For the below question, consider a consumer that consumes two goods, x and z wit
ID: 1217861 • Letter: F
Question
For the below question, consider a consumer that consumes two goods, x and z with the following utility function.
U =( x^.25)(z^.75)
1. What is the marginal rate of subsitution?
a. -3x/z
b.-x/3z
c. -3z/x
d. -z/3x
2. Derive the demand function for good x
a. 3y/4px
b. y/4px
c. 3px/4y
d. px/4y
3. Suppose initial values for income and the prices of goods x and z are Y=100, px=5, and pz=15 respectively, then the price of good x falls to px' = 2. What is the magnitude of the Total Effect and then calculate the magnitude of the income and substitution effects with respect to good x.
Explanation / Answer
U = x0.25z0.75
(1) (d)
Marginal rate of substitution (MRS) = - MUx / MUz
MUx = dU / dx = 0.25. (z / x)0.75
MUz = dU / dz = 0.75. (x / z)0.25
MRS = - (0.25 / 0.75). (z / x) = - z / 3x
(2) (b)
Budget line: y = x.px + z.pz
Optimality condition requires that: MUx / MUz = px / pz
z / 3x = px / pz
z.pz = 3x.px
Substituting in budget line,
y = x.px + 3x.px = 4x.px
x = y / 4px [Demand function for x]
Note: First 2 questions are answered.