Question

Question 1 (20 points). The utility function of the consumer is u(x1, x2) = x1 + x2. a) Let pı = 2 ,P2 = 20 and m = 24. Calculate the optimal quantity demanded of good 1 and 2. (7 points) b) Let p1 = 1,P2 = 4 and m = 100. Calculate the optimal quantity demanded of good 1 and 2. (6 points) c) Let P1 = 1, p2 = 4 and m = 4. Compared to point b), by how much would the consumer decrease the quantity demanded of good 1 and good 2? (7 points)

Answer 1

Look at the adjoined sheets for the solution:

The budget constraint is of the form:

p1* x1 + p2 * x2 = M, where p1 and p2 are prices of x1 and x2 respectively, and x1 and x2 are the quantities and M = income.

A consumer maximizes his utility when the MRS is equal to the price ratio, because at this point the slope of the indifference curve which is equal to the ratio of marginal utility of good x1 and marginal utility of good x2 is equal to the slope of the budget constraint i.e. price ratio, i.e. dividing price of x2 by price of x1. The absolute values are taken.

At this point of tangency, the points are optimal, and when we substitute it into the budget constraint, we find the optimal quantities of the two goods.

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So, we see that in part c) the quantity demanded of good 1 does not change, however, as the M decreases from 100 to 4, the quantty demanded of good 2 drops considerably from 24.99 to 0.996. This may be because p2>p1, so the consumer would want to consume much lesser quantity of the relatively expensive good as his income has reduced.