Henri's utility function is min{x, 5y +2z}.The price of x is $1, the price of y is $15, and the price of z is $7.Henri's income is $44.How many units of x does Henri demand?
A)9.78
B)11
C)5
D)3
E)None of the above.

Question

Quizplus · Accepted Answer

Henri's income is $44. He will spend all his income on goods x, y, and z. So, we have the budget constraint:1x + 15y + 7z = 44Henri's utility function is min{x, 5y +2z}. So, he will consume goods x, y, and z in such a way that his utility is maximized subject to his budget constraint.To find the optimal consumption bundle, we can use the Lagrangian method. The Lagrangian is:L = min{x, 5y +2z} + \lambda(44 - 1x - 15y - 7z)Taking the partial derivatives of L with respect to x, y, z, and \lambda, we get:\frac{\partial L}{\partial x} = -1 + \lambda = 0\frac{\partial L}{\partial y} = -5\lambda = 0\frac{\partial L}{\partial z} = -2\lambda = 0\frac{\partial L}{\partial \lambda} = 44 - 1x - 15y - 7z = 0From the first three equations, we get:\lambda = 1x = 1y = 0z = 0Substituting these values into the fourth equation, we get:44 - 1(1) - 15(0) - 7(0) = 44So, Henri will consume 1 unit of x, 0 units of y, and 0 units of z.

Henri's Utility Function Is Min{x, 5y +2z}