Suppose that a small economy's production possibility frontier for oil (X) and food (Y) is given by the equation X2 + Y2 = 3200. a) Sketch this PPF in a diagram. Write down an expression for the slope of the curve, in terms of X and Y. b) If every individual in this economy has preferences summarized by the utility function U (X ,Y ) = XY , then how much of each good will be produced in an autarky competitive equilibrium? What do we know about the relationship between the prices of oil and food, PX and P1,, in this equilibrium? Illustrate your answer as point A in your diagram from part a). c) Now suppose food and oil are traded on international markets at the price ratio PX {FY = 4. Assuming that this country produces the quantities you calculated in part b, describe how it can benet from fully engaging in international trade. Which product (and how much) will this country export, and which product (and how much) will it import? Indicate this country's optimal consumption choice on your diagram from part a, labeling it as point T. Quantity the gains from trade in dollar terms, using an approach similar to that mentioned in the Irwin article (Le. at the autarky prices, how large a proportional increase in income would be equivalent to the welfare gain from trade)? d) Repeat your analysis from parts b) and c) for the case where every individual in this country has preferences summarized by U (X ,Y) = Min(X,Y). Show your work, and draw a new diagram showing points A and T. e) Repeat your analysis once again, now for the case where every individual in this country has preferences summarized by U(X,Y) = X + Y. Show your work, and draw a new diagram showing points A and T. i) In words, give a brief intuitive explanation of the relationship between the extent of a country's gains from trade and the degree to which its consumers are able to substitute among goods. g) Finally, suppose this country adjusts its domestic production of oil and food in response to world prices. How much will it choose to produce of each good? Does your answer depend upon the utility function you specify? Show your work; you do not need to draw a new diagram (although you can do so if it helps you visualize the answer)