Finding the competitive equilibrium in an exchange economy with two perfect complements

External question — Economics Stack Exchange Author: Jovan Jezdic Original post: https://economics.stackexchange.com/questions/57633 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I am currently in a Microeconomics class and have come across the problem described below. I have tried to solve the problem algebraically but only get to the intercept (x1a,x2a)=(8,4). I know that this is not the only solution so I would appreciate any help! Consider an exchange economy with two goods (1 and 2) and two consumers ( A and B). The preferences of the two consumers are: Consumer A is endowed with k1 of good 1 and k2 of good 2. Consumer B owns only 12-k1 of good 1 and 12-k2 of good 2. First, solve the utility maximization problem of each consumer. Then, find all competitive equilibria of this economy for any k1, k2 between 0 and 12. Finally, characterize the contract curve of this economy.

External answer — Economics Stack Exchange Author: Amit Original post: https://economics.stackexchange.com/a/57639 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Given a pure-exchange economy with $u_A(x_A,y_A)=\min(x_A,2y_A)$ , $u_B(x_B,y_B)=\min(2x_B,y_B)$ Endowment of A is $(k_X,k_Y)$ and of B is $(12-k_X,12-k_Y)$ Set of feasible allocations is $\mathcal{F} = \{((x_A,y_A),(x_B,y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=y_A+y_B=12\}$ Here is the Edgeworth box representation of feasible allocations and set of efficient allocations: To determine the competitive equilibrium, we can consider the following cases for the endowments. Case 1 : $k_X\leq 8, k_Y\geq 4, (k_X,k_Y)\neq (8,4)$ In this case, given $(k_X,k_Y)$ , there is a unique competitive equilibrium allocation $((x_A,y_A),(x_B,y_B))=((8,4),(4,8))$ supported by prices $(p_X,p_Y)=(k_Y-4,8-k_X)$ . Case 2 : $(k_X,k_Y)= (8,4)$ In this case, there is a unique competitive equilibrium allocation $((x_A,y_A),(x_B,y_B))=((8,4),(4,8))$ and it can be supported by any pair of prices from the set $\{(p_X,p_Y)\in\mathbb{R}^2_+|p_X+p_Y=1\}$ . Case 3 : $k_X\geq 8, k_Y \leq 4, (k_X,k_Y)\neq (8,4)$ In this case, given $(k_X,k_Y)$ , there are three sets of competitive equilibria: (i) $((x_A,y_A),(x_B,y_B))=((8,4),(4,8))$ and it is supported by prices $(p_X,p_Y)=(4-k_Y,k_X-8)$ . (ii) Allocations in the set $\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|y_A=k_Y, 2k_Y\leq x_A\leq \frac{k_Y+12}{2}\}$ are supported by the prices $(p_X,p_Y)=(0,1)$ (iii) Allocations in the set $\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|x_A=k_X, \frac{k_X}{2}\leq y_A\leq 2k_X-12\}$ are supported by the prices $(p_X,p_Y)=(1,0)$ Case 4 : $k_X> 8, k_Y > 4$ In this case, given $(k_X,k_Y)$ , allocations in the set $\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|x_A=k_X, \frac{k_X}{2}\leq y_A\leq 2k_X-12\}$ are supported by the prices $(p_X,p_Y)=(1,0)$ Case 5 : $k_X< 8, k_Y < 4$ In this case, given $(k_X,k_Y)$ , allocations in the set $\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|y_A=k_Y, 2k_Y\leq x_A\leq \frac{k_Y+12}{2}\}$ are supported by the prices $(p_X,p_Y)=(0,1)$

External answer — Economics Stack Exchange Author: user51592 Original post: https://economics.stackexchange.com/a/59426 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. LoL, this was the exam question I set... Here is the solution: Consumers utility function is quasi-concave and satisfy LNS as $u^{A}(y^{A})>u^{A}(x^{A})$ for all $y^{A}>>x^{A}$ . Thus, the Marshallian demands of consumers will necessarily satisfy the budget constraint at equality. When $p>>0$ , a simple solution exists, namely \begin{gather*} x^{A}(p)=\left( \frac{2m^{A}}{2p_{1}+p_{2}},\frac{m^{A}}{2p_{1}+p_{2} }\right) \text{ \ and \ }x^{B}(p)=\left( \frac{m^{B}}{p_{1}+2p_{2}} ,\frac{2m^{B}}{p_{1}+2p_{2}}\right) \text{.} \end{gather*} The latter is a solution for $A$ , since for any other consumption bundle $y^{A}$ such that $p_{1}y_{1}^{A}+p_{2}y_{2}^{A}=m^{A}$ and $\varepsilon>0$ sufficiently small, \begin{gather*} u^{A}(y_{1}^{A}-p_{2}\varepsilon/p_{1},y_{2}^{A}+\varepsilon)=2(y_{2} ^{A}+\varepsilon)>2y_{2}^{A}=u^{A}(y^{A})\text{ \ when }y_{1}^{A}>2y_{2} ^{A}\text{,}\\ u^{A}(y_{1}^{A}+\varepsilon,y_{2}^{A}-p_{1}\varepsilon/p_{2})=y_{1} ^{A}+\varepsilon>y_{1}^{A}=u^{A}(y^{A})\text{ \ when }y_{1}^{A}<2y_{2} ^{A}\text{.} \end{gather*} The latter is a solution for $B$ , since for any other consumption bundle $y^{b}$ such that $p_{1}y_{1}^{B}+p_{2}y_{2}^{B}=m^{B}$ and $\varepsilon>0$ sufficiently small, \begin{gather*} u^{B}(y_{1}^{B}-p_{2}\varepsilon/p_{1},y_{2}^{B}+\varepsilon)=y_{2} ^{B}+\varepsilon>y_{2}^{B}=u^{B}(y^{B})\text{ \ when }2y_{1}^{B}>y_{2} ^{B}\text{,}\\ u^{B}(y_{1}^{B}+\varepsilon,y_{2}^{B}-p_{1}\varepsilon/p_{2})=2(y_{1} ^{B}+\varepsilon)>2y_{1}^{B}=u^{B}(y^{B})\text{ \ when }2y_{1}^{B}<y_{2} ^{B}\text{.} \end{gather*} When $p_{1}=0$ and $p_{2}>0$ , consumers naturally spend all of their budget on the expensive good and demands become correspondences, since they only care about getting enough of the free good, but do not mind about purchasing more of that. Therefore, \begin{gather*} x^{A}(p)=\left( \left[ \frac{2m^{A}}{p_{2}},\infty\right) ,\frac{m^{A}% }{p_{2}}\right) \text{ \ and \ }x^{B}(p)=\left( \left[ \frac{m^{B}}{2p_{2}% },\infty\right) ,\frac{m^{B}}{p_{2}}\right) \text{.} \end{gather*} Similarly, when $p_{1}>0$ and $p_{2}=0$ , \begin{gather*} x^{A}(p)=\left( \frac{m^{A}}{p_{1}},\left[ \frac{m^{A}}{2p_{1}}% ,\infty\right) \right) \text{ \ and \ }x^{B}(p)=\left( \frac{m^{B}}{p_{1}% },\left[ \frac{2m^{B}}{p_{1}},\infty\right) \right) \text{.}% \end{gather*} Of course if both prices were equal to $0$ , demand would be unbounded for any one of the two goods. To find CEs, begin by focusing on CEs in which both prices are strictly positive. By Walras Law we know than in these CE, $z_{1}(p)=z_{2}(p)=0$ . Further, we can normalize any one price to $1$ . So set $p_{1}=1$ and clear the market for good $1$ \begin{gather*} x_{1}^{A}(p)+x_{1}^{B}(p)=\frac{2\left( k_{1}+p_{2}k_{2}\right) }{2+p_{2} }+\frac{\left( 12-k_{1}+p_{2}(12-k_{2})\right) }{1+2p_{2}}=12\text{.} \end{gather*} The latter yields $p_{2}=\frac{k_{1}-8}{4-k_{2}}$ . In order for $p_{2}>0$ , this requires that either $k_{1}>8$ and $4>k_{2}$ \ or\ $k_{1}<8$ and $4<k_{2} $ . So in all these circumstances this competitive equilibrium exists. Next, look for CE in which $p_{1}=0$ and $p_{2}>0$ . By Walras Law we know than in these CE, $z_{2}(p)=0$ . Further, we can normalize $p_{2}$ to $1$ . So set $p_{2}=1$ and clear the market for good $2$ , \begin{gather*} x_{2}^{A}(p)+x_{2}^{B}(p)=m^{A}+m^{B}=12\text{.} \end{gather*} Then, verify that there is excess supply in the market for the free good. Note that \begin{gather*} x_{1}^{A}(p)+x_{1}^{B}(p)\geq2m^{A}+\frac{m^{B}}{2}=2k_{2}+\frac{12-k_{2}} {2}=6+\frac{3k_{2}}{2}\text{.} \end{gather*} For there to be excess supply then $6+\frac{3k_{2}}{2}\leq12$ ore equivalently $k_{2}\leq4$ . So in all these circumstances this competitive equilibrium exists. Finally, look for CE in which $p_{1}>0$ and $p_{2}=0$ . By Walras Law we know than in these CE, $z_{1}(p)=0$ . Further, we can normalize $p_{1}$ to $1$ . So set $p_{1}=1$ and clear the market for good $1$ , \begin{gather*} x_{1}^{A}(p)+x_{1}^{B}(p)=m^{A}+m^{B}=12\text{.} \end{gather*} Then, verify that there is excess supply in the market for the free good. Note that \begin{gather*} x_{2}^{A}(p)+x_{2}^{B}(p)\geq\frac{m^{A}}{2}+2m^{B}=\frac{k_{1}}{2} +24-2k_{1}=24-\frac{3k_{1}}{2}\text{.} \end{gather*} For there to be excess supply then $24-\frac{3k_{1}}{2}\leq12$ ore equivalently $k_{1}\geq8$ . So in all these circumstances this competitive equilibrium exists. An allocation $x=(x_{1}^{A},x_{2}^{A},x_{1}^{B},x_{2}^{B})\in R_{+}^{4}$ is Pareto efficient if and only if it is feasible and there is no other feasible allocation $\hat{x}$ such that: (a) $U^{j}(\hat{x}^{j})\geq U^{j}(x^{j})$ for any $j\in\left\{ A,B\right\} $ ; (b) $U^{j}(\hat{x}^{j})>U^{j}(x^{j})$ for some $j\in\left\{ A,B\right\} $ . The contract curve identifies the set of Pareto efficient allocations of an economy. \begin{gather*} x_{2}^{A}\in C_{2}^{A}(x_{1}^{A})=\left\{ \begin{array} [c]{lll}% \lbrack0,x_{1}^{A}/2] & \text{if} & x_{1}^{A}\in\lbrack0,6]\\ \lbrack2x_{1}^{A}-12,x_{1}^{A}/2] & \text{if} & x_{1}^{A}\in\lbrack6,8]\\ \lbrack x_{1}^{A}/2,2x_{1}^{A}-12] & \text{if} & x_{1}^{A}\in\lbrack8,12] \end{array} \right. \end{gather*} To show the latter, note that for any point $x^{A}$ on the conjectured contract curve, the intersection between the upper-contour sets of both players has an empty interior. Formally, let $x^{A}$ satisfy $x_{2}^{A}% =C_{2}^{A}(x_{1}^{A})$ , let \begin{align*} U_{+}^{A}(x^{A}) & =\left\{ x\in\mathbb{R}_{+}^{2}\text{ }|\text{ }% \min\{x_{1},2x_{2}\}\geq\min\{x_{1}^{A},2x_{2}^{A}\}\right\} \text{,}\\ U_{+}^{B}(x^{A}) & =\left\{ x\in\mathbb{R}_{+}^{2}\text{ }|\text{ }% \min\{24-2x_{1},12-x_{2}\}\geq\min\{24-2x_{1}^{A},12-x_{2}^{A}\}\right\} \text{,}% \end{align*} and note that $int(U_{+}^{A}(x^{A})\cap U_{+}^{B}(x^{A}))=\emptyset$ . Thus, it is impossible to benefit a player without hurting the other player. For instance for $x_{1}^{A}\in\lbrack0,6]$ and $x_{2}^{A}\in\lbrack0,x_{1}^{A}% /2]$ , these sets become \begin{align*} U_{+}^{A}(x^{A}) & =\left\{ x\in\mathbb{R}_{+}^{2}\text{ }|\text{ }% \min\{x_{1},2x_{2}\}\geq2x_{2}^{A}\right\} \text{,}\\ U_{+}^{B}(x^{A}) & =\left\{ x\in\mathbb{R}_{+}^{2}\text{ }|\text{ }% \min\{24-2x_{1},12-x_{2}\}\geq12-x_{2}^{A}\right\} \text{,}% \end{align*} and we cannot have that $x_{2}>x_{2}^{A}$ and $12-x_{2}>12-x_{2}^{A}$ holding at once. So, level curves are tangent and we cannot benefit a player without hurting the other one. Finally, for any point $x^{A}$ which is not on the conjectured contract curve, the intersection between the upper-contour sets of both players has a non-empty interior. Thus, given that both players have monotonic preferences, it is possible to benefit a player without hurting the other one.