Slutsky Compensated Demand Curve (With Diagram) Theorem and Derivation of Demand Curve

The compensated demand curve shows the quantity of a good which a consumer would buy if he is income-compensated for a change in the price of that good. In other words, the compensated demand curve for a good is a curve that shows how much quantity would be purchased at the changed price by the consumer if the income effect is eliminated. The compensated demand curve can be explained in terms of both the Hicks and Slutsky approaches to the substitution effect. The two-storey Figure 45(A) illustrates the construction of the Hicks and Slutsky compensated demand curves and the uncompensated (or ordinary or Marshallian) demand curve.

The upper portion of the figure shows the substitution effects of the Hicks and Slutsky analyses and the combined price effect. 

The lower portion of the figure shows the derivation of the Hicks and Slutsky compensated demand curves and the ordinary demand curve. First consider the lower diagram (B) where the price of good X is taken on the vertical axis. Point P is an arbitrary point on this axis which shows the price of X when the budget line is PQ in the upper diagram. The fall in the price of X as shown by the budget line PQ1 is reflected in point P1 in the lower diagram.

The Marshallian Uncompensated Demand Curve:

First we explain the derivation of the Marshallian uncompensated demand curve. Suppose the initial equilibrium of the consumer is at point R where the budget line PQ is tangent to the indifference curve I₁, and OA of good X is bought by the consumer in the tipper diagram.

Let the price of X fall. As a result, the budget line PQ extends to PQ₃ and the consumer is at a higher point of equilibrium T on the indifference curve I₃ .The movement from R to T is the price effect which includes both the substitution effect and the income effect. This is shown by the D₃ curve in the lower portion of the figure. This is the uncompensated (or ordinary or Marshallian) demand curve which shows that when the price of good X falls from P to P1 its quantity demanded increases from OA to OD.

The Hicksian Compensated Demand Curve:

Since the compensated demand curve is based on the substitution effect of a change in the price of good X, we carry the above analysis further and derive the Hicks substitution effect. Let us take away the increase in the real income of the consumer due to the fall in the price of X equal to PM of good Y and Q₁ N of X good by drawing a compensated budget line MN parallel to the budget line PQ₁.

This line MN is tangent to the original indifference curve I₁, at point H. The movement from point R to H on the I₁, curve is the substitution effect which traces out the demand curve D₁, in the lower portion of the figure when the demand for X increases from OA to OB with the fall in its price from P to P1.

The Slutsky Compensated Demand Curve:

In order to derive the Slutsky substitution effect, let us take away the increase in the apparent real income of the consumer equal to PM_x of Y and Q₁N₁ of X by drawing the Slutsky compensated budget line M₁N₁, parallel to PQ which passes through the original point R on the I₁, curve where he will buy the same quantity OA of X. But since the price of X has fallen, he will buy more of it so that he moves to point S on the higher indifference curve I₂, which is tangent to the budget line M₁N₁ Thus the movement from R to S traces out the Slutsky compensated demand curve D₂ in the lower part of the figure.

This curve shows that with the fall in the price of good X from P to P₁ its demand increases from OA to OB.

A perusal of the compensated demand curve D₁ of Hicks and D₂ of Slutsky shows that the curve D₂ is more elastic than D₁ .This is because the total expenditure on the purchase of good X is greater in the Slutsky approach than in the Hicks approach. While the conventional demand curves D₃ is more elastic than even the Slutsky demand curve D₂.

Another important point to be noted is that the compensated demand curve, whether of Hicks or Slutsky, always slopes downward because it is so drawn that the substitution effect only is in operation and the income effect is altogether eliminated through compensating variation in income. But the ordinary demand curve may or may not slope downward. In the case of the ordinary demand curve like D, both the substitution and income effects are in operation and they explain the downward slope of the curve.

In case X is an inferior good, the ordinary demand curve will slope downward but will be elastic than the compensated demand curves D₁ and D₂ because the substitution effect is stronger than the income effect in the case of the ordinary demand curve. But if X happens to be a Giffen good, the ordinary demand curve will slope from left to right upward i.e. it will have a positive slope because the income effect is stronger than the substitution effect. On the other hand, the compensated demand curves will have a negative slope because they are not affected by the income effect.

The Slutsky Substitution Effect – Explained

The concept of substitution effect put forward by J.R. Hicks. There is another important version of substitution effect put forward by E. Slutsky. The treatment of the substitution effect in these two versions has a significant difference. Since Slutsky substitution effect has an important empirical and practical use, we explain below Slutsky’s version of substitution effect in some detail.

In Slutsky’s version of substitution effect when the price of good changes and consumer’s real income or purchasing power increases, the income of the consumer is changed by the amount equal to the change in its purchasing power which occurs as a result of the price change. His purchasing power changes by the amount equal to the change in the price multiplied by the number of units of the good which the individual used to buy at the old price. In other words, in Slutsk’s approach, income is reduced or increased (as the case may be), by the amount which leaves the consumer to be just able to purchase the same combination of goods, if he so desires, which he was having at the old price.

That is, the income is changed by the difference between the cost of the amount of good X purchased at the old price and the cost of purchasing the same quantity if X at the new price. Income is then said to be changed by the cost difference. Thus, in Slutsky substitution effect, income is reduced or increased not by compensating variation as in case of the Hicksian substitution effect but by the cost difference.

Slutsky Substitution Effect for a fall in Price:

Slutsky substitution effect is illustrated in Fig. 9B.1 With a given money income and the given prices of two goods as represented by the price line PL, the consumer is in equi­librium at Q on the indifference curve IC1 buying OM of X and ON of Y. Now suppose that price of X falls, price of Y and money income of the consumer remaining un­changed. As a result of this fall in price of X, the price line will shift to PL’ and the real income or the purchasing power of the con­sumer will increase. Now, in order to find out the Slutsky substitution effect, consumer’s money income must be reduced by the cost- difference or, in other words, by the amount which will leave him to be just able to pur­chase the old combination Q, if he so desires.

This appendix is meant for B.A (Hons.) and B.Com. (Honours) classes and should therefore be omitted by B.A. (Pass) Course students. For this, a price line GH parallel to PL’ has been drawn which passes through the point Q. It means that income equal to PG in terms of Y or L’H in terms of X has been taken away from the consumer and as a result he can buy the combination Q, if he so desires, since Q also lies on the price line GH.

Actually, he will not now buy the combination Q since X has now become relatively cheaper and Y has become relatively dearer than before. The change in relative prices will induce the consumer to rearrange his purchases of X and Y. He will substitute X for Y. But in this Slutsky substitution effect, he will not move along the same indifference curve IC₁, since the price line GH, on which the consumer has to remain due to the new price-income circumstances is nowhere tangent to the indif­ference curve IC₂.

The price line GH is tangent to the indifference curve IC₂ at point S. Therefore, the consumer will now be in equilibrium at a point S on a higher indifference curve IC₂. This move­ment from Q to S represents Slutsky substitution effect according to which the consumer moves not on the same indifference curve, but from one indifference curve to another. A noteworthy point is that movement from Q to S as a result of Slutsky substitution effect is due to the change in relative prices alone, since the effect due to the gain in the purchasing power has been eliminated by making a reduction in money income equal to the cost-difference. At S, the consumer is buying OK of X and O W of Y; MK of X has been substituted for AW of Y. Therefore, Slutsky substitution effect on X is the increase in its quantity purchased by MK and Slutsky substitution effect on Y is the decrease in its quantity purchased by NW.

Slutsky Substitution Effect for a Rise in Price:

We have graphically explained above Slutsky substitution effect for a fall in price of good X. It will be instructive to explain it also for a rise in price of X. This is demonstrated in Fig. 9B.2. Initially, the consumer is in equilibrium at point Q on the indifference curve IC1, prices of the two goods and his money income being given. Now sup­pose that price of good X rises, price of Y remaining unchanged. As a result of the rise in price of X, bud­get line will shift downward to PL” and consumer’s real income or purchasing power of his given money income will fall. Further, with this price change, good X has become relatively dearer and good Y relatively cheaper than before.

In order to find out Slutsky sub­stitution effect in this present case, consumer’s money income must be increased by the ‘cost-difference’ cre­ated by the price change to compensate him for the rise in price of X. In other words, his money income must be increased to the extent which is just large enough to permit him to purchase the old combina­tion Q, if he so desires, which he was buying before. For this, a budget line GH has been drawn which passes through point Q. It will be evident from the figure that, PG (in terms of Y) or L” H (in terms of X) represents ‘cost difference’ in this case. With budget line GH he can buy if he so desires the combination Q, which he was buying at the previous price of X.

But actually he will not buy combination Q, since on budget line GH, X is relatively dearer than before, he will therefore replace some X by Y (i.e., he will substitute of Y for X). As is shown Fig. 9B.2, with budget line GH he is in equilibrium position at S on a higher indifference curve of IC2 and is buying OK of X and OW of Y. MK of X has been replaced by AW of Y. Movement from point Q to S is the result of Slutsky substitution effect; the effect due to the fall in purchasing power has been cancelled by giving him money equal to PG of Y or L” H of X. In this present case of stipulated rise in price of X, Slutsky substitution effect on X is the fall in its quantity bi ought by MK and Slutsky substitution effect an Y is the increase in its quantity brought by NW.

From the above analysis it is clear that whereas Hicks-Allen substitution effect takes place on the same indifference curve, Slutsky substitution effect involves the movement from one indiffer­ence curve to another curve, a higher one. The difference between the two versions of the substitu­tion effect arises solely due to the magnitude of money income by which income is reduced or increased to compensate for the change in income. The Hicksian approach just restores to the con­sumer his initial level of satisfaction, whereas the Slutsky approach “over-compensates” the con­sumer by putting him on a higher indifference curve.

Merits and Demerits of Hicksian and Slutsky Methods:

Prof. J.R. Hicks points out that the method of adjusting the level of money income by the compensating variation has the merit that on this interpretation, the substitution effect measures the effect of change in relative price, with real income constant, the income effect measures the, effect of the change in real income. Thus the analysis which is based upon the compensating variation is a resolution of the price change into two fundamental economic ‘directions’, we shall not encounter a more fundamental distinction upon any other route. But Slutsky method has a distinct advantage in that it is easier to find out the amount of income equal to the ‘cost difference’ by which income of the consumer is to be adjusted. On the other hand, it is not so easy to know the compensating variation in income.

Thus, the cost-difference method has the advantage of being dependent on observable market data, while for knowing the amount of compensating variation in income, knowledge of indifference curves (that is, tastes and preferences of the consumer between various combinations of goods is required. It follows from what has been said above that both the cost-difference and compensating varia­tion methods have their own merits. While the law of demand can be easily and adequately estab­lished by the method of cost-difference, method of compensating variation is very useful for the analysis of consumer’s surplus and welfare economics. With the help of the cost-difference, the income effect can be easily separated from the substitution effect but the substitution effect so found out involves some gain in real income (since it causes movement from a lower indifference curve to a higher indifference curve). It is because of this that, on cost-difference method, substitution effect is not a theoretically distinct concept.

A. Numerical Examples:

Let us explain the concept of cost-difference and Slutsky substitu­tion effect with a numerical example stated below:

When the price of petrol is Rs. 20.00 per litre, Amit consumes 1,000 litres per year. The price of petrol rises to Rs. 25.00 per litre. Calculate the cost difference equal to which the Government should give him extra money income per year to compensate him for the rise in price of petrol. Will Amit be better off or worse off after the price rise plus the cash compensation equal to the cost difference than he was before? What will happen to petrol consumption?

As explained above, the cost difference is equal to AP.Q where AP stands for the change in price of a good and Q stands for the quantity of commodity he was consuming prior to the change in price. Thus, in our above example.

P = Rs. 25 – 20 = Rs. 5

Q = 1,000 litres per year

Cost-difference = P.Q

= Rs. 5 x 1,000 =Rs. 5,000.

Now, with higher price of petrol of Rs. 25.00 per litre and cash compensation of Rs. 5,000 equal to the cost difference he can buy, if he so desires, the original quantity of 1,000 litres of petrol per year. However, he may not buy this original quantity of petrol in the new price-income situation it his satisfaction is maximum at some other point. Consider Figure 9B.3. Where we measure petrol on the X- axis and money income representing other goods on the F-axis.Suppose, BL1 is the initial budget line when price of petrol is Rs. 20.00 per litre and consumer is in equilibrium at point Q on the indifference curve IC1 where he is consuming 1,000 litres of petrol per year. Now, with the rise in price of petrol to Rs. 25.00 per litre, suppose the budget line shifts to BL2.

Now, if to com­pensate for the rise in price, his, money income is raised by Rs. 5,000, that is, equal to the cost difference, the budget line shifts in a parallel manner to the left so that it reaches the position GH which passes through the original point of consumption Q. A glance at Fig. 9B.3, will reveal that the consumer with higher price of petrol and having received monetary compensation equal to the cost difference of Rs. 5,000 will not be in equilibrium at the original point Q and instead he will maximising his satisfac­tion in the new situation at point S on a higher indifference curve IC2 where his con­sumption of petrol has decreased to ON litres (that is, the decrease in consumption of petrol by MN is the Slutsky substitution effect.) Since, with the rise in price and simultaneous increase in his income equal to the cost-difference has enabled him to attain a higher indifference curve, he has become better off than before the rise in price.

Price Effect Broken Up into Income and Substitution Effects: Slutsky Method:

In our discussion of substitution effect we explained that Slutsky presented a slightly different version of the substitution and income effects of a price change from the Hicksian one. In fact it was Slutsky who first of all divided the price effect into income and substitution ef­fects. His way of breaking up the price effect is shown in Fig. 9B.4. With a certain price- income situation, the consumer is in equilib­rium at Q on indifference curve IC1.

With a fall in price of X, other things remaining the same, budget line shifts to PL2. With budget line PL2, the consumer would now be in equilibrium at R on the indifference curve IC3. This move­ment from Q to R represents the price effect. As a result of this he buys MN quantity of good X more than before. Now, in order to find out the substitution effect his money income be reduced by such an amount that he can buy, if he so desires, the old combination Q.

Thus, a line AB, which is parallel to PL2, has been so drawn that it passes through point Q. Thus PA in terms of good Y represents the cost difference. With budget line AB, the consumer can have combination Q if he so desires, but actually he will not buy combination Q because X is now relatively cheaper than before. It will pay him to substitute X for Y. With budget line AB he is in equilibrium at S on indifference curve IC2. The movement from Q to S represents Slutsky substitution effect which induces the consumer to buy MH quantity more of good X. If now the money taken away from him is restored to him, he will move from S on indifference curve IC2 to R on indifference curve IC3.

This movement from S to R represents income effect. Thus, movement from Q to R as a result of price effect can be divided into two steps. First, movement from Q to S as a result of substitution effect and secondly, movement from S to R as a result of income effect. It may be pointed out here again that, unlike the Hicksian method, Slutsky substitution effect causes movement from a lower indifference curve to a higher one. While separately discussing substitution effect above, we pointed out the merits and demerits of the Hicksian and Slutskian methods of breaking up the price effect.

Slutsky Equation:

We have graphically shown above how the effect of change in price of a good can be broken up into its two component parts, namely, substitution effect and income effect. The decomposition of price effect into its two components can be derived and expressed mathematically. Suppose price of good X falls, its substitution effect on quantity demanded of the good arises due to substitution of the relatively cheaper good X for the now relatively dearer good Y and as a result in the Hicksian method the consumer moves along the same indifference curve so that his level of utility remains constant.

The overall effect of change in its own price on the quantity demanded can be expressed as dqx/dpx and the substitution effect can be expressed ∂px/∂px|u=u. The term ∂qx/∂px|u=u shows change in quantity demanded resulting from a relative change in price of X while utility or satisfaction of the consumer remains constant. However, expressing income effect of the price change mathematically is rather a ticklish affair. Suppose a unit change in income (∂ I) causes a (∂ qx) change in quantity demanded of the good. This can be written as ∂qx/∂I. But how much income changes due to a change in price of the good is determined by how much quantity of the good (qx) the consumer was purchasing on the one hand and change in price of the good (∂ px) that has taken place on the other. The change in income due to a change in price can be measured by qx (∂ px). How much this change in income will affect the quantity demanded of the good X is determined by ∂qx/∂I which shows the effect of a unit change in income on the quantity demanded of the good X.

Thus the overall effect of change in price of the good X on its quantity demanded can be expressed by the following equation which is generally called Slutsky equation because it was Russian economist E. Slutsky who first of all divided the price effect into substitution effect and income effect.

∂qx/∂px = ∂qx/∂px|u=u + qx .∂px .∂qx/∂I

The first term on the right hand side of the equation represents the substitution affect obtained after income of the consumer has been adjusted to keep his level of utility constant. The second term on the right hand side of the equation shows the income effect of the fall in price of the good. The term qx. ∂ Px measures the increase in income or purchasing power caused by the fall in price and ∂qx/∂I measures the change in quantity demanded resulting from a unit increase in income (I). Therefore, income affect of the price change is given by qx .∂px .∂qx/∂I. Since the fall in price increases income or purchasing power of the consumer which in case of normal goods leads to the increase in quantity demanded of the good, sign of the income affect has been taken to be positive.

Further, a point needs to be clarified. In the above analysis of Slutsky equation, we have con­sidered the substitution effect when with a change in price, the consumer is so compensated as to keep his real income or purchasing power constant. In obtaining Slutsky substitution effect, income of the consumer is adjusted to keep his purchasing power (i. e. real income) constant so that he could buy the original combination of goods if he so desires. On the other hand, in the Hicksian substitu­tion effect, with a change in price of a good money income with the consumer is so adjusted that his satisfaction remains constant.

In fact, Hicks interprets real income in terms of satisfaction obtained by a consumer. This difference was later emphasised by J.R. Hicks, but since it was Slutsky who first of all split up the price effect into substitution effect and income effect the above equation is popu­larly known as Slutsky equation. It is proper to call it Slutsky-Hicks equation. An important result follows from the Slutsky equation. If the commodity is a normal good, then ∂qx/∂I is positive by definition. It follows that a fall in price will lead to the increase in income causing increase in quantity demanded of the good and therefore the expression for income effect of the price change qx .∂px × (∂qx/∂I) is taken to be positive in the Slutsky equation (i) above. Besides, since the substitution effect is always negative, a fall in the relative price of a good will cause the increase in its quantity demanded. Therefore, Slutsky equation tells us that when commodity X is normal, the price effect dqx/dpx is necessarily negative implying that fall in price will cause quantity demanded of the good to increase. Thus, in case of normal goods both the substitution effect and income effect work in the same direction and reinforce each other. Thus in case of normal goods a fall in price of a commodity leads to the increase in quantity demanded due to both the substitution effect and income effect.

On the other hand, if price of the commodity rises, then due to the negative substitution effect, the consumer will buy less of the good, his pruchasing power remaining the same. Therefore, in case of rise in price of a good, the first term in the right side of Slutsky equation, namely, ∂qx/∂px| u=u, will have a negative sign. Further, rise in price of a good causes income of the consumer to fall, and income effect will lead to the decrease in quantity demand of good and therefore, the second term (qx .∂px .∂qx/∂I) on the right hand side of the equation will have a negative sign in case of normal goods. Thus, in case of rise in price of a good, both the substitution effect and also income effect (if it is a normal good) will work in the same direction to reduce the quantity demanded of the good whose price rises. The second important conclusion which follows from Slutsky equation is that as the quantity of commodity (qx) consumed becomes smaller and smaller, the income effect of the price change will becomes smaller and smaller. Thus, if the quantity consumed of a commodity is very small, then the income effect is not very significant.