Question

7. Jay's Utility function is given by U(x,z) = 3x₁^0.2 x₂^0.8 and P₁=$2 and P₂=$4 and his budget is $200.

• Write out the Lagrange but DO NOT solve it
• Find the utility maximizing values of x₁and x₂

8. What does the substitution effect cause a consumer to do if the price of a good increases?

9. What does the income effect cause a consumer to do if the price of a good increases? What else is needed here?

10. What can we say about the substitution and income effects on a decrease in wages? What is an Engel curve and how does it relate here? If leisure is viewed as an inferior good, how would the labor demand curve look?

11. Suppose m=120, p1= 5, and p2=6.

• What is the budget equation?
• What is the slope of the budget line?
• How does the budget set change with the following:
  • a 20% increase in m
  • a 2$ decrease in p2

Answer 1

7. The consumer's problem is:
$\dpi{80} \fn_phv \small max \ 3x^{0.2}y^{0.8} \ s.t. \ 2x+4y=200$

The Lagrange is:
$\dpi{80} \fn_phv \small \mathbb{L}=3x^{0.2}y^{0.8} -\lambda(2x+4y-200)$

At equilibrium, marginal rate of substitution is equal to the price ratio:
$\dpi{80} \fn_phv \small \frac{0.6x^{-0.8}y^{0.8}}{2.4x^{0.2}y^{-0.2}}=\frac{y}{4x}=\frac{2}{4}\rightarrow y=2x$

Substituting this into the consumer's budget equation:
$\dpi{80} \fn_phv \small 2x+4(2x)=200\rightarrow x=20, y=40$