Definition: the compensated demand curve is a demand curve that ignores the income effect of a price change, only taking into account the substitution effect. To do this, utility is held constant from the change in the price of the good. In this section, we will graphically derive the compensated demand curve from indifference curves and budget constraints by incorporating the substitution and income effects, and use the compensated demand curve to find the compensating variation.

• Let us consider a price increase for a normal good, a good whose demand increases as income increases. In Figure 7.e.1, assume that the price of Y (PY) is $1, and that the individual has an income of $100. The initial price of X (PX) is $1, so the individual’s initial budget constraint is therefore BC1, with a vertical intercept of 100, and a horizontal intercept of 100. The individual reaches his optimum (maximizes utility) at point A, where his initial budget constraint BC1 is tangent to the indifference curve IC1. Let’s say that at this point, he maximizes his utility by consuming 43 units of good X.

If PX increases from $1 to $2, his budget constraint will rotate inward until it reaches BC2 where there is now a horizontal intercept of 50. The individual now reaches his new optimum where the indifference curve IC2 is tangent to BC2 at the point B, where he maximizes his utility by consuming 18 units of good X. We can use these points to plot a demand curve for good X: According to Figure 7.e.1, when PX is $1, the individual maximizes utility at point A where he consumes 43 units of X. This information can be replotted on a curve showing the relationship between the price of X and the quantity of X consumed (figure 7.e.2). At a price of $1, the individual will consume 43 units of X, so the point A will replot on figure 7.e.2 as the point A’. Similarly at point B, at a price of $2, the individual will consume 18 units of X, so the point B will replot on figure 7.e.2 as the point B’. If we connect A’ and B’ together, we will get the ordinary demand curve for good X

In order to obtain the compensated demand curve, we must first observe 2 effects that take place as PX increases:

Substitution Effect: when Px increases from $1 to $2, X becomes relatively more expensive than Y, so the individual consumes less X. To show the substitution effect, we must hold the individual’s utility constant. To do this, we draw a budget constraint BC3 that is parallel to BC2 and shift it up until it is just tangent to a point on his original indifference curve (IC1). This occurs at point C, where the consumer is consuming 29 units of X. The substitution effect is the movement from point A to C

Income Effect: because Px has increased, the individual’s purchasing power has decreased, and thus has less money to spend on both X and Y. Because X is a normal good, the individual will consume more as his income increases. The individual will reach an optimum at point B where he will consume 18 units of X. The income effect is the movement from point C to B

To summarize,

Total effect = Substitution Effect + Income Effect

= A to C +C to B

We have already found the ordinary demand curve by replotting points A and B as points A’ and B’. In essence, this is the total effect of the increase in PX. Because the compensated demand curve assumes that utility is held constant, it only shows the substitution effect. Therefore, we simply have to replot points A and C. We have already determined that point A replots as A’ at a price of $1 and a quantity of 43. At point C, the individual consumes 29 units at a price of $2; so we can replot this point as point C’ on figure 7.e.2. If we connect these 2 points together, we get the compensated demand curve.

We can prove that good X is a normal good. One way to do it is to look at Figure 7.e.1 and notice that between points B and...