Question

Question 1

[30 marks] Consider a consumer who has preferences over two goods, x and 3/7 4/7 y, represented by the utility function u(x, y) = x y .

a) [5 marks] Find equations for this consumer’s indifference curves for utility levels u = 1/2 and u = 1, and represent them graphically.

b) [5 marks] Find this consumer’s marginal rate of substitution (MRS). Then, using this result and supposing that this consumer has money income m, and faces prices px and py, solve the system of equations consisting of the optimization principle and the budget constraint to find functions x(px, py, m) and y(px, py, m). These two equations, relating quantities bought of both goods to prices and income, are Marshallian demands.

c) [10 marks] Solve her utility maximization problem by the Lagrangian method, and again find her Marshallian demands for both goods, as well as the optimal value of the Lagrangian multiplier. For the demand functions, you should obtain the same result as in b). [Please refer to the Handout on (Constrained) Optimization for more information and a solved example, as well as the lecture slides.]

d)[3marks] Let px =6, py=4,m=50.Based on your answers in c),solve for the optimal values of x∗, y∗ and for the marginal utility of money – i.e. λ∗, the optimal value of the Lagrangian multiplier for these parameter values.

e) [7 marks] Suppose instead that this consumer has the utility function v(x1,x2) = (3/7) ln x1 + (4/7) ln x2. Solve her utility maximization problem, either as in b) or c). What are her new Marshallian demands ? What do you notice, and why ?

Answer 1

The question can be solved by following general rules of optimisation.

B)

C) Using Lagrangian Method :

D)

The new Marshallian demand functions obtained in part e) are same as the ones obtained before. It is so because the utility function that we have in e) is just a logarithmic version of the original utility function. This transformation doesn't change the MRS and the preferences that were being presented earlier. Therefore , the Marshallian Demand Functions are the same.