Question

Qin has the utility function U(x1, x2) = x1 + x1x2, where x1 is her consumption of good 1 and x2 is her consumption of good 2. The price of good 1 is p1, the price of good 2 is p2, and her income is M.

Setting the marginal rate of substitution equal to the price ratio yields this equation: p1/p2 = (1+x2)/(A+x1) where A is a number. What is A?

Suppose p1 = 11, p2 = 3 and M = 63. What quantity of good 2 will Qin demand?

Answer 1

Given:

Utility function of Qin, U (x1, x2) = x1 + x1x2

Budget constraint of Qin, p1x1 + p2x2 = M

dU (x1, x2)/ d(x1) = 1 + x2

dU (x1, x2)/ d(x2) = x1

MRS x1x2 = [dU (x1, x2)/ d(x1)] / [dU (x1, x2)/ d(x2)]

= (1 + x2)/ x1

Slope of budget constraint = p1/p2

Thus, at equilibrium,

(1 + x2)/ x1 = p1/p2

Now, comparing this to the given optimal equilibrium = p1/p2 = (1+x2)/(A+x1)

we find that A = 0

Now suppose p1 = 11, p2 = 3, and M = 63

Then, putting these values in budget constraint gives:11(x1) + 3(x2) = 63…………(1)

And, putting these values in equilibrium condition gives (1+x2)/x1 = 11/3

Or, x1 = 3 (1+x2) / 11

Now putting the value of x1 in (1), we find

11* 3 (1+x2)/11 + 3 (x2) = 63

Or, (1+x2) + x2 = 21

Or, 2(x2) = 20,

Or, x2 = 10

Hence Qin will demand 10 quantity of x2.