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3. Problem Set 1 International Equilibrium Explain how it is possible for a production function to have the property of constant or increasing returns to scale and still satisfy the Law of Diminishing Returns. If a production function has only one input, say labor, and displays constant returns to scale, does it then violate the Law of Diminishing Returns? Suppose that a production fimction hm two inputs1 K and L, and that1 contrary to the Law of Diminishing lileturns1 an increase in L alone always causes output to rise by the same proportion that L has increased. Show that the production function must therefore display increasing returns to scale. Differentiate the denition of \"Homogeneous of Degree it\" with respect to it... and evaluate the result at i=1. Then use this result to show that1 if factors of production are paid the value of their marginal products as assumed under perfect competition1 then factor payments will exactly equal the value of output if remms to scale are constant, but will exceed it if returns to scale are increasing. 1|iii'l'iat does the latter result tell you about the compatibility of perfect competition with increasing returns to scale? In the 2x2 production model with constant returns to scale1 show that if the two industries have the same factor intensity for all factor prices, then the production possibility frontier {PPF} is a straight line. b. Use the Edgeworth-Bowley Box diagram to illustrate the e'ects on production possibilities of an increase in the endowment of capital holding the endowment of labor constant. Show how the changes in the box diagram translate to changes in the PPF. In the 2:2 model ofproduction and trade. suppose that a country is initially exporting good I and that its production possibilities then expand {we don't know why}1 making it possible for it to produce more of both goods. Assuming that the country is small enough that the world market price does not change as a result1 which of the following must be one. which of the following might or might not be true, and which of the following cannot be true? It will produce more of good X. It will produce more of good Y. Its income will rise. It will import more of good Y. Problem Set 2 Gains from Trade and the Ricardian Model 1. Use community indifference curves as your indicator of national welfare in order to evaluate the following claim: "An improvement in the terms of trade increases welfare only if the country increases its quantity of exports in response. If a country is unwilling or unable to increase exports when their price rises, then the price increase does it no good." 2. Consider an economy that does not produce goods, but is simply endowed with certain amounts of them, and in which the population consists of two groups: farmers who own only wheat and weavers who own only cloth. (Never mind that these people don't actually farm and weave - the names are just for convenience.) There are equal numbers of people in both groups, and they all share identical homothetic preferences for consuming wheat and cloth. a. Draw the production possibility frontier for this economy, and illustrate the autarky equilibrium. b. Is it possible to say who is better off in autarky, the farmers or the weavers? On what does that depend? C. Suppose now that the country opens up to free trade at a relative price of wheat that is higher than its autarky price. Show the new equilibrium. d. Construct the utility possibility frontiers for autarky and free trade, and indicate how the actual equilibria along these frontiers will compare. Who gains and who loses from trade? In what sense, if any, are there gains from trade in this case? 3. Which of the following characterize the Ricardian Model? a. Perfect mobility of factors across industries b. Perfect mobility of factors across countries c. Constant returns to scale d. The law of diminishing returns e. Identical technologies across industries f. Identical technologies across countries g. Cournot competition h. Perfect competition4. 5. Juana-unh- Suppose that a small open economy has Elli} workers and that its technology requires 1 worker-hour per unit of food (thus 1 unit of food pet worker-hour} and 3 worker- hours per unit of cloth {thus output per worker-hour = NJ). in autarky, it employs lllll workers in each of the two industries. With free trade, it faces world prices of $lll per unit of food and $21} perunit ofcloth. a. Suppose that in autarky, workers in both industries are paid $8 per hour. What are the autarlcy prices of food and cloth? b. When the country opens to free trade, under the normal assumptions of the Ricardian model, what will it produce, import, and export? From the information given, can you determine the quantities of any of these? What is the country's national income with trade, measured in dollars? c. Suppose, contrary to the normal Ricardian Model assumptions, that when trade is opened, workers are unwilling or unable to change occupations, so that we continue to have lll workers in each industry. What, then, is the national income of the country, in dollars, and how does it compare to the national income you got in part (b) when workers were mobile? d. What are the wages of the two groups of workers in part {c}? e. Does the country gain from trade in part {c}? Who gains and who loses within it? Using the same assumptions at the start of problem 4, but returning to the usual assumption that labor can move between industries, suppose now that the wage of labor in the country in antarlry had been $15 per hour instead of $3. a. Now what would be the antarky prices of Food and Cloth? b. When the country opens to free trade, what will happen? How will your answers to part {b} of problem 4 be changed? What is the wage of labor with free trade, in dollars? d. What has happened, as a result of trade, to the real wages of labor? Use the E-country Ricardian Model with free trade to work out the effects of the following changes (one at a time) on the Home cotmuy's terms of trade and welfare. Assume that the Home countryr has a comparative advantage in Food (the other good being Cloth}, and that in the initial equilibrium, both countries specialize completely. You may assume, if it is helpful, that preferences are identical and homothetic in the two countries. a. An increase in the Foreign labor force. b. A increase, by the same percentage in both industries, in the Home country's productivity in producing both Food and Cloth