Question

d

04 Question (2 points) See page 78 In addition to finding the optimal bundles given prices and income, utility maximization can be used to find individual demand functions at any prices and income. Setting up the problem and solving it are the same, except that the prices of each good and the income will be left in variable form (economists dub these parameters or exogenous variables). 1st attempt See Hint Suppose utility for an average consumer over food and clothing is represented by u(x, y) = 100xy. Find the optimal values of x and y as a function of the prices px and py and the income level m. х*(рх» Ру, т) %3 Choose one: A O B. OC. D. O E. Pxt2 2m Px+2py F y*(px, Py, m) Choose one: A O B. O D. OE. OF.

Answer 1

u(x,y) = 100xy

$\partial$u/$\partial$x = MUx = 100y

$\partial$u/$\partial$y = MUy = 100x

MRSx,y   = MUx/MUy

= 100y/100x

= y/x

Budget constraint

P_xx + P_yy = m (i)

At optimal choice

MRSx,y = Px/Py

y/x = Px/Py

y = (Px/Py)x

substitute    y = (Px/Py)x in equation (i)

P_xx + P_yy = m

P_xx + Py(Px/Py)x = m  

P_xx + P_xx = m

2P_xx = m

x = m/2P_x  

x*(Px,Py,m) =   m/2P_x  

y = (Px/Py)x

=    (Px/Py)( m/2P_x )

= m/2Py  

y^*(Px,Py,m) =   m/2Py