We found in class that when individual agents in the economy have identical preferences but different endowments, international trade creates gainers and losers; viz, agents whose endowments are valued less under trade versus autarky are worse off and agents whose endowments are valued more under trade versus autarky are better off. However, those who gain can compensate those who lose, at least in principle, so that trade makes the overall economy better off (i.e., there are gains from trade).
Interestingly, the same is true when agents have identical endowments but different preferences. Agents with a stronger preference for a good that has a higher relative price under trade versus autarky are worse off with trade, but agents with a stronger preference for a good that has a lower relative price under trade versus autarky are better off with trade. But it is still true that there are gains from trade, i.e. those who gain from trade can in principle compensate those who lose such that the overall economy is better off. The following homework problem asks you to investigate this case.
Assume that there are two agents C (Crusoe) and F (Friday) and two goods X (coconuts) and Y (fish). Each agent has an endowment of X = 2, Y = 2 prior to exchange (i.e., self-sufficiency). Assume that C's utility function is given as UC = XC2/3YC1/3 and F's utility function is given as UF = XF1/3YF2/3. The marginal rates of substitution for C and F are obtained, respectively, as MRSC = 2YC/XC and MRSF = YF/2XF.
a) Compute the utility and MRS of C and F prior to exchange. Based on these MRS values, can C and F gain from exchange? If so, briefly explain how they would exchange.
b) Derive C's demand curve for X and F's supply curve for X. Let p = PX/PY stand for the relative price of good X. Hint: To obtain these curves, use the utility maximizing conditions MRSC = p and MRSF = p, as well as the fact that pX = Y under balanced trade.
c) Graph these curves in the usual way, i.e., put p on the vertical axis and X on the horizontal axis. Note that both of these curves intersect the vertical axis. Label these intercepts and interpret their meaning in words.
d) Find the equilibrium price and quantity under autarky (exchange prior to international trade) and label in your diagram. Compare the utilities of C and F under autarky with those under self-sufficiency to show that both C and F are better off with exchange. [You'll need a calculator for this]
e) Suppose that the relative price of X in international trade is pT = 1.5. Determine the economy's export (or import) of X. Compute the utilities of C and F under trade. Who gains and who loses from trade versus autarky? Is either C or F worse off under trade versus self-sufficiency?
f) Compute the gains from trade in utility terms for the economy. Illustrate these gains as a change in social surplus in your diagram.