6.1 Charlie is back - still consuming apples and bananas. His utility function is U(xA; xB) = xAXB. We want to find his demand function for apples, xA(pA; pB;m), and his demand function for bananas, xB(pA; pB;m). (a) When the prices are pA and pB and Charlie's income is m, the equation for Charlie's budget line is pAxA+pBxB = m. The slope of Charlie's indifference curve at the bundle (xA; xB) is -MU1(xA; xB)/MU2(xA; xB) = -xB/xA. The slope of Charlie's budget line is -pA/pB. Charlie's indifference curve will be tangent to his budget line at the point (xA; xB) if the following equation is satisfied: pA/PB = xB/XA. (b) You now have two equations, the budget equation and the tangency equation, that must be satisfied by the bundle demanded. Solve these two equations for xA and xB. Charlie's demand function for apples is xA(pA; pB;m) = m/2pA, and his demand function for bananas is xB(pA; pB;m) = m/2pB. (c) In general, the demand for both commodities will depend on the price of both commodities and on income. But for Charlie's utility function, the demand function for apples depends only on income and the price of apples. Similarly, the demand for bananas depends only on income and the price of bananas. Charlie always spends the same fraction of his income on bananas. What fraction is this? 1/2. 6.2 Douglas Cornfield's preferences are represented by the utility function u(x1; *2) = x1 X2 . The prices of x1 and x2 are p1 and p2. (a) The slope of Cornfield's indifference curve at the point (x1; x2) is -2x2/3X1