Question

1. (24 total points) Suppose a consumer’s utility function is given by U(X,Y) = X1/2*Y1/2. Also, the consumer has $72 to spend, and the price of Good X, PX = $4. Let Good Y be a composite good whose price is PY = $1. So on the Y-axis, we are graphing the amount of money that the consumer has available to spend on all other goods for any given value of X.

a) (2 points) How much X and Y should the consumer purchase in order to maximize her utility?

b) (2 points) How much total utility does the consumer receive?

c) (2 points) Now suppose PX increases to $9. What is the new bundle of X and Y that the consumer will demand?

d) (2 points) How much additional money would the consumer need in order to have the same utility level after the price change as before the price change? (Note: this amount of additional money is called the Compensating Variation.)

e) (2 points) Of the total change in the quantity demanded of Good X, how much is due to the substitution effect and how much is due to the income effect? (Note: since there is an increase in the price of Good X, these values will be negative).

f) (14 points) In the space below, draw on a graph the • original budget constraint (draw this in black) • new budget constraint (draw this in green) • compensated budget constraint (draw this in red) Also, on your graph, indicate the optimal bundle on each budget constraint. • Label the optimal bundle on the original budget constraint X* and Y* • Label the optimal bundle on the new budget constraint X** and Y** • Label the optimal bundle on the compensated budget constraint XC and YC In order to receive full credit, your graph must be neat, accurate, and fully labeled!!

Answer 1

Z U= x lyk @ MRSM Consumer's Income & 72 Px = 4 Pyal At optimality MRS = Px / Py x = 4 17 = 4x put inbudget 4x+y=72 (X=72 94 4.36) - Point A Total Wtility = 36th ar- 6x3 = 18 If Px increases to $9. 869)?? & Co put in uneleting Compensated Innome At new price , Z =9 (9=9x U=18= x 24 t = 3x [(x=6) 4 14 = 54] Point B e) Income required = 6(a)+54=/08 compensating Varichan = 108-72=36 when one = 108 thin x = log 108 Ceint A to B =) Substitution affect Can't B to co nome Effect. font A toc lice Effect.