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 = (c_{1},c_{2}) =
c_{1}^{α},c_{2}^{1-}^{α}

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

** I** ≥p

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

At the optimum, the consumer spends the fraction
p_{1}c_{1}^{*}/** I**
of his income on good 1 and
p

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.

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