C}{n}(\text { case } 1) \ &=-\left(\sum_{j \in N \backslash{1}} m_{j}-\frac{n-1}{n} C\right) \ &=u_{1}\left(0, \sum_{j \in N \backslash{1}} m_{j}-\frac{n-1}{n} C, \ldots\right) \end{aligned}

## Existence of the Walras equilibrium|ECON经济作业代写ECONOMICS代考

EXERCISE XIX.3. Consider a market where the excess demand of three individuals 1,2 , and 3 is given by
$$z_{1}(p)=\frac{8}{p}-4, z_{2}(p)=\frac{4}{p}-2, z_{3}(p)=\frac{12}{p}-2 .$$
Find the market-clearing price. Is individual 3 a buyer or a seller?
EXERCISE XIX.4. Abba (A) and Bertha (B) consider buying two goods 1 and 2, and face the price $p$ for good 1 in terms of good 2. Think of good 2 as the numéraire good with price 1 . Abba’s and Bertha’s utility functions $u_{A}$ and $u_{B}$, respectively, are given by $u_{A}\left(x_{1}^{A}, x_{2}^{A}\right)=\sqrt{x_{1}^{A}}+x_{2}^{A}$ and $u_{B}\left(x_{1}^{B}, x_{2}^{B}\right)=$ $\sqrt{x_{1}^{B}}+x_{2}^{B}$. Endowments are $\omega^{A}=(18,0)$ and $\omega^{B}=(0,10)$. Find the bundles demanded by these two agents. Then find the price $p$ that fulfills $\omega_{1}^{A}+\omega_{1}^{B}=x_{1}^{A}+x_{1}^{B}$ and $\omega_{2}^{A}+\omega_{2}^{B}=x_{2}^{A}+x_{2}^{B} .$

In the above exercise, what if only market 1 is cleared? The following lemma shows that local nonsatiation excludes this possibility.

## Existence of the Nash equilibrium|ECON经济作业代写ECONOMICS代考

Example: The Cobb-Douglas exchange economy with two agents. We remember from chapter VI that income $m$ and Cobb-Douglas utility function
$$u\left(x_{1}, x_{2}\right)=x_{1}^{a} x_{2}^{1-a}$$
implies the household optimum
\begin{aligned} &x_{1}=a \frac{m}{p_{1}} \ &x_{2}=(1-a) \frac{m}{p_{2}} . \end{aligned}
Consider, now, individual 1 with Cobb-Douglas utility function $u^{1}$ and parameters $a_{1}($ for good 1$)$ and $1-a_{1}$ (for good 2). The initial endowment of individual 1 equals $\omega^{1}=(1,0)$. Individual 2 possesses a Cobb-Douglas utility function $u^{2}$ with parameters $a_{2}$ (for good 1$)$ and $1-a_{2}$ (for good 2). His initial endowment is $\omega^{2}=(0,1)$. Parameters $a_{1}$ and $a_{2}$ obey the following conditions: $0<a_{1}<1$ and $0<a_{2}<1$. Both goods are desired and the preferences obey local nonsatiation and weak monotonicity. According to lemma XIX.4, the market is in equilibrium only if it is cleared. Substituting the value of the endowment for income, we get individual 1’s demand for $\operatorname{good} 1$

## Existence of the Walras equilibrium.|ECON经济作业代写ECONOMICS代考

2.5.3. Proof of the existence theorem XIX.1. In order to apply Brouwer’s fixed-point theorem to theorem XIX.1, we first construct a convex and compact set. The prices of the $\ell$ goods are normed such that the sum of the nonnegative (!, we have strict

monotonicity) prices equals 1. Just divide all prices by the sum of the prices. We can restrict our search for equilibrium prices to the $\ell-1$ – dimensional unit simplex:
$$S^{\ell-1}=\left{p \in \mathbb{R}{+}^{\ell}: \sum{g=1}^{\ell} p_{g}=1\right}$$
$S^{\ell-1}$ is nonempty, compact (closed and bounded as a subset of $\mathbb{R}^{\ell}$ ), and convex.
EXERCISE XIX.7. Draw $S^{1}=S^{2-1}$
The idea of the proof is as follows: First, we define a continuous function $f: S^{\ell-1} \rightarrow S^{\ell-1}$. Brouwer’s theorem

says that there is at least one fixed point of this function. Second, we show that such a fixed point fulfills the conditions of the Walras equilibrium.
The continuous function is defined by
$$f=\left(\begin{array}{c} f_{1} \ f_{2} \ \cdot \ \cdot \ \cdot \ f_{\ell} \end{array}\right): S^{\ell-1} \rightarrow S^{\ell-1}$$
and
$$f_{g}(p)=\frac{p_{g}+\max \left(0, z_{g}(p)\right)}{1+\sum_{g^{\prime}=1}^{\ell} \max \left(0, z_{g^{\prime}}(p)\right)}, g=1, \ldots, \ell$$

## EXCHANGE EDGEWORTH BOX: PRICES AND EQUILIBRIA|ECON经济作业代写ECONOMICS代考

p1X1一种+p2X2一种=p1ω1一种+p2ω2一种

## DEFINITION OF AN EXCHANGE ECONOMY|ECON经济作业代写ECONOMICS代考

$$\mathcal{E}=\left(N, G,\left(\omega^{i}\right) {i \in N},\left(\precsim^{i}\对) {i \in N}\right)$$

• 代理集ñ=1,2,…,n,
• 有限的商品集G=1,…,ℓ，
并且对于每个代理一世∈ñ
• 禀赋ω一世=(ω1一世,…,ωℓ一世)∈R+ℓ， 和
• 偏好关系≾.
因此，每个代理人对捐赠都有财产权。交换经济的总禀赋由下式给出ω=∑一世∈ñω一世. 家庭的消费可能性由预算描述。我们请读者参考第六章。

## EXISTENCE OF THE NASH EQUILIBRIUM|ECON经济作业代写ECONOMICS代考

X1=一种米p1 X2=(1−一种)米p2.

## EXISTENCE OF THE WALRAS EQUILIBRIUM.|ECON经济作业代写ECONOMICS代考

2.5.3. 存在定理 XIX.1 的证明。为了将 Brouwer 不动点定理应用到定理 XIX.1，我们首先构造一个凸紧集。的价格ℓ商品经过规范，使得非负（！，我们有严格的单调性）价格之和等于 1。只需将所有价格除以价格之和即可。我们可以将我们对均衡价格的搜索限制在ℓ−1– 维数单位单纯形：

S^{\ell-1}=\left{p \in \mathbb{R} {+}^{\ell}：\sum {g=1}^{\ell} p_{ g}=1\右}
$小号ℓ−1$一世sn○n和米p吨和,C○米p一种C吨(C一世○s和d一种ndb○你nd和d一种s一种s你bs和吨○F$Rℓ$),一种ndC○nv和X.和X和RC一世小号和X一世X.7.Dr一种在$小号1=小号2−1$吨H和一世d和一种○F吨H和pr○○F一世s一种sF○一世一世○在s:F一世rs吨,在和d和F一世n和一种C○n吨一世n你○你sF你nC吨一世○n$F:小号ℓ−1→小号ℓ−1$.乙r○你在和r′s吨H和○r和米s一种和s吨H一种吨吨H和r和一世s一种吨一世和一种s吨○n和F一世X和dp○一世n吨○F吨H一世sF你nC吨一世○n.小号和C○nd,在和sH○在吨H一种吨s你CH一种F一世X和dp○一世n吨F你一世F一世一世一世s吨H和C○nd一世吨一世○ns○F吨H和在一种一世r一种s和q你一世一世一世br一世你米.吨H和C○n吨一世n你○你sF你nC吨一世○n一世sd和F一世n和db和
f=\左（F1 F2 ⋅ ⋅ ⋅ Fℓ\right): S^{\ell-1} \rightarrow S^{\ell-1}

f_{g}(p)=\frac{p_{g}+\max \left(0, z_{g}(p)\right)}{1+\sum_{g^{\prime}=1}^ {\ell} \max \left(0, z_{g^{\prime}}(p)\right)}, g=1, \ldots, \ell

matlab代写请认准UprivateTA™.