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Chapter 13: Problem 5

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

Expert verified
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

01

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.
02

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).
03

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.
04

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.
05

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).
06

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.

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Most popular questions from this chapter

In Example 13.3 we showed how a Pareto efficiency exchange equilibrium can be described as the solution to a constrained maximum problem, In this problem we provide a similar illustration for an economy involving production. Suppose that there is only one person in a two-good economy and that his or her utility function is given by \(U(x, y)\). Suppose also that this economy's production possibility frontier can be written in implicit form as \(T(x, y)=0\) a. What is the constrained optimization problem that this economy will seek to solve if it wishes to make the best use of its available resources? b. What are the first-order conditions for a maximum in this situation? c. How would the efficient situation described in part (b) be brought about by a perfectly competitive system in which this individual maximizes utility and the firms underlying the production possibility frontier maximize profits. d. Under what situations might the first-order conditions described in part (b) not yield a utility maximum?

Suppose there are only three goods \(\left(x_{1}, x_{2}, x_{3}\right)\) in an economy and that the excess demand functions for \(x_{2}\) and \(x_{3}\) are given by \\[ \begin{array}{l} E D_{2}=-\frac{3 p_{2}}{p_{1}}+\frac{2 p_{3}}{p_{1}}-1 \\ E D_{3}=-\frac{4 p_{2}}{p_{1}}-\frac{2 p_{3}}{p_{1}}-2 \end{array} \\] a Show that these functions are homogencous of degree 0 in \(p_{1}, p_{2},\) and \(p_{3}\) b. Use Walras' law to show that, if \(E D_{2}=E D_{3}=0,\) then \(E D_{1}\) must also be \(0 .\) Can you also use Walras' law to calculate \(E D_{1} ?\) c. Solve this system of equations for the equilibrium relative prices \(p_{2} / p_{1}\) and \(p_{3} / p_{1}\). What is the equilibrium value for \(p_{3} / p_{2} ?\)

The relationship between social welfare functions and the optimal distribution of individual tax burdens is a complex one in welfare economics. In this problem, we look at a few elements of this theory. Throughout we assume that there are \(m\) individuals in the economy and that each individual is characterized by a skill level, \(a_{p}\), which indicates his or her ability to earn income. Without loss of generality suppose also that individuals are ordered by increasing ability. Pretax income itself is determined by skill level and effort, \(c_{b}\) which may or may not be sensitive to taxation. That is, \(I_{i}=I\left(a_{b} c_{i}\right) .\) Suppose also that the utility cost of effort is given by \(\psi(c), \psi^{\prime}>0, \psi^{\prime \prime}<0, \psi(0)=0 .\) Finally, the government wishes to choose a schedule of income taxes and transfers, \(T(I),\) which maximizes social welfare subject to a government budget constraint satisfying \(\sum_{i=1}^{m} T\left(I_{i}\right)=R\) (where \(R\) is the amount needed to finance public goods). a Suppose that each individual's income is unaffected by effort and that each person's utility is given by \(u_{i}=u_{i}\left[I_{i}-T\left(I_{i}\right)-\right.\) \(\psi(c)]\). Show that maximization of a CES social welfare function requires perfect equality of income no matter what the precise form of that function. (Note: for some individuals \(T\left(I_{i}\right)\) may be negative.) b. Suppose now that individuals' incomes are affected by effort. Show that the results of part (a) still hold if the government based income taxation on \(a_{i}\) rather than on \(I_{i}\) c. In general show that if income taxation is based on observed income, this will affect the level of effort individuals undertake d. Characterization of the optimal tax structure when income is affected by effort is difficult and often counterintuitive. Diamond \(^{25}\) shows that the optimal marginal rate schedule may be U-shaped, with the highest rates for both low- and highincome people. He shows that the optimal top rate marginal rate is given by \\[ T^{\prime}\left(I_{\max }\right)=\frac{\left(1+e_{L, w}\right)\left(1-k_{i}\right)}{2 c_{L, w}+\left(1+e_{L, w}\right)\left(1-k_{i}\right)} \\] where \(k,\left(0 \leq k_{l} \leq 1\right)\) is the top income person's relative weight in the social welfare function and \(e_{L w}\) is the elasticity of labor supply with respect to the after-tax wage rate. Try a few simulations of possible values for these two parameters, and describe what the top marginal rate should be. Give an intuitive discussion of these results.

Suppose that Robinson Crusoe produces and consumes fish ( \(F\) ) and coconuts (C). Assume that, during a certain period, he has decided to work 200 hours and is indifferent as to whether he spends this time fishing or gathering coconuts. Robinson's production for fish is given by \\[ F=\sqrt{I_{F}} \\] and for coconuts by \\[ C=\sqrt{I_{C}} \\] where \(l_{F}\) and \(l_{C}\) are the number of hours spent fishing or gathering coconuts. Consequently, \\[ l_{c}+l_{F}=200 \\] Robinson Crusoe's utility for fish and coconuts is given by \\[ \text { utility }=\sqrt{F \cdot C} \\] a. If Robinson cannot trade with the rest of the world, how will he choose to allocate his labor? What will the optimal levels of \(F\) and \(C\) be? What will his utility be? What will be the \(R P T\) (of fish for coconuts)? b. Suppose now that trade is opened and Robinson can trade fish and coconuts at a price ratio of \(p_{F} / P_{C}=2 / 1 .\) If Robinson continues to produce the quantities of \(F\) and \(C\) from part (a), what will he choose to consume once given the opportunity to trade? What will his new level of utility be? c. How would your answer to part (b) change if Robinson adjusts his production to take advantage of the world prices? d. Graph your results for parts (a), (b), and (c).

The construction of the production possibility curve shown in Figures 13.2 and 13.3 can be used to illustrate three important "theorems" in international trade theory. To get started, notice in Figure 13.2 that the efficiency line \(O_{x}, O_{y}\) is bowed above the main diagonal of the Edgeworth box. This shows that the production of good \(x\) is always "capital intensive" relative to the production of good \(y\). That is, when production is efficient, \(\left(\frac{5}{7}\right)_{x}>\left(\frac{5}{1}\right)\), no matter how much of the goods are produced. Demonstration of the trade theorems assumes that the price ratio, \(p=p_{x} / p_{y}\) is determined in international markets-the domestic economy must adjust to this ratio (in trade jargon, the country under examination is assumed to be "a small country in a large world"). a Factor price equalization theorem: Use Figure 13.4 to show how the international price ratio, \(p\), determines the point in the Edgeworth box at which domestic production will take place. Show how this determines the factor price ratio, \(w / v\). If production functions are the same throughout the world, what will this imply about relative factor prices throughout the world? b. Stolper-Samuelson theorem: An increase in \(p\) will cause the production to move clockwise along the production possibility frontier \(-x\) production will increase and \(y\) production will decrease. Use the Edgeworth box diagram to show that such a move will decrease \(k / l\) in the production of both goods. Explain why this will cause \(w / v\) to decrease. What are the implications of this for the opening of trade relations (which typically increases the price of the good produced intensively with a country's most abundant input). c. Rybczynski theorem: Suppose again that \(p\) is set by external markets and does not change. Show that an increase in \(k\) will increase the output of \(x\) (the capital-intensive good) and reduce the output of \(y\) (the labor-intensive good).
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