Question

The utility function is u = x₁^½ + x₂, and the budget constraint is m = p₁x₁ + p₂x₂.

1. Derive the optimal demand curve for good 1, x₁(p₁, p₂), and good 2, x₂(m, p₁, p₂).

2. Looking at the cross price effects (∂x₁/∂p₂ and ∂x₂/∂p₁) are goods x₁ and x₂ substitutes or complements? Looking at income effects (∂x₁/∂m and ∂x₂/∂m) are goods x₁ and x₂ inferior, normal or neither?

3. Assume m=100, p₁=0.5 and p₂=1. Using the demand function you derived in part a, what is the consumption of x₁ and of x₂?

4. Consider a price drop for only good 1 to p₁’=0.25. Calculate the new demands for good 1 and good 2 and label them x₁^M and x₂^M. Plot in a graph with p₁ on the vertical axis the x₁ you found in part c, and x₁^M. Graph the resulting Marshallian demand curve.

5. Assume you are endowed with the amounts x₁ and x₂ you found in part c. What is their total worth, given the new prices? Define this as m^S = p₁’x₁ + p₂x₂. If m^S was your original budget constraint and you had to maximize your utility, what would be the resulting optimal consumption x₁^S(p₁’,p₂) and x₂^S(m^S,p₁’,p₂)?

   f. Plot the original x₁ from part c and x₁^S from part e in a graph with p₁ on the vertical axis. Draw the Slutsky demand curve. How does it differ from the Marshallian demand curve? What is the substitution effect and the income effect of this price change?

Answer 1

It's mandatory to answer only first 4 parts

& UCx1, x2) = NX1 + X2 = Pixi + P2X2 a) at eam MRS112 = PilP2 MRS=MUL = 1 x at Eqm ax = 1/2 = x1 = (P2) 2 MO 2 from B.C. x2 * = m- pi x = m - 1 por la P2 b.) Now axi = P2 - 2P2 = +12 70, 80 two Goods are 21, 2 4 P12 substitutes. Ang ax"=0, (Neither, Goody demand is independent am of Income) on = VP2 70, Noomal good 1 C-) m=100, P = 05, P2=1, Xi= 11 x2 = 100 = 99.5 4•5)2 (X, 9X 2)" = (1, 99.5) Ang - 99.5 4(05)2 = d) *25, *'* *1992 X2' = 100 = 99 + Ang Graph →

(15) ad curve (4,•25) Xim.