Question

Consider the utility function ? = min(?, ?). Graphically, decompose the effect of an increase in the price of x on demand for x into substitution and income effects. What is the relative importance of each effect?

Answer 1

Suppose the L shaped indifference curve IC1 is tangent to the budget line AB at point M with respect to which the initial consumption bundle is (X0,Y0). Now suppose the price of X increases and the budget line rotates inward from AB to AC and the new budget line becomes tangent to a new indifference curve IC2 at point O with respect to which the final consumption bundle will be (X1,Y1). Now the decline in the consumption of X from X0 to X1 can be decomposed into a substitution effect and an income effect. '

To understand the substitution effect, we consider the real income is fixed and hence the consumer will stay on the same indifference curve IC1 but as the price ratio have changed due to increase in the price of X, the slope of the budget line has changed and hence consumer will stay on the budget line with its slope equal to the slope of the new budget line AC. Hence we draw a hypothetical budget line DE which is tangent to IC1 at point M and is parallel to the new budget line DE. Now as in case of perfect complement utility function, consumer has to consume X and Y in fixed proportion, hence here consumer cannot substitute one good for other as the consumer consumes both good in a specific ratio. Hence here the hypothetical budget line is tangent to IC1 at initial tangency point M and hence here substitution effect is zero.

Now to understand the income effect, we consider real income as variable. Now, as price of X increases, consumer's real income or purchasing power M/Px falls and consumer feels poorer and the budget line shifts leftward from DE to AC which becomes tangent to a new indifference curve IC2 at new bundle (X1,Y1). Hence the decline in the consumption of X from X1 to X2 is due to the income effect. Hence in this case the total effect of this price increase of X is due to the income effect only as the substitution effect here is zero.