Answer to Class Example re: EDGEWORTH BOX

Given the following, find the equilibrium price of good X (= PX) and the optimal quantities of X and Y for both person A and person B:

A’s utility function: XY + 12X + 3Y

A’s original endowment: X=8 & Y=30

B’s utility function: XY + 8X + 9Y

B’s original endowment: X=10 & Y=10

Assume the Price of good Y = $1

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We do this by dealing with each person in turn; for each individual, first calculate a formula for his/her demand for good X as a function of PX.

First for person A:

We know person A must consume along his/her “budget line” – that is, for a given PX, the dollar value of A’s original endowment is 8•PX + 30•1, and this must be equal to the value of the final (post-trade) endowment of X•PX + Y•1. Of all the (X,Y) combinations that lie on A’s budget line, the one that would give maximum utility is the one where the marginal rate of substitution is equal to the price ratio (= PX/1). Given A’s utility function, the MRS is ∂U/∂X divided by ∂U/∂Y, or (Y + 12)/(X + 3), and this will equal PX/1.

Manipulate this equation to isolate Y:

(Y + 12)/(X + 3) = (PX/1); (Y + 12) = (PX/1)(X + 3) = X•PX+ 3•PX; Y = X•PX + 3•PX – 12.

Next, replace Y in the budget line equation with this just-calculated expression:

8•PX + 30•1 = X•PX + (X•PX + 3•PX – 12)(1).

Then collect like terms, with the ultimate goal of getting X by itself on the left hand side of the equal sign: 5PX + 42 = 2XPX; 2X = 5 + 42/PX; X = 2.5 + 21/PX. This is person A’s demand for good X for any given PX.

Now do the same thing for person B:

B’s MRS = (Y + 8)/(X + 9), and the value of B’s initial endowment = 10PX + 10.

MRS = Price ratio  (Y + 8)/(X + 9) = PX; isolating Y by cross-multiplication and subtraction gives Y = X•PX + 9•PX – 8.

Substitute this expression into the budget line equation:

10PX + 10 = XPX + XPX + 9PX – 8.

Manipulate to isolate X:

2XPX = PX + 18; X = 0.5 + 9/PX.

Final steps - We know equilibrium is defined by demand equal to supply. Therefore, form the equation for (total demand = total supply) to solve for PX.

Total Demand = the sum of the individual demands of A & B at a given price:

(2.5 + 21/PX) + (0.5 + 9/PX) = 3 + 21/PX + 9/PX. (Note this aggregate demand is inversely related to PX).

Total Supply is fixed at the sum of the initial endowments:

8 + 10 = 18.

Therefore, the PX that solves the equation 3 + 21/PX + 9/PX = 18 is the equilibrium price of X:

3 + 21/PX + 9/PX = 18  21/PX + 9/PX = 15  21 + 9 = 15PX (multiplying all terms by PX);

Since 30 = 15PX, 2 = PX.

Individual demands:

A wants 2.5 + 21/2 = 13 units of X, and B wants 0.5 + 9/2 = 5 units of X.

(What about good Y?) From the respective budget lines:

Person A: 8(2) + 30 = 13(2) + Y(1); YA = 46 – 26 = 20;

Person B: 10(2) + 10 = 5(2) + Y(1); YB = 30 – 10 = 20.

Last but not least, do the two marginal rates of substitution equal the price ratio?

Person A: (Y + 12)/(X + 3) = (20 + 12)/(13 + 3) = 32/16 = 2;

Person B: (Y + 8)/(X + 9) = (20 + 8)/(5 + 9) = 28/14 = 2. YES!