Compensated Demand Curve

The Compensated Demand Curve 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 constrai nt 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 t he 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 CIncome 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. and notice that between po ints B and C, as income increases, the consumption of good X increases, which fits the definition of a normal good. Another way is to look at the compensated demand curve and compare it with the ordinary demand curve. The compensated demand curve in figure 7. e. 2 is steeper than the ordinary demand curve. When this condition holds, good X is a normal good. We can also use the compensated demand curve to find the compensating variation. The compensating variation is the amount of money required to restore an individual to his original utility level when prices change.In figure 7. e. 2, it is represented by the area between the two prices, and left of the compensated demand curve – it is the sum of areas S and T. Meanwhile the change in consumer surplus is simply the area between the two prices and left of the ordinary demand curve – it is the area S ———————————————à ¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€œ †¢ Next, consider a price decrease for an inferior good, a good whose demand decreases as income increases.In Figure 7. e. 3, 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 $2, so the individual’s initial budget constraint is therefore BC1, with a vertical intercept of 100, and a horizontal intercept of 50. 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 17 units of good X.If PX decreases from $2 to $1, his budget constraint will rotate outward until it reaches BC2 where there i s now a horizontal intercept of 100. 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 28 units of good X. Using the same method as described in figure 7. e. 1 and figure 7. e. 2, we can replot A and B on figure 7. e. 3 as A’ and B’ on figure 7. e. 4. If we connect these points together, we will get the ordinary demand curve for good XIn order to obtain the compensated demand curve, we must first observe 2 effects that take place as PX increases: Substitution Effect: when Px decreases from $2 to $1, X becomes relatively cheaper than Y, so the individual will consume more 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 down until it is just tangent to a point on his original indifference curve (IC1). This occurs at point C, where the consumer is consuming 33 units of X.The substitution effect is the movement from point A to C Income Effect: Px has decreased, so the individual’s purchasing power has increased, and thus has more money to spend on both X and Y. Because X is an inferior good, the individual will consume less as his income increases. The individual will reach an optimum at point B where he will consume 28 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 Using the same method as described in figure 7. . 1 and figure 7. e. 2, we can replot A and C on figure 7. e. 3 as A’ and C’ on figure 7. e. 4. If we connect these points together, we will get the compensated demand curve for good X We can prove that good X is an inferior good. One way to do it is to look at Figure 7. e. 3 and notice that between points B and C, as income increases, the consumption of good X decreases, which fits the definition of an inferior good. Another way is to look at the compensated demand curve and compare it with the ordinary demand curve.The compensated demand curve in figure 7. e. 4 is flatter than the ordinary demand curve. When this condition holds, good X is an inferior good. Again, we can also use the compensated demand curve to find the compensating variation. It is the area between the two prices, and left of the compensated demand curve – it is the sum of areas S and T ——————————————————————————————————————————————– †¢ Let us now consider a price decrease for an extreme case: a giffen good.A giffen good violates the law of demand and results in an upward s loping demand curve. In Figure 7. e. 5, 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 50. 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 37 units of good X.If PX decreases from $2 to $1, his budget constraint will rotate outward until it reaches BC2 where there is now a horizontal intercept of 100. 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 30 units of good X. The total consumption of good X has actually decreased; let us decompose this. Using the same method as described in figure 7. e. 1 and figure 7. e. 2, we can replot A and B on figure 7. e. 5 as A’ and B’ on figure 7. e. 6.The shape of the ordinary demand curve for a giffen good is as follows: between the points A and B, it is upward sloping (known as the â€Å"Giffen Range†), and at any price above or below points A and B, respectively, the demand curve is downward sloping. This results in a backward-bending ordinary demand curve W In order to obtain the compensated demand curve, we must first observe 2 effects that take place as PX increases: Substitution Effect: when Px decreases from $2 to $1, X becomes relatively cheaper than Y, so the individual will consume more 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 down until it is just tangent to a point on his original indifference curve (IC1). This occurs at point C, where the consumer is consuming 47 units of X. The sub stitution effect is the movement from point A to C Income Effect: Px has decreased, so the individual’s purchasing power has increased, and thus has more money to spend on both X and Y. Because X is a giffen good, the individual will consume less as his income increases; also note that the income effect is stronger than the substitution effect.This results in the individual reaching an optimum at point B where he will consume 30 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 Using the same method as described in figure 7. e. 1 and figure 7. e. 2, we can replot A and C on figure 7. e. 5 as A’ and C’ on figure 7. e. 6. If we connect these points together, we will get the compensated for good X Note that the compensated demand curve is still downward sloping.This is because the substitution effect always works in one direction, while the income effect can work in both directions Study Questions 1) Redraw figure 7. e. 1 and figure 7. e. 2 for a decrease in the price of a normal good. Shade the area representing the compensation variation. 2) Redraw figure 7. e. 3 and figure 7. e. 4 for an increase in the price of an inferior good. Shade the area representing the compensation variation. 3) Redraw figure 7. e. 5 and figure 7. e. 6 for an increase in the price of a giffen good. Shade the area representing the compensation variation.