Question

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.

(4) Derive the PCC (price consumption curve) of this consumer as the price of good X changes .

(3) 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

(4) When the price of one of the two products changes, the consumer has to readjust and find a new equilibrium. If the price of a product falls, the consumer will move on to a higher indifference curve and vice versa.

Starting with a situation of consumer’s equilibrium, we assume that the price of X falls successively. As a result, the budget line will shift away from the point of origin on the X-axis and become flatter and flatter with every fall in the price of the product X.

The price effect can be defined as an effect of a change in price of a product on consumer’s equilibrium and on the quantity consumed of that product by the consumer, price of other product and the income of the consumer remaining the same.

In the above figure, When the price of the product X falls, the budget line shifts from point B on the X-axis to point B1 and becomes flatter. This implies that at a lower price, the consumer is capable of buying a larger quantity of X (OB1) than before (OB). Since the price of Y remains unchanged, there will be no change in point A on the Y-axis.

As such, the new budget line assumes the shape AB1 and the consumer finds his equilibrium at point S on IC2. At this point, the consumer opts for OX2 of X. Such an effect is also in accordance with the law of demand which implies a larger demand of the product at a lower price.

A further fall in the price of X will shift the budget line further away from the point of origin on the X-axis to point B2 forming a new budget line AB2 and the consumer’s equilibrium at point T on the IC3. The consumption of X, as a result, further increases to OX3. Thus, as the price of X continues to fall, the budget line will keep on moving away from the point of origin and the consumer’s equilib­rium will shift representing higher and higher consumption of X.

If all the points of equilibrium i.e., R, S, T, and V, are joined, the curve so formed is termed as price consumption curve (PCC). Thus, the PCC shows a relationship between changes in con­sumption as a result of the change in the price of one of the two products.

(3) The demand curve for this consumer could be derived through lagrangian multiplier,

Max L = XY + [400 - PX - 2Y]

Differentiating above equation with respect to X, Y and respectively, we get

X = 400/2P = 200/P

(3) As the price of good X falls, this consumer’s demand becomes less elastic only when there will be upward sloping PCC.

An upward sloping PCC indicates that a fall in the price of X will result in an increase in the consumption of both X and Y by the consumer. The price of Y remaining the same, an increase in consumption of Y means more expenditure on it. As such, consumer’s expenditure on product X will fall since the total expenditure of the consumer is given. As per the total expenditure method of the elasticity of demand, if the total expenditure declines following a fall in the price of the product, the demand will be less elastic.

when the PCC is horizontal, the demand will be unit elastic.

In the case of a backward sloping PCC, as in case of Giffen goods, the demand will be more elastic.