Charlie has a utility function U(x_A, x_B)=x_Ax_B, the price of apples is $1, and the price of bananas is $2.If Charlie's income were $120, how many units of bananas would he consume if he chose the bundle that maximized his utility subject to his budget constraint? A)30 B)15 C)60 D)6 E)90

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The optimal bundle is found by solving the following problem:max U(x_A, x_B) = x_Ax_Bsubject to p_Ax_A + p_Bx_B = mwhere p_A = $1, p_B = $2, and m = $120.The Lagrangian for this problem is:L = x_Ax_B + \lambda(m - p_Ax_A - p_Bx_B)Taking the partial derivatives of L with respect to x_A, x_B, and \lambda, we get:\frac{\partial L}{\partial x_A} = x_B - \lambda p_A = 0\frac{\partial L}{\partial x_B} = x_A - \lambda p_B = 0\frac{\partial L}{\partial \lambda} = m - p_Ax_A - p_Bx_B = 0Solving these equations, we get:x_A = \frac{m}{p_A + p_B} = \frac{120}{1 + 2} = 40x_B = \frac{m}{p_B + p_A} = \frac{120}{2 + 1} = 60Therefore, Charlie would consume 60 units of bananas if he chose the bundle that maximized his utility subject to his budget constraint.

Charlie Has a Utility Function U(xA, XB)=xAxB, the Price of Apples

Charlie Has a Utility Function U(x_A, X_B)=x_Ax_B, the Price of Apples