Question

A consumer likes two goods; good 1 and good 2. the consumer’s preferences are described the by the cobb-douglass utility function

U = (c₁,c₂) = c₁^α,c₂^1-α

Where c₁ denotes consumption of good 1, c₂ denotes consumption of good 2, and parameter α lies between zero and one; 1>α>0. Let I denote consumer’s income, let p₁ denotes the price of good 1, and p₂ denotes the price of good 2. Then the consumer can be viewed as choosing c₁ and c₂ to maximize utility subject to the budget constraint

I ≥p₁c₁ + p₂c₂

1. Define the lagrangian for this constrained optimization problem
2. Write down the full set of conditions that, according to Kuhn-tucker theorem must be satisfied by the consumer’s optimal choice c₁^* and c₂^* together with the associated value of the lagrange multiplier
3. Use your results from above to solve for the optimal choice c₁^* and c₂^* in terms of model’s four parameters.

At the optimum, the consumer spends the fraction p₁c₁^*/I of his income on good 1 and p₂c₂^*/I of his income on good 2. What do your results tell you about the implications of the cobb-douglass utility function for these optimal expenditure shares?

Answer 1

L=c1^a.c2^(1-a)+ T1(I-p1c1-p2c2)+T2(c1)+T2(c2)

FOC must be satisfied and also c1 and C2 should be at least 0. This implies additional langrange multipliers which are T2 and T3. We will use the complementary slackness condition.

As it is a multiplicative utility function, c1* and C2* should be greater than 0.So complementary slackness implies T2 and T3 are 0. Also the utility function is increasing in both c1 and C2. So the budget constraint will be binding. So T1>0.

Foc

Differentiation wrt c1.

ac1^(a-1)*c2^(1-a)- T1p1=0.

So ac1^(a-1)*c2^(1-a)=T1p1

Similarly, c1^a(1-a)c2^-a=T1p2.

Dividing the two,

(1-a)c1/ac2=p2/p1.

So c2p2=p1c1(1-a)/a.

In the budget,

P1c1(1+1-a/a)=I. So c1*=aI/p1.

Similarly C2=(1-a)I/p2.

Expenditure shares.

Share of c1= a. Share of C2= (1-a)

Therefore in Cobb Douglas functions, the optimal expenditure shares are the indices of the good in the utility function.