Question

Suppose that a consumer’s utility function is U(x,y)=xy＋10y. the marginal utilities for this utility function are MUx=y and MUy=x＋10. The price of x is P_x and the price of y is P_y, with both prices positive. The consumer has income I. (this problem shows that an optimal consumption choice need not be interior, and may be at a corner point.)

1. Assume first that we are at an interior optimum. Show that the demand schedule for x can be written as x=I／(2Px)－5.
2. Suppose now that I=100. Since x must never be negative, what is the maximum value of P_x for which this consumer would ever purchase any x?

Answer 1

a. At Optimal point, MUx/MUy=Px/Py

y/(x+10)=Px/Py

y=Px*(x+10)/Py... put this in Budget constraint

X*Px+Y*Py=I

X*Px+Py*(Px*(x+10)/Py)=I

X*Px+X*PX+10Px=I

2XPx=I-10Px

X*=(I-10Px)/2Px

X*= I/2Px-5

b. X*=100/2Px -5= 50/Px-5

For X* to be positive, maximum value of Px should be less than 10.

When Px=10, X*=0.