Question

1. A consumer spends all of his income only on two goods, X and Y. His utility function is given by U=XY. The price of good X is $P and the price of good Y is $2. His income is $400.
   1. Derive the PCC (price consumption curve) of this consumer as the price of good X changes .
   2. Derive this consumer’s demand function for good X.
   3. As the price of good X falls, this consumer’s demand becomes less elastic. True or False? Explain.

Answer 1

Given utility function is U=XY, this represents a special case of Cobb-douglas production function

Budget (M) =XPx+YPy

$400=X$P+Y$2

Slope of Budget constraint=P/2

Slope of indifference curve=-Y/X

With utility maximization, Slope of Budget constraint=slope of indifference curve

P/2=-Y/X

(a) When the price of a good changes, the consumer would be either better off or worse off than before, depending upon whether the price falls or rises.

In other words, as a result of change in price of a good, his equilibrium position would lie at a higher indifference curve in case of the fall in price and at a lower indifference curve in case of the rise in price.

Price consumption curve (PCC) is sloping downward. Downward-sloping price consumption curve for good X means that as price of good X falls, the consumer purchases a larger quantity of good X and a smaller quantity of good Y. This is quite evident from the above figure. We obtain downward-sloping price consumption curve for good X when demand for it is elastic (i.e., price elasticity is greater than one). But downward sloping is one possible shape of price consumption curve. Price consumption curve can have other shapes also.

(b) Consumer's demand function for good X:

We know P/2=-Y/X

then Y=-XP/2

Substituting this into budget constraint,

Budget (M) =XPx+YPy

$400=X$P+Y$2

$400=X$P+(-XP/2)$2

$400=3X$P/4

X=(400*4)/3P=1600*P/3

Y=-XP/2=-(1600P/3)P/2=-800P2/3

U=XY

U=(1600*P/3)(-800P2/3)

(c)

Here upward- sloping price consumption curve is shown. Upward-sloping price consumption curve for X means that when the price of good X falls, the quantity demanded of both goods X and Y rises. We obtain the upward-sloping price consumption curve for good X when the demand for good is inelastic, (i.e., price elasticity is less than one).