# Common sense, and the model that we have used, suggest that an increase in p means a decrease in demand, but this is not the only possibility.

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1 Lecture 6: Income and Substitution E ects c 2009 Je rey A. Miron Outline 1. Introduction 2. The Substitution E ect 3. The Income E ect 4. The Sign of the Substitution E ect 5. The Total Change in Demand 6. Examples 7. Another Substitution E ect 8. Compensated Demand Curves 9. OPTIONAL: Calculus Derivation of the Slutsky Decomposition 1 Introduction The single most important issue in consumer theory is how demand responds to changes in the economic environment, such as changes in the prices of the goods. Common sense, and the model that we have used, suggest that an increase in p means a decrease in demand, but this is not the only possibility. Recall that, for a Gi en good, an increase in price can mean an increase in demand. The classic Gi en good example potatoes in Ireland might seem somewhat special. There are other settings, however, where these perverse e ects are plausible and natural. These include: 1) the e ect of wage rates on labor supply 2) the e ect of interest rates on savings 1

2 In all these cases, a similar mechanism is at work. And, in all cases, a change in the relevant price can have ambiguous e ects. Think about this intuitively for a moment: if my wage is higher, I will want to work more; but, because I feel richer, I might want to work less. Similarly, if the interest rate is higher, I will want to save more, but the higher interest rates means the return on savings is higher, so perhaps I will want to save less. A related issue, that arises even when the direction of the price e ect is unambiguous, is understanding when we would expect the price e ect to be large or small. The next section examines in more detail exactly how price should a ect demand. We will do it rst with graphs, then re-do a bit of it with equations. 2 The Substitution E ect The reason for the ambiguity in the e ect of price on demand is that a price change really consists of two separate e ects. First, a price change causes a substitution e ect. If good 1 becomes cheaper, for example, you have to give up less of good 2 to get some of good 1. You face a di erent rate of exchange of good 1 for good 2 in the marketplace. Second, there is an income e ect. Because the price of good 1 is lower, you can buy more of it with a given amount of income. The purchasing power of your existing dollars has increased. These are rough de nitions for the moment. exactly what is going on. Let s look in more detail to see We will break the price movement into two steps. First, we will let relative prices change, and adjust money income to keep something called purchasing power constant. 2

3 Then, we will let purchasing power adjust while holding relative prices constant. To see this, consider the graph below. This graph shows the e ect of a decline in p 1. 3

4 Graph: The Substitution E ect: Pivot and Shift x A B C p1 goes down x1s goes up x1 Point A is the original optimizing bundle; it lies on the original budget constraint. The decrease in the price of good 1 rotates the budget line outward, holding the x 2 intercept xed. Point B is the bundle this consumer would choose if, in response to the decline in the price of good 1, we reduced income just enough so that the consumer could just a ord the original consumption bundle, but now faced relative prices determined by the new p 1. The "pivot" is a movement where the slope of the budget line changes, yet purchasing power stays constant. Point C is the bundle the consumer will choose if we hold the new ratio of p 1 to p 2 constant but add back the income we hypothetically took away from the consumer in going from A to B. The "shift" is a movement where the slope of the budget line stays constant and the purchasing power changes. 4

5 This gets us to the new budget line, and C is the point the consumer will choose given the new p 1. This decomposition is hypothetical; in reality, a consumer only observes a change in the price and chooses a new bundle in response. The pivoted and shifted budget lines, however, have meaningful economic interpretations that we can examine with equations and graphs. The pivoted line has same slope (tradeo between good 1 and good 2) as the nal budget line. The total money income implied by this pivoted line is di erent from that of the original budget line. However, purchasing power has not changed; the consumer can still buy original bundle. We can, in fact, calculate how much we have to decrease money income to keep the old bundle just a ordable. Let m 0 be the amount necessary to just make the original consumption bundle just a ordable. Since (x 1 ; x 2 ) is a ordable at both (p 1 ; p 2 ; m) and (p 0 1; p 2 ; m 0 ), we have m 0 = p 0 1x 1 + p 2 x 2 m = p 1 x 1 + p 2 x 2 Subtracting the second from the rst gives m 0 m = x 1 (p 0 1 p 1 ) This says that the reduction in money income necessary to make the old bundle just a ordable at the new prices is simply the original amount of consumption of good 1 times the change in prices. Let and p 1 = p 0 1 p 1 5

6 m = m 0 m So, m = x 1 p 1 Note that the change in income and the change in price always move in the same direction (in this case, negative). Now, note that although we have worked out how much we would have to reduce income to keep purchasing power constant, the consumer does not usually want to stick with the original consumption bundle. Remember our original graph: 6

7 Graph: Substitution and Income E ects x A B C p1 goes down x1s goes up x1 On the pivoted budget line that goes through the original bundle, the consumer would normally move to a di erent point, such as B. This is the substitution e ect. This shows how the consumer would respond to a change in relative price, assuming we keep purchasing power constant. Using our equations, the substitution e ect, x s 1, is the change in demand for good 1 when p 1 changes to p 0 1, and m changes to m 0 : x s 1 = x 1 (p 0 1; m 0 ) x 1 (p 1 ; m) To determine the substitution e ect, we must use the consumer s demand function to calculate the optimal choices at (p 0 1; m 0 ) and (p 1 ; m). The substitution e ect is also known as the change in compensated demand. This name explicitly represents the fact that the consumer is compensated for a price rise/fall by adjusting income enough to purchase the old consumption bundle. 7

8 3 The Income E ect We turn next to the second component of the price adjustment the shift movement. This is straightforward: a parallel shift is the movement that occurs when income changes while relative prices stay xed. This is called the income e ect. We change the consumer s income from m 0 to m, keeping prices at p 0 1; p 2. In the previous gure, this moves is from B to C. More precisely, the income e ect, x n 1 is the change in demand for good 1 when we change income from m 0 to m, holding the price of good 1 xed at p 0 1: x n 1 = x 1 (p 0 1; m) x 1 (p 0 1; m 0 ) The income e ect can operate in either direction, as noted before. Consider what happens for a normal versus an inferior good. 4 Sign of the Substitution E ect The sign of the substitution e ect always has to be negative (i.e., the opposite direction of the change in price). That is, if price goes down, the substitution e ect must result in quantity demanded going up. Why is this? Let s go back to the previous gure. 8

9 Graph: Why Substitution E ect Must be Negative x substitution effect must be negative because the consumer could have chosen the shaded portion at the intial price 6 5 A x1 Consider the points on the pivoted budget line where the amount of good 1 consumed is less than at A. These bundles (with lower amounts of good 1) were all a ordable at the old prices, but they were not chosen. Instead, A was chosen. If the consumer always chooses the best bundle, then A must be preferred to all of the bundles on the part of the pivoted line that lies inside the original budget set. This means that the optimal choice must not be one of the bundles that lies underneath the original budget line. The optimal choice on the pivoted line would have to be either A, or some point to the right of A. Thus, demand for x 1 goes up. 9

10 5 The Total Change in Demand It is useful to develop an equation that puts these results together. The total change in demand resulting from a price change is x 1 = x 1 (p 0 1; m) x 1 (p 1 ; m) That is, we change p and hold m constant. This total change can be broken into two pieces: or x 1 = x s 1 + x n 1 x 1 (p 0 1; m) x 1 (p 1 ; m) = [x 1 (p 0 1; m 0 ) x 1 (p 1 ; m)] + [x 1 (p 0 1; m) x 1 (p 0 1; m 0 )] In words, this equation says that the total change in demand equals the subtitution e ect plus the income e ect. This equation is called the Slutsky Identity or the Slutsky Decomposition. Note that this equation is true by construction. We just added and subtracted the rst and fourth terms on the right hand side. The real content of this equation comes in the interpretation of the two di erent terms, the substitution and income e ects. We can now use what we know about the signs of the pieces to determine the sign/size of the total e ect. The substitution e ect must always be negative, while the income e ect can be positive or negative. Thus, the total e ect can go either way. For normal goods, the total e ect of an increase in price must be negative (since the substitution and income e ects are both negative). For an inferior good, though, the two e ects go in opposite directions. 10

11 So, the total e ect could be perverse in that an increase in price could lead to an increase in demand. Thus, the Gi en good case can only arise for an inferior good; these two concepts are related. A Gi en good implies an inferior good; however, an inferior good does not necessarily imply a Gi en good. 6 Examples of Income and Substitution E ects 6.1 Perfect Complements 11

12 Graph: Slutsky Decomposition for Perfect Complements x the pivot has no effect x1 The solid line is the original budget line. Pivoting this line to re ect the lower price of good 1 has no e ect on the optimal choices, so the substitution e ect is zero. The entire e ect of the decrease in price is the income e ect. 6.2 Perfect Substitutes 12

13 Graph: Slutsky Decomposition for Perfect Substitutes x IC as the budget line gets flatter, there is no effect until it gets flat enough to switch choice to only x x1 With perfect substitutes, the consumer chooses to consume only one of the two goods depending on the slope of the budget line versus the slope of the ICs. If the consumer was already consuming all good 1, then the fall in the price of good 1 has no e ect. If the consumer was consuming all good 2, the fall in the price has no e ect until the budget line gets at enough so that the consumer switches to only good 1. In this case the entire e ect is the substitution e ect. 6.3 Quaslinear Preferences Graph: Slutsky Decomposition for Quasilinear Preferences 13

14 y A the key point is that point C lies directly above point B C B x The entire e ect is the substitution e ect, so the income e ect on the demand for good 1 is zero. 7 The Total Change in Demand: Review We have derived several expressions. as The total change in demand due to a change in price from p to p 0 can be written x 1 = x 1 (p 0 1; m) x 1 (p 1 ; m) That is, we change p and hold m constant. This total change can be broken into two pieces: x 1 = x s 1 + x n 1 or x 1 (p 0 1; m) x 1 (p 1 ; m) = [x 1 (p 0 1; m 0 ) x 1 (p 1 ; m)] + [x 1 (p 0 1; m) x 1 (p 0 1; m 0 )] 14

15 We have shown that the rst term has to have the opposite sign from that of the price change. We also know that the second term can have either the same or opposite sign. If good 1 is a normal good, then the second term is the opposite sign. That is, making p 1 lower makes x 1 cheaper; this provides more purchasing power, and therefore means more demand for x 1 than would occur at the purchasing-powerconstant income, m 0. So, when we write that the substitution e ect is negative, we mean that it is always the opposite sign from a change in p 1. The e ect can be either an increase or a decrease, depending on whether the change in p is positive or negative. 15

16 Graph: Another Graph of Slutsky Decomposition x m/p1 m/p1' m'/p1' x1 8 Another Substitution E ect There are actually several substitution e ects. The one we have examined so far is the Slutsky substitution e ect. Another one is called the Hicks substitution e ect. Suppose that instead of pivoting the budget line around the original consumption bundle, we roll the budget line around the indi erence curve through the original consumption bundle: 16

17 Graph: Hicks Substitution E ect x A p1 goes down A to B is the Hicks substitution effect C 4 3 B x1 Thus, the Hicks approach keeps utility constant rather than keeping purchasing power constant. This approach gives the consumer just enough income to get back to the old indi erence curve. The Hicks substitution e ect must be negative. The proof can be seen by revealed preference (see the algebra in the text). We can also de ne a Hicks income e ect - this is just the residual. For small prices changes, the Hicks substitution e ect is the same as the Slutsky substitution e ect; we can sort of see this by drawing examples. The theorem that guarantees this is known as the envelope theorem. We will not cover that, but it s an interesting result. 17

18 9 Compensated Demand Curves The analysis above shows that we can de ne three di erent demand curves: the standard one, which holds income xed; the Slutsky substitution e ect, which holds purchasing power xed; and the Hicks substitution e ect, which holds utility xed. The last of these, the Hicks substitution e ect, turns out to be useful in theoretical analysis because it allows one to say cleanly how much consumers would require in payment to accept a particular change in price; that is, it shows how much extra or less income they would need to leave their utility constant. The Hicks substitution e ect is also known as a compensated demand curve, meaning the demand for a good as price changes assuming the consumer is compensated in a way that leaves utility constant. It might seem annoying that one can de ne both the Slutsky and Hicks substitution e ects, since it is not immediately obvious whether holding purchasing power constant or utility constant is the right thought experiment. For in nitessimal changes in price, however, these two e ects are the same. 10 OPTIONAL: Calculus Derivation of Slutsky Equation We are now going to derive the Slutsky equation using calculus. In the Slutsky de nition of the substitution e ect, income is adjusted so as to give the consumer just enough to buy the original bundle. Call this bundle and income (x 1 ; x 2 ) m 18

19 If the prices are (p 1 ; p 2 ), then the consumer s actual choice with this adjustment will depend on both these prices and the initial bundle. We call this relationship the Slutsky demand function, and write it as x s 1(p 1; p 2 ; x 1 ; x 2 ) x 1 (p 1; p 2 ; p 1 x 1 + p 2 x 2 ) This equation says that the Slutsky demand at prices (p 1 ; p 2 ) is the amount the consumer would demand if he had enough income to purchase the original bundle (x 1 ; x 2 ). Now di erentiate this identity with respect to p 1 s 1(p 1; p 2 ; x 1 ; x 2 ) 1(p 1; p 2 ; m) 1;p 2 ; We can rearrange this to 1 (p 1; p 2 ; m) 1(p 1; p 2 ; x 1 ; x 2 1 (p 1; p 2 ; This says that the total e ect of a price change is composed of a substitution e ect (where income is adjusted to keep the bundle (x 1 ; x 2 ) a ordable) and an income e ect. We explained using a graph that the substitution e ect is negative. The income e ect can have either sign. Assuming a normal good (i.e., one for which an increase in income increases demand), the second partial derivative is positive, so the e ect of the price change via income is also negative (i.e., the opposite sign from the change in price). Note the advantages of deriving the Slutsky equation using calculus. It is easier to prove, with just a few steps. Also, this approach make the role of the change in p explicit, so confusion about signs is not likely to occur. One plus of using calculus is that it s a cleaner, more parsimonious notation. We can also derive a version of the Slutsky equation for the Hicks substitution e 1 (p 1; p 2 ; m) 1(p 1; p 2 ; 1 (p 1; p 2 ; 19 x 1 x 1 x 1

20 The proof relies on the fact h 1(p 1; p 2 ; 1 1(p 1; p 2 ; x 1 ; x 2 1 for in nitesimal changes in price. 20

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