Question

A consumer has utility functionU(q1, q2) =q1+√q2, incomeY= 3, and faces pricesp1= 4 andp2= 1. Find all consumption bundles that satisfy the necessary condition fora utility maximizing choice. Then determine which of these is optimal.

Answer 1

Consider the given problem here the utility function of the consumer is, “U=q1 + q2^1/2”, with price “P1=4”, “P2=1” and money income “Y=3”.

So, here the budget constraint of the consumer is given below.

=> P1*q1 + P2*q2 = Y, => 4*q1 + q2 = 3”.

So, the lagrange function of the consumer is given below.

=> L = q1 + q2^1/2 + λ*(3 – 4q1 – q2).

So, given the lagrangian function for maximization the “FOC” necessary condition for maximization is given below.

=> δL/δq1 = δL/δq2 = δL/δλ = 0.

=> δL/δq1 = 0, => 1 + λ(– 4) = 0, => 4λ = 1 …………(1).

=> δL/δq2 = 0, => (1/2)q2^(-1/2) + λ(– 1) = 0, => (1/2)q2^(-1/2) = λ …………(1).

=> δL/δλ = 0, => 3 – 4q1 – q2 = 0 ……………(3).

Now, form (1), we get “λ=1/4”. Now, by substituting the value of “λ” into (2) we will get the optimum choice of “q2”.

=> (1/2)q2^(-1/2) = λ = ¼, => q2^(-1/2) = 2/4 = ½, => q2^(1/2) = 2, => q2 = 2^2=4.

Now, we can see that the optimum “q2=4”, => the total money required to purchase “q2” is “4*1 = 4 < 3”. So, the total expenditure is more than the given income, => here the consumer will totally spend on “q2” and not on “q1”. So, the optimum consumption choice is given by, “q1=0” and “q2=3”.