Question

For the utility function U(x, z) x+z derive expressions for the optimal levels of good x and good z as functions of the price of good x, py, the price of good z, p7, and income, Y. For simplicity, assume that the price of good z is normalized to be one: PZ 1 1 In your answer, use 1 for "p2" and the relationship o 1- The optimal value of good z is (Properly format your expression using the tools in the palette. Hover over tools to see keyboard shortcuts. E.g., a subscript can be created with the_ character.) z =

Answer 1

PP x + 2 = 0(3,2) = a +2 Pix + f2 = Y. As l = 1. So, fai x + 2 = y Setting up lagrange de a + 2 + 1(y - fax - 2) Pod = 8289-1 Pa = 0 - © ze a faltas 0.- zdy - P4.8 - 2 0 - ( anar, - From ® and (i fast a fizet!. Pe

Using it in equation in => Y - fizioten. Z o => Y-Retz - 4:0 y a patriz ta. >> y = 2 ( + 1) a = 2.8%- - Now, If r= -1 samo 146 = IES TO types of CM 1+ r
So, Optimal Values of z and x :-

$\large z^{*} = \frac{Y}{P_{x}^{1+\sigma} +1}\;\;\;\;\;\;\;\;\;\;\;\; and \;\;\;\;\;\;\;\;\;\;\;\;\; x^{*} = \frac{Y.P_{x}^{\sigma}}{P_{x}^{1+\sigma} +1}$

So, Optimal value of good z is :-

$\large z^{*} = \frac{Y}{P_{x}^{1+\sigma} +1}$