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 \(U_{\mathrm{s}}=m\) in \((H, C / 2)\) 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 possibilities for exchange in this situation. What is the only exchange ratio that can prevail in any equilibrium? b. Suppose Smith initially had \(40 H\) and \(80 C\). 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

In this situation, the fixed proportion Smith needs to consume his food is given by (H, C/2). The general utility function of Jones is given by 4H + 3C. a. The only exchange ratio that can prevail in any equilibrium is 2 units of cheese for 1 unit of ham. b. With an initial allocation of 40 H and 80 C for Smith, the equilibrium position will be where Smith receives 40 H and 80 C, and Jones receives 60 H and 120 C. c. With an initial allocation of 60 H and 80 C for Smith, the equilibrium position will be where Smith receives 40 H and 80 C, and Jones receives 60 H and 120 C. d. If Smith does not follow the fixed proportion rule, the final equilibrium position could be anything that maximizes the utilities of both parties while adhering to the total endowment constraints. It may not be possible to find a unique equilibrium position in this case.

Step-by-step solution

Utility Functions

We know Smith's utility function is given by \(U_s = m\) in \((H, C/2)\). Jones’ utility function is given by \(U_j = 4H + 3C\). Total endowment on the island is 100 slices of ham and 200 slices of cheese.

Edgeworth Box Diagram

Let's plot the consumption choices for Smith (with proportions) and Jones using the endowments as diagram limits. To find the pareto efficient allocation (trade), we need to identify the points where the indifference curves for both the utility functions are tangent.

Exchange Ratio

Since Smith consumes 2 slices of cheese for 1 slice of ham, the only exchange ratio that can prevail will be always in this proportion: 2 units of cheese for 1 unit of ham. #b. Suppose Smith initially had \(40 H\) and \(80 C\). What would the equilibrium position be?#

Allocation

Starting with an initial endowment of 40 H and 80 C for Smith, let's calculate the initial allocation for Jones: He has 60 H and 120 C.

Pareto Efficient Allocation

Since Smith has a fixed proportion of 2 units of cheese for 1 unit of ham, his consumption can only occur at the usual proportion (assuming an ideal demand): 40 ham and 80 cheese. Now, we examine Jones' point of preference. We need to find the point where Jones' utility function is maximized under the given constraint.

Equilibrium Position

Because Smith's consumption is fixed on the proportion of 2 units of cheese for 1 unit of ham, a pareto efficient allocation will be where Smith receives 40 H and 80 C and Jones receives 60 H and 120 C. Smith's utility is at a maximum satisfying his fixed proportion constraint, while Jones' utility reaches its highest level with the remaining resources available. #c. Suppose Smith initially had \(60 H\) and \(80 C\). What would the equilibrium position be?#

Allocation

Starting with an initial endowment of 60 H and 80 C for Smith, let's calculate the initial allocation for Jones: He has 40 H and 120 C.

Pareto Efficient Allocation

We need to redistribute the endowment between Smith and Jones, so that Smith continues to consume in his fixed proportion of 2 units of cheese for 1 unit of ham while maintaining optimum utility level for Jones.

Equilibrium Position

Smith can only consume at a fixed proportion of 2C:1H, so redistributing the initial endowment, we have Smith with 40 H and 80 C and Jones with 60 H and 120 C. Once again, this allocation will satisfy the constraint of Smith's consumption while maximizing the overall benefit for both players. #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?#

Smith’s Decision

If Smith decides not to play by the rules and neglects the fixed proportion of consumption, the equilibrium must then be found by maximizing both Smith's and Jones's utilities without the constraint of fixed proportions.

Unconstrained Optimization

We need to solve an optimization problem, focusing on maximizing both Smith's and Jones' utilities without adhering to the fixed proportions of 2 units of cheese for 1 unit of ham.

Equilibrium Position

It may not be possible to find a unique equilibrium position, as there could potentially be several allocations that can maximize both players' utilities without taking the fixed proportions constraint into account. Depending on the negotiation and power play between Smith and Jones, the final equilibrium position could be anything that maximizes the utilities of both parties, subject to the total endowment constraints.