Perfect complement preferences in an exchange economy

Question

So I have an exam in a bit, I understand that to find the optimal choice you have to equate tangent of the two indifference curves. However, if the other indifference curve is a perfect complement, what is MRS? I calculated the MRS as infinity/infinity.

Take a look at the question below:

Suppose Jane has an endowment of 2 units of x and 2 units of y, and has preferences given by utility function $u_J(x_J , y_J) = x_J^{2/3} \cdot y_J^{1/3}$ . Suppose Derek has an endowment of 4 units of x and 1 unit of y, and has preferences given by the utility function $u_D(x_D, y_D) = \min(x_D, 2 y_D)$.

1. On an Edgeworth box diagram, indicate the set of Pareto efficient allocations. Explain how it is determined.
2. Compute the competitive equilibrium (prices and quantities) for the exchange economy

Was I right about the MRS being infinity over infinity? If yes, then how would I compute the Pareto Efficient outcome when I cannot equate the MRSs? I was thinking that the solution will be on one of the vertexes of Derek's indifference curve but I'm not sure (because kinks are not diff).

Any ideas?

Answer 1

Set of Pareto Efficient Allocations consists of feasible allocations $((x_J, y_J), (x_D, y_D))$ satisfying $y_J=\displaystyle\frac{x_J}{2}$.

Competitive Equilibrium is the price $(p_x, p_y=1)$ satisfying the following conditions:

• Budget Requirement: $p_xx_J+ y_J = 2p_x + 2$ and $p_xx_D+ y_D = 4p_x + 1$
• Equilibrium Conditions: $\displaystyle\frac{2y_J}{x_J} = p_x$ and $y_D=\displaystyle\frac{x_D}{2}$

Solving the system of equations we get equilibrium price vector as: $(p_x, p_y) = (1,1)$ and the equilibrium allocation is $((x_J, y_J), (x_D, y_D)) = \displaystyle \left(\left(\frac{8}{3}, \frac{4}{3}\right), \left(\frac{10}{3}, \frac{5}{3}\right)\right)$

Answer 2

Perfect complements is equivalent to Leontief utility:

$U(x,y) = min(x/a_x, y/a_y)$

The MRS is defined as:

$MRS_x,y = MU_x / MU_y$

Since this utility function is not differentiable the concept of marginal substitution is not well defined for Derek. However, we don't need marginal arguments for Derek to solve the problem. A function doesn't have to be differentiable to have a unique maximum. Derek has Leontief preferences and so wants to consume $x_d = 2 y_d$ for all prices. He exhausts his wealth (his endowment): $4 p_x + 1 p_y = w = x_d p_x + y_d p_y$

We can use this to solve for $y_d = (4 p_x + 1 p_y) / (p_y + 2 p_x)$ (eqn 1) and $x_d = 2 (4 p_x + 1 p_y) / (p_y + 2 p_x)$ (eqn 2).

Jane has Cobb-Douglas preferences and so wants to spend constant budget shares on x: $x_j p_x / (2 p_x + 2 p_y) = 2/3$ and $y_j p_y / (2 p_x + 2p_y) = 1/3$. We can also solve these for $x_j = (2/3) / (p_x / (2 p_x + 2 p_y) )$ (eqn 3) and $y_j = (1/3) / (p_y / (2 p_x + 2p_y))$ (eqn 4).

However, we also know that in equilibrium $y_j + y_d = 3$ (eqn 5) and $x_j + x_d = 6$ (eqn 6) This is 6 equations and 6 unknowns ($p_x, p_y, x_d, y_d, x_j, y_j$) and can be solved for the equilibrium prices and allocation.