Part-1

Assume preferences can be represented by the following utility function:

u(x₁, x₂) = 3 ln (x₁) + ln (x₂)

a.  Is the utility function monotonic? Justify.

b. How can this person be better off than by consuming the bundle (x₁, x₂) = (10,10)?

c. Set up the utility maximization problem for the consumer, when facing prices p₁ = 6, p₂ = 2 and income m = 32.

d.  Solve the problem by finding (x₁*, x₂*).

e.  Graph the budget set, a couple of indifference curves and the optimal choice.

Part-2

Assume preferences can be represented by the following utility function:

u(x₁, x₂) = -x₁² + 150x₁ - 2x₂² + 100x₂ + x₁x₂

a. Is the utility function monotonic? Justify.

b. How can this person be better off than by consuming the bundle (x₁, x₂) = (100,100)?

a Set up the utility maximization problem for the consumer, when facing:

prices p₁ = 2, p₂ = 1 and income m = 30.

d. Solve the problem by finding (x₁*, x₂*).

Part-3

Assume preferences can be represented by the following utility function:

u(x₁, x₂) = 4ln (x₁) + x₂

a. Is the utility function monotonic? Justify.

b. Set up the consumer's utility maximization problem for prices pi, p2 and income m (the general case)

C Solve the problem. You will obtain solutions x₁* (p₁, p₂, m), x₂*  (p₁, p₂, m) in terms of the parameters of the model  (p₁, p₂, m).