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

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.

• "I have tried to solve the problem algebraically but only get to the intercept" Can you please edit the question so that it includes your calculations? It is not trivial to see where you get stuck. Commented Feb 12 at 6:27

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 $$(p_X,p_Y)=(0,1)$$

• Thanks for this answer, this clears things up a bit, but I still have two more questions: In Case 4: Can't the competitive equilibrium also be ya=k2, xa=6+k2/2 supported by prices (px,py)=(0,1). At this point, the utility of both consumers would be maximized and the outcome would be Pareto efficient. Furthermore, in Case 1 why can't there be 3 sets of equilibria as in Case 3, I'm not sure I understand what prevents the equilibrium from being for example : (xA,yA)=(kx, 0<yA<kx/2) supported by prices (px,py)=(1,0) Commented Feb 12 at 17:45