Question

Bert and Ernie each have preferences over two goods: X-men comics (2) and yo-yos (y). Suppose the total number of X-men comics available is 10, and the total number of yo-yos available is also 10. Suppose Bert's utility function is Uᴮ (Xᵦ, Yᵦ) = Xᵦy³ᵦ and Ernie's is Uᴱ (Xₑ,Yₑ)= √Xₑ + 3√Yₑ. Derive the contract curve for this economy. Sketch the contract curve in an Edgeworth box where Bert is in the lower left hand corner and Ernie is in the upper right hand corner.

Answer 1

To derive the contract curve for Bert and Ernie's preferences over X-men comics and yo-yos, we need to find the points where their marginal rate of substitution of one good for the other is equal. By equating their respective MRS expressions, we can solve for Xₑ and Xᵦ to find the contract curve.

Step-by-step explanation:

The contract curve in an Edgeworth box represents the set of allocations that are mutually beneficial for both Bert and Ernie. To derive the contract curve, we need to find the points where each individual's marginal rate of substitution (MRS) of one good for the other is equal. For Bert, his MRS of X-men comics for yo-yos is given by the derivative of his utility function with respect to yo-yos, dUᴮ/dYᵦ = 3Xᵦy²ᵦ. For Ernie, his MRS of X-men comics for yo-yos is given by the derivative of his utility function with respect to X-men comics, dUᴱ/dXₑ = 1/(2√Xₑ). Equating the two MRS expressions and solving for Xₑ and Xᵦ, we can find the contract curve.

Let's start by finding Bert's MRS:

dUᴮ/dYᵦ = 3Xᵦy²ᵦ

Now let's find Ernie's MRS:

dUᴱ/dXₑ = 1/(2√Xₑ)

By equating the MRS expressions and solving for Xₑ and Xᵦ, we can find the contract curve.