Perfect complement preferences in an exchange economy

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?

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 $((x_J, y_J), (x_D, y_D)) = \displaystyle \left(\left(\frac{8}{3}, \frac{4}{3}\right), \left(\frac{10}{3}, \frac{5}{3}\right)\right)$

• What if the Leontief preferences are min(x,y)? Will the Pareto set only include the line joining the kinks (starting from D's origin)? Or would J's origin (bottom left) be included in the the Pareto set? – PGupta Feb 1 '19 at 18:17

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.