Answer to Question #247119 in Microeconomics for Chilu

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 US = min(H, C/2). Jones is more flexible in his dietary tastes and has a utility function given by UJ = 4H+ 3C. 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 40H and 80C. What would the equilibrium position be?


c. Suppose Smith initially had 60H and 80C. 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?

Answer 1

(a)

In an economy, there are only two goods – Ham and Cheese, and only two persons – Person S and Person J. Person S consumes the two goods in fixed proportion: 2 slices of cheese to 1 slice of ham. The utility function of Person S is given by:

$U_s=min(H_s,\frac{C_s}{2})$

The utility function of Person J is given by:

$U_j=4H_j+3C_j$

Furthermore, there is a total of 100 slices of ham and 200 slices of cheese in the economy; that is,

$H_s+H_j=100 C_s+C_j=200$

An Edgeworth box representing the possibilities of exchange in required to be drawn.

The Edgeworth box is shown in Figure 1.

The L-shaped blue curves are indifference curves of Person S, which show that Person S consumes 2 slices of cheese with every slice of ham.

The orange straight lines represent the indifferences curves of Person J. These are the indifference curves of Person J because the slope of these indifference curves is,

$- \frac{4}{3}=_ \frac{Vertical \; intercept \; of \; any \; orange \; line}{Horizontal intercept \; of \; that \; orange \; line}; for \;example, - \frac{66.67}{50}$

which is the same as the slope that can be calculated from Person J’s utility function, as shown below by differentiating the utility function of Person J with respect to:

$H_j \\ 0=4(1)+3\frac{dC_j}{dH_j} \\ 3 \frac{dC_j}{dH-J}=-4 \\ \frac{dC_j}{dH_j}=-\frac{4}{3}$

The black dots where the blue curves touch the orange curves are pareto efficient allocations. These are pareto efficient allocations because at any of those allocations, no person can be made better off without making the other worse off.

All the pareto efficient allocations lie on a straight line, which is the contract curve. So, the contract curve is a straight line that intersects the corners of the L-shaped curves. The contract curve is therefore represented by the following equation:

$H_s=\frac{C_s}{2} \\ C_s=2H_s$

Person J has a linear utility function, which means that the two goods are perfect substitutes for Person J. In the case of perfect substitutes, there is usually a corner solution, but the person may choose a bundle that does not lie on either of the axes only if the price ratio is the same as absolute value of the slope of his indifference curves.

Therefore, Person J will choose an allocation inside the Edgeworth box optimal only and only if the price ratio ($\frac{P_H}{P_s}$ ) equals $\frac{4}{3}$ .

Hence, the equilibrium price ratio is $\frac{4}{3}$ .

(b)

The initial allocation is given by:

H_s=40 and C_s=80

Substitute 40 for H_s in the equation of the contract curve, if that gives C_s as 80, then, this means the initial allocation is Pareto optimal.

$C_s=2H_s \\ C_s=2(40) \\ C_s=80$

Hence, the initial allocation is Pareto optimal and the equilibrium position will be the same as the initial allocation:

$H_s=40, \; C_s=80$

(c)

The initial allocation is given by:

$H_s=60, \; C_s=80$

Substitute 60 for H_s in the equation of the contract curve, if that gives C_s as 80, then, this means the initial allocation is Pareto optimal.

$C_s=2H_s \\ C_s=2(60) \\ C_s= 120$

Hence, the initial allocation is not Pareto optimal and the equilibrium position and the initial allocation will be different.

Substitute $H_s= 60, \; C_s=80$ in Person S’s utility function.

$U_s=min(H_s, \frac{C_s}{2})=min(60, \frac{80}{2})=min(60,40)=40$

Hence Person S’s initial utility function is given by:

$40=min(H_s, \frac{C_s}{2})$

Substitute the contract curve equation C_s=2H_s in the above equation:

$40=min(H_s, \frac{C_s}{2}) \\ 40=min(H_s, \frac{2H_s}{2}) \\ min(H_s,H_s)=40 \\ H_s=40$

Substitute 40 for H_s in the contract curve equation.

$C_s=2H_s \\ C_s=2(40) \\ C_s= 80$

Therefore, in the range of possible equilibrium positions, one end is given by the allocation:$H_s=40, \; C_s=80$ . This allocation lies on the contract curve such that Person S’s utility is unchanged.

The next step is to find an allocation that lies on the contract curve, such that the Person J’s utility is unchanged.

Since H_s + H_j=100 fnd C_s+C_j=200, the initial allocation H_s=40, C_s=120 also means H_j=40, C_j=120

Substitute H_j=40, C_j=120 in Person J’s utility function.

$U_j= 4H_j +3C_j \\ U_j=4(40) +3(120) \\ U_j=160 + 360 \\ U_j=520$

Hence Person J’s initial utility function is given by:

520= 4H_j +3C_j

Since H_s +H_j=100 and C_s +C-J=200, write the above equation as follows:

$520=4(100-H_s)+3(200-C_s) \\ 520=400 — 4H_s + 600 — 3C_s \\ 4H_s +3C_s=480$

Now substitute the contract curve equationC_s=2H_s in the above equation:

$4H_s +3(2H_s)=480 \\ 10H_s=480 \\ H_s=48$

Substitute 48 for H_s in the contract curve equation.

$C_s=2H_s \\ C_s=2(40) \\ C_s= 96$

Therefore, in the range of possible equilibrium positions, the other end is given by the allocation:H_s=48, C_s=96

Conclusion for part (c):

The equilibrium position will lie on the contract curve between H_s=48, C_s=80 and H_s=48, C_s=96.

(d)

The two people are stranded on an island and there is no government to make sure that they both abide by the rules of exchange. The rules of exchange will prevail if both are equally strong. But if Person S is stronger of the two and decides not to abide by the rules of exchange, then Person S will take all the ham and cheese. Person S’s consumption will be 100 slices of ham and 200 slices of cheese, and Person J will have nothing.