Consider two countries, Home and Foreign, each of which produces two goods, butter (B) and guns (G), using labour and capital. The production possibility frontier at Home is given by B2 + G2 = 1, while that at Foreign is 2B2 + 1 G2 = 1. The preferences of consumers in both 2vp countries are identical and captured by the following utility function: U = BG. Let B = p pG be the relative price of butter.
(a) Compute the equilibrium levels of B and G consumed and produced in the Home country under autarky. What is the corresponding value of p?
(b) Compute the equilibrium levels of B and G consumed and produced in the Foreign country under autarky. What is the corresponding value of p?
(c) Based on your answers to parts (a) and (b), if the two countries open up, what will be the pattern of trade?
(d) Derive the export-supply curve of B for the B-exporting country as a function of p.
(e) Derive the import-demand curve of B for the B-importing country as a function of p.
(f) What is the equilibrium p under free trade? How much B does Home import/export under free trade?
Hint : To find p, you need to solve a non-linear equation in p. You may not be able to do this using pencil and paper. You can use MS-Excel to do this. For example, suppose you have an equation that looks like this: ap3 + bp2 + cp + d = 0, where a, b, c and d are constants. Then pick a value for p. Create separate columns for ap3, bp2, cp and d, and another column that has their sum. Then vary the value for p until the sum is equal to zero. The corresponding p value is a solution. Keep in mind that non-linear equations typically have more than one solution.