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 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 \(U_{S}=\min (H, C / 2)\) 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 \(40 \mathrm{H}\) and \(80 \mathrm{C}\). What would the equilibrium position be? c. Suppose Smith initially had \(60 \mathrm{H}\) and \(80 \mathrm{C}\). 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

a. Exchange ratio is 2C:1H. b. Equilibrium is Smith's initial endowment. c. Equilibrium is Smith (40H, 80C), Jones (60H, 120C). d. Smith can keep an optimal ratio, leaving Jones with minimal satisfaction.

Step-by-step solution

Understanding Utility Functions

Smith's utility function is \(U_{S} = \min(H, C/2)\), meaning he only gets utility when the ratio of ham to cheese is 1:2. Jones's utility function is \(U_{J} = 4H + 3C\). Total available resources are 100 slices of ham and 200 slices of cheese.

Drawing Edgeworth Box

The Edgeworth box represents all possible distributions of ham and cheese between Smith and Jones. With Smith starting at the origin (0,0) for his allocations, the total resources (100H, 200C) represent the corner opposite to Smith's origin. Smith can only consume in a 1:2 cheese to ham ratio, creating an L-shaped utility space, while Jones has linear indifference curves.

Finding Exchange Ratios in Equilibrium

At equilibrium, Smith's marginal rate of substitution (MRS), which is zero because he only consumes at his corner point (1H:2C), must equal the relative price of ham to cheese to be willing to trade. Since Jones values goods differently, for equilibrium, the trade ratio must align with Smith's preferences. Thus, the equilibrium ratio of consumption is 2 slices of cheese per slice of ham.

Initial Endowments at 40H, 80C for Smith

Smith starts with exactly his preferred proportion for all his cheese and ham (40H, 80C), aligning perfectly with his utility function \(U_{S} = \min(H, C/2) = 40\). There would be no incentive for Smith and Jones to trade as Smith is already fully satisfied. Thus, equilibrium stays at Smith's initial endowment.

Initial Endowments at 60H, 80C for Smith

Smith starts with extra ham (60H) compared to his ideal of 1:2, since \(80C/2 = 40H\) for perfect utility balance. He will trade 20H with Jones. In exchange, he needs 40C to make it \(20H\) to \(40C\). End result: Smith ends with (40H, 80C) and equilibrium is reached as previously calculated.

Smith Trading Based on Strength

If Smith doesn't adhere to fair trading and exploits his strength, he could possibly take more cheese than required by his utility function from initial exchanges, leaving Jones with minimum utility satisfaction. The arbitrary power dynamics mean Smith could enforce his ideal utility conditions while restricting Jones's satisfaction below fair exchange levels.

Key concepts

Edgeworth Box

An Edgeworth Box is a visual tool that helps us understand trade-offs and exchanges between two parties. It's like a game board where each player has their own strategy to maximize their utility. In our scenario with Smith and Jones, the Edgeworth Box represents the total resources of 100 slices of ham and 200 slices of cheese available between them.
Think of the Edgeworth Box as a confined space that shows every possible way Smith and Jones can divide ham and cheese. This box is drawn such that Smith's starting point of consumption is at one corner (usually the lower left), and the total available resources sit at the opposite corner. As they trade, they move around this box.

• Smith's initial allocation: For example, if he starts with 40 slices of ham and 80 slices of cheese, his consumption point starts from here.
• Jones's utility: Given Jones's desire for lots of ham and cheese based on his utility function, he aims to maximize his total utility within this box.

The box helps illustrate the constraints and possibilities in their exchanges. In equilibrium, they should ideally meet at a point where neither can be better off without making the other worse off.

Marginal Rate of Substitution

The Marginal Rate of Substitution (MRS) is a crucial concept to understand how much of one good an individual is willing to give up for an additional unit of another good while maintaining the same level of utility. For Smith, who consumes ham and cheese in strict proportions through his utility function \(U_{S} = \min(H, C/2)\), the MRS is different because he needs exactly two slices of cheese for each slice of ham to be satisfied.
In mathematical terms, his MRS is 0 whenever he is consuming at his optimal ratio because he cannot substitute one good for another without losing satisfaction. Contrast this with Jones, who has a more flexible approach, so his MRS may vary as he balances his desire for both ham and cheese.

• Smith's MRS: Since he won't exchange ham for cheese unless it's in his fixed 1 ham:2 cheese ratio, he doesn't substitute them freely, essentially having an indifference curve that appears 'L' shaped.
• Jones's flexibility: Jones's MRS can be calculated based on his linear utility function \(U_{J} = 4H + 3C\). Jones may substitute them more flexibly according to price and need.

In equilibrium, Smith's stringent MRS leads to a specific set of conditions where any trade benefiting him must meet his needs exactly.

Equilibrium Exchange

Equilibrium Exchange is achieved when neither Smith nor Jones can improve their utility without worsening the other's utility. This is where both players are maximizing their gains given their initial resources and preferences.
For any exchange to reach equilibrium, it requires the trade ratios to reflect both parties’ utility functions. In this case, Smith's strict ratio of 1 slice of ham to 2 slices of cheese dictates the prevailing exchange terms. If Smith starts perfectly balanced, as in the initial setup with 40H and 80C, both achieve equilibrium without needing exchange.
However, if initial endowments are unbalanced, trades can occur. Consider when Smith starts with 60H and 80C. He has more ham than necessary for his cheese, ending up wanting to trade 20H for 40C to meet his strict consumption preferences. Once he hits this point, both Smith and Jones will not wish to change further as any alteration means losing utility for one of them.
The equilibrium plays out within the Edgeworth Box visually as a central point where both are satisfied, highlighting that precise ratios and adherence to utility preferences guide these exchanges.