Question

Excercise 1 （2excercise are related）

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

Exercise 2

a) suppose price of good 1 is 2, p1 = 2, and p2 = 1, use the results from exercise 1, part (c) to answer: How many units of good 1 and good 2 does an individual consumer demand?

b)still assume p1 = 2 and p2 = 1. what is the aggregate demand of good 1? what is the aggregate demand of good 2? how many units of good 1 (good 2, respectively) do all 1000 consumers demand together? (sum up all the individual demands from previous part (a))

c) the aggregate demand function of a good takes the market prices as an argument and maps them into the total units that are demanded. For given prices that the aggregate demand function of a good tell us how many units of that good are demanded by all 1000 consumers together. Use your results from Excercise 1, part (c) to derive that aggregate demand function for each good.

Answer 1

Exercise -1

Given the utility function = U = ln(x₁³x₂³) and the budget constraint p₁x₁+p₂x₂=1000

a) The Lagrangian is given by maximizing the given function subject to the constraint, using as the Lagrangian multiplier.

Therefore, the Lagrangian

b)

For the system of first order conditions we will partially differentiate the Lagrangian with respect to x₁ and x₂.

Differentiating with respect to x₂

c)

To solve the system of the above first order conditions, we will equate the value of found in both the equations. Therefore,

Using the above equality, we will substitute it in the given budget constraint p₁x₁+p₂x₂=1000

Therefore,

p₁x₁+p₂x₂=1000

p₁x₁+p₁x₁=1000

2p₁x₁=1000

p₁x₁=500

x₁=500/p₁

Therefore, p₂x₂=500

x₂=500/p₂

d)

In order to prove that the values of x₁ and x₂ found in part (c) are the maximizing bundle, we need to prove that the second order conditions are less than 0.

Therefore,

Therefore,

Since both the second order conditions are less than 0, therefore, the solved value of x₁ and x₂ are the utility maximizing bundle.