Question

Smith and Jones are stranded on a desert island. Each has in his possession some slices of ham \((H)\) and cheese (C). Smith is a choosy eater and will eat ham and cheese only in the fixed proportions of 2 slices of cheese to 1 slice of ham. His utility function is given by \(U_{s}=\min (H, C / 2)\) Jones is more flexible in his dietary tastes and has a utility function given by \(U_{j}=4 H+3 C\). Total endowments are 100 slices of ham and 200 slices of cheese. a. Draw the Edgeworth box diagram that represents the possibilitics for exchange in this situation. What is the only exchange ratio that can prevail in any equilibrium? b. Suppose Smith initially had \(40 \mathrm{H}\) and \(80 \mathrm{C}\). What would the equilibrium position be? c. Suppose Smith initially had \(60 H\) and \(80 C\). What would the equilibrium position be? d. Suppose Smith (much the stronger of the two) decides not to play by the rules of the game. Then what could the final equilibrium position be?

Short answer

Answer: The equilibrium position would be where Smith has 40 slices of ham and 80 slices of cheese, and Jones has the remaining 60 slices of ham and 120 slices of cheese.

Step-by-step solution

Analyzing Smith and Jones' utility functions

Smith's utility function is given by \(U_s=\min(H, C/2)\). This means that Smith only gains utility from consuming 2 slices of cheese for every 1 slice of ham. On the other hand, Jones' utility function is given by \(U_j=4H+3C\), which means Jones always gains utility from consuming both ham and cheese.

Creating an Edgeworth Box

The total endowment of goods is 100 slices of ham and 200 slices of cheese. In this scenario, the Edgeworth box has dimensions 100 units for the length (ham) and 200 units for the height (cheese).

a. Identifying the Exchange Ratio

Since Smith only gains utility from consuming 2 slices of cheese for every 1 slice of ham, the only exchange ratio that can prevail in any equilibrium is 2 slices of cheese for 1 slice of ham. This exchange ratio can be represented as a line with slope -2 in the Edgeworth box diagram.

b. Equilibrium when Smith has \(40H\) and \(80C\) initially

Smith must consume ham and cheese in a 1:2 ratio for him to gain any utility. So if Smith has 40 slices of ham, he needs exactly 80 slices of cheese to maximize his utility. Given the initial endowment, the equilibrium position would be where Smith has 40 slices of ham and 80 slices of cheese, and Jones has the remaining 60 slices of ham and 120 slices of cheese.

c. Equilibrium when Smith has \(60H\) and \(80C\) initially

In this scenario, Smith can only consume 40 slices of ham alongside 80 slices of cheese. This means that 20 slices of ham will be left unused since Smith cannot consume more due to the fixed ratio of consumption. The equilibrium position would be where Smith has 40 slices of ham and 80 slices of cheese, and Jones has the remaining 60 slices of ham and 120 slices of cheese (with 20 slices of ham being unused by Smith).

d. Final equilibrium if Smith does not play by the rules

Suppose Smith decides to consume more ham and cheese without following the 1:2 ratio. In this case, the exact equilibrium position would depend on the choices Smith makes. It should be noted that Smith would not be maximizing his utility if he chooses not to follow the 1:2 ratio of consuming ham and cheese. The final equilibrium position could essentially be any allocation of ham and cheese between Smith and Jones, as long as Smith is not consuming ham and cheese according to the 1:2 ratio.