Consider a consumer with utility function u = U(x, y) defined on the nonnegative

Question

Consider a consumer with utility function u = U(x, y) defined on the nonnegative orthant facing positive prices p^o and q^o for x and y by, respectively. State restrictions on U sufficient to imply that x = g(y, u) is a continuously differentiable function of y and u such that u U(g(y, u), y). Consider the problem of minimizing spending while attaining utility level u^o. Assume g exists and use the method of substitution to eliminate x as a choice variable, and assume that a unique value y* minimizes spending. State restrictions on U sufficient to imply that y* is a continuously differentiable function at (p^o, q^o, u^o).

Explanation / Answer

a) u = U(x,y)

Or x = g(u) + h(y)

i.e x will be a function in u and y

x = g(y,u)

For x to be continuous and differentiable g(u) and h(y) need to be continuous and differentiable everywhere

B) we have to minimize p0x + q0y

=> p0 g(y,u) + q0y

=> p0 g(y,u0) + q0y

For its minimization,

d/dy = 0

=> d(p0 g(y*,u0) + q0y* )/dy = 0

Or the function should be differentiable in p0 , q0, u0

Again u = U(g(y,u),y)

Or optimized u0 = U(g(y*,u0),y*)

y* = f(u0)

For min y* d/du should exist or the function should be differentiable at u0

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