Question

Each individual consumer takes the prices as given and chooses her consumption bundle,(x1,x2)ER^2, by maximizing the utility function: U(x1,x2) = ln(x1^3,x2^3), subject to the budget constraint p1*x1+p2*x2 = 1000

a) write out the Lagrangian function for the consumer's problem

b) write out the system of first-order conditions for the consumer's problem

c) solve the system of first-order conditions to find the optimal values of x1, x2. your answer might depend on p1 and p2.

d) check if the critical point satisfies the second-order condition

Answer 1

a) We will have to

Maximize U= log( x1³x2³)

subject to p1x1 +p2x2 = 100

The lagrangian equation will be;

b) The first order conditions of maximization are:

i) ...........(i)

ii) ............(ii)

iii) ..........(iii)

c) Equating equation i and ii, we get

Soling this we get

Putting this value of x2 in equation iii we get

similarly,