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  Income and Substitution Effects

Income and Substitution Effects:

A price increase of single good, holding tastes, prices of other goods and money income constant, has two effects on demand for the good. The first effect is the substitution effect; i.e., if the price has increased the good has become more expensive relative to other goods and the consumer wants to substitute away from this good towards nearby substitute goods. The second effect is an income effect. The price increase has decreased the consumer’s real income, or purchasing power, since money income is constant as well as all other prices. If the good is a normal good demand will decrease, if it is an inferior good, demand will increase. For normal goods, the two effects go in the same direction, towards a decrease in the quantity demanded for a price increase, and towards an increase in the case of a price decrease. If the good is an inferior good, the two effects go in different directions and the overall effect of a price change cannot be determined a priori.

Figure shows the effect of a decrease in the price of good 1. At the initial prices and money income, the consumer maximizes her income at point A on the budget line named “budget set 0”, and she gets utility U0. Now the price of good 1 drops and her budget line “swings” out to “budget set 2”; she will choose point C and get the final utility U1. However, the move from A to C can be divided into a move from A to B′ and from B′ to C. The first move is the substitution effect and the second is the income effect.

“Budget set 1” is a hypothetical budget line which would be valid if the government, for example, stepped in and taxed the consumer’s income so that she would reach utility level U0 at the new price on good 1. She would in that case be as well off as she was initially. We call this type of income change “income compensation”, even though in this case it is a tax, which is a negative compensation. However, the main point is that we take budget set 2 and shift it inwards until it just touches the initial indifference curve. In reality, no such compensation (or tax) is usually applied and the consumer ends up at point C. But we can theoretically divide the total change in the quantity demanded into these two effects.

Example:

Assume that the consumer’s initial income is: Y0 = 10, the initial goods prices are: p01 = p02 = 1, the consumer’s utility function is: U(Q) = Q1 . Q2. In this (Cobb-Douglas) case, the demand functions are: Q1(p1, Y ) = Y/(2.p1) and Q2(p2, Y ) = Y/(2.p2) Putting in the numbers this implies that the initial bundle is: (QA1 , QA2 ) = (5, 5) and the initial utility: U0 = 5 . 5 = 25. The price of good 1 now rises to, p11 = 2, to find the necessary compensation for the price increase, we should find a bundle which gives the same utility as the initial bundle, i.e.: QB1 . QB2 = 25. How can we find (QB1 ,QB2)? Well, we know the form of the demand functions and the new prices, but we do not know the new income Y1H. But we can actually solve for it by substituting the demand functions into the utility function and set the utility function equal to U0 = 25 :

1329_income & substitution effects1.jpg

1980_income & substitution effects2.jpg

 The income needed to get the consumer back to the original utility level is Y1H = 14.142, and the necessary compensation is: ?YH = 14.142 − 10 = 4.142.

We can note in passing that this type of compensation is called Hicksian compensation (after Sir John Hicks, hence the super- and subscripts H above). Another type of compensation is called Slutsky compensation. In that case the consumer receives enough income after a price increase (or is taxed after a price decrease) so that she can afford to buy her initial bundle.

That corresponds to the dashed budget line and the choice of bundle B in Figure. In general, if the consumer gets Slutsky compensation after a price increase she will be overcompensated, because she will be substituting away from the more expensive good and she will be able to reach a higher utility level than U0. In the case of a price decrease (as is illustrated in Figure) the government tax is less than with Hicks compensation and the consumer is therefore better off with Slutsky compensation.

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