The utility function is u x 1 x 2 , and the budget constraint is m p 1 x 1 p 2 x 2 Derive the optimal demand curve for good 1, x 1 (p 1 , p 2 ), and good 2, x 2 (m, p 1 , p 2 ) Looking at the cross price effects (x 1 p 2 and x 2 p 1 ) are goods x 1 and x 2 substitutes or complements Looking at inco...

The utility function is u = x 1 + x 2 , and the budget constraint

Fantastic news! We've Found the answer you've been seeking!

Question:

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

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

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?

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?

Consider a price drop for only a good 1 to p₁’=0.25. Calculate the new demands for good 1 and good 2 and label them x₁^M and x₂^M. The 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.

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 of 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?