Smith and Jones are stranded on a desert island. Each has in his possession some slices of ham (//) and cheese (C). Smith is a very 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 (\mathrm{H}, \mathrm{C} / 2)\\] . Jones is more flexible in his dietary tastes and has a utility function given by \(U j=\) \(4 / /+3 G\) 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 \(40 / /\) and \(80 \mathrm{C}\). What would the equilibrium position be? c. Suppose Smith initially had \(60 /\) 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?

To establish the Edgeworth box diagram, we first find the total resources available. On the island, ham (H) and cheese (C) are 100 and 200 slices respectively. Smith consumes ham and cheese in the fixed proportions 1:2, i.e., for each slice of ham, he needs two slices of cheese. This means that any equilibrium position must satisfy: $$ \frac{C_S}{H_S} = \frac{C_J}{H_J} = 2 $$ Where \(C_S\) and \(H_S\) are the quantities of cheese and ham consumed by Smith, and \(C_J\) and \(H_J\) are the quantities consumed by Jones.

To draw the Edgeworth box, we need to plot the indifference curves of Smith and Jones given their utility functions: $$ U_S = \min(H_S, \frac{C_S}{2}) $$ $$ U_J = 4H_J + 3C_J $$ The total sum of H and C must equal the total endowments: $$ H_S + H_J = 100 $$ $$ C_S + C_J = 200 $$ To find the contract curve, or set of all possible equilibrium allocations, we need to equate the marginal rates of substitution, i.e., the rate at which each individual is willing to give up cheese for ham: $$ \frac{dU_S}{dH_S} = \frac{dU_J}{dH_J} $$ Inserting the utility functions for Smith and Jones, we get that the equilibrium exchange ratio must satisfy: $$ \frac{C_S}{2H_S} = \frac{3}{4} $$ b. Suppose Smith initially had 40H and 80C. What would the equilibrium position be?

Given the initial situation of Smith with 40 slices of ham and 80 slices of cheese, we look for the equilibrium by solving for \(C_S\) and \(H_S\) while respecting the exchange ratio and the total resource constraints: 1. Set up the equilibrium exchange ratio equation: $$ \frac{C_S}{2H_S} = \frac{3}{4} $$ 2. Utilize the total resource constraint equations: $$ H_S + H_J = 100 $$ and $$ C_S + C_J = 200 $$ Since Smith initially has 40H and 80C, the remaining resources are 60H (for Jones) and 120C (for Jones). Now, we can solve for the equilibrium position: 1. From the total resource constraints, $$ H_J = 100 - H_S $$ $$ C_J = 200 - C_S $$ 2. Plug these into the equilibrium exchange ratio equation: $$ \frac{C_S}{2H_S} = \frac{200 - C_S}{4(100 - H_S)} = \frac{3}{4} $$ 3. Solve for \(H_S\) and \(C_S\): $$ H_S \approx 33.33 $$ $$ C_S \approx 66.67 $$ Therefore, in the equilibrium position, Smith has about 33.33 slices of ham and 66.67 slices of cheese. c. Suppose Smith initially had 60H and 80C. What would the equilibrium position be? For the initial endowment, Hamilton has 60H and 80C; we can repeat the steps as above to find the equilibrium position: 1. Set up the equilibrium exchange ratio equation: $$ \frac{C_S}{2H_S} = \frac{3}{4} $$ 2. Utilize the total resource constraint equations: $$ H_S + H_J = 100 $$ and $$ C_S + C_J = 200 $$ Since Smith initially has 60H and 80C, the remaining resources are 40H (for Jones) and 120C (for Jones). Now, we can solve for the equilibrium position: 1. From the total resource constraints, $$ H_J = 100 - H_S $$ $$ C_J = 200 - C_S $$ 2. Plug these into the equilibrium exchange ratio equation: $$ \frac{C_S}{2H_S} = \frac{200 - C_S}{4(100 - H_S)} = \frac{3}{4} $$ 3. Solve for \(H_S\) and \(C_S\): $$ H_S \approx 42.86 $$ $$ C_S \approx 85.71 $$ Therefore, in the equilibrium position, Smith has about 42.86 slices of ham and 85.71 slices of cheese. 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? In this case, since Smith is stronger, he can choose to take as much of the resources as he wants to maximize his utility. However, since he has specific preferences for the ratio of ham and cheese, he will still consume them in a 1:2 proportion. If Smith decides to consume all the ham, he would need 200 slices of cheese as well to maintain his consumption ratio. In this scenario, there are no remaining resources for Jones, and the equilibrium position cannot be reached.