Chapter 13

Suppose the production possibility frontier for guns \((x)\) and butter \((y)\) is given by \\[ x^{2}+2 y^{2}=900 \\] a. Graph this frontier b. If individuals always prefer consumption bundles in which \(y=2 x,\) how much \(x\) and \(y\) will be produced? c. At the point described in part (b), what will be the \(R P T\) and hence what price ratio will cause production to take place at that point? (This slope should be approximated by considering small changes in \(x\) and \(y\) around the optimal point. d. Show your solution on the figure from part (a).

Consider an economy with just one technique available for the production of each good. $$\begin{array}{lcc} \hline \text { Good } & \text { Food } & \text { Cloth } \\ \hline \text { Labor per unit output } & 1 & 1 \\ \text { Land per unit output } & 2 & 1 \\ \hline \end{array}$$ a. Suppose land is unlimited but labor cquals 100 . Write and sketch the production possibility fronticr. b. Suppose labor is unlimited but land equals \(150 .\) Write and sketch the production possibility frontier. c. Suppose labor equals 100 and land equals \(150 .\) Write and sketch the production possibility frontier. Hint: What are the intercepts of the production possibility fronticr? When is land fully employed? Labor? Both? d. Explain why the production possibility frontier of part (c) is concave. e. Sketch the relative price of food as a function of its output in part (c). f. If consumers insist on trading 4 units of food for 5 units of cloth, what is the relative price of food? Why? g. Explain why production is exactly the same at a price ratio of \(p_{F} / p_{C}=1.1\) as at \(p_{F} / p_{C}=1.9\) h. Suppose that capital is also required for producing food and clothing and that capital requirements per unit of food and per unit of clothing are 0.8 and \(0.9,\) respectively. There are 100 units of capital available. What is the production possibility curve in this case? Answer part (e) for this case.

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{l_{F}} \\] and for coconuts by \\[ C=\sqrt{l_{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{\boldsymbol{F} \cdot \boldsymbol{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).

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?

In the country of Ruritania there are two regions, \(A\) and \(B\). Two goods \((x \text { and } y)\) are produced in both regions. Production functions for region \(A\) are given by \\[ \begin{array}{l} x_{A}=\sqrt{l_{x}} \\ y_{A}=\sqrt{l_{y}} \end{array} \\] here \(l_{x}\) and \(l_{y}\) are the quantities of labor devoted to \(x\) and \(y\) production, respectively. Total labor available in region \(A\) is 100 units; that is, \\[ l_{x}+l_{y}=100 \\] Using a similar notation for region \(B\), production functions are given by \\[ \begin{array}{l} x_{B}=\frac{1}{2} \sqrt{l_{x}} \\ y_{B}=\frac{1}{2} \sqrt{l_{y}} \end{array} \\] There are also 100 units of labor available in region \(B:\) \\[ l_{x}+l_{y}=100 \\] a. Calculate the production possibility curves for regions \(A\) and \(B\). b. What condition must hold if production in Ruritania is to be allocated efficicntly between regions \(A\) and \(B\) (assuming labor cannot move from one region to the other)? c. Calculate the production possibility curve for Ruritania (again assuming labor is immobile between regions \() .\) How much total \(y\) can Ruritania produce if total \(x\) output is \(12 ?\) Hint: \(\mathrm{A}\) graphical analysis may be of some help here.

The country of Podunk produces only wheat and cloth, using as inputs land and labor. Both are produced by constant returns-to-scale production functions. Wheat is the relatively land-intensive commodity. a. Explain, in words or with diagrams, how the price of wheat relative to cloth ( \(p\) ) determines the land-labor ratio in cach of the two industries. b. Suppose that \(p\) is given by external forces (this would be the case if Podunk were a "small" country trading freely with a "large" world). Show, using the Edgeworth box, that if the supply of labor increases in Podunk then the output of cloth will rise and the output of wheat will fall. Note: This result was discovered by the Polish economist Tadeusz Rybczynski. It is a fundamental result in the theory of international trade.

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 homogeneous 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 usc 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 \(p_{3} / p_{2} ?\)