Prof. Olivier Bochet Room A.34 Phone 3 63 476 E-mail olivier.bochet@vwi.unibe.ch Webpage http//sta.vwi.unibe.ch/bochet Advanced Microeconomics Fall 2 Lecture Note Choice-Based Approach Price e ects, Wealth e ects and the WARP Basic notations and de nitions Our consumer makes choices in a given consumption set. Here we assume that there are ` = ; ; L goods that are in nitely divisible. The consumer s consumption set is thus R L +. That is, the consumer s choices are made within the non-negative orthant. Properties of R L + R L + is a closed set. R L + is a convex set R L + is bounded below notice that the point of L coordinates (; ; ) is part of R L +
A point x 2 R L + is called a bundle. Formally, it is a column vector x = B @ I will sometimes write x = (x ; ; x L ) without the use of a transpose notation x T, this should cause no confusion. Ultimately, the only thing we will observe are the choices that the consumer is making in R L +. Obviously, not all choices are possible. We assume that the consumer has a given level of wealth w > and faces prices p 2 R L + where p = B @ The consumer s choices are bounded by the constraints imposed to him. These constraints are fully determined by w and p. The consumer s choices must then take place in his respective budget set denoted B p;w, where x x 2 x L p p 2 p L C A C A B p;w = fx 2 R L + p x wg Notice that the budget set is bounded above by the budget hyperplane fx 2 R L + px = wg. When there are only two goods, the budget hyperplane is simply called the budget line. Properties of B p;w B p;w is non-empty since w >. B p;w is a convex set B p;w is a compact set it is closed and both bounded below and above. 2
As I have emphasized in the lecture, one of the main di erence between the approach presented here and the utility-based approach that you saw in undergrad micro is that here we only observe nal choices made by our consumer. We do not have any information regarding his preferences. However, we would like to make some statements regarding the rationality of our consumer. Is he someone that is maximizing some utility function subject to his budget constraint when making his choices? Basically, are his choices consistent with rationality? At each price-wealth situation (p; w), the consumer s choices are summarized in a demand system that we call x(p; w), the Walrasian demand function (or simply demand function), where x(p; w) = B @ x (p; w) x L (p; w) C A In order to answer the question raised above, we are going rst to make a few assumptions regarding the demand function x(p; w). 2 A few basic restrictions on x(p; w) Assumptions 8(p; w) 2 R L ++ R ++, x(p; w) is single-valued 8(p; w) 2 R L ++ R ++, x(p; w) satis es homogeneity of degree 8(p; w) 2 R L ++ R ++ ; x(p; w) satis es Walras law Homogeneity of degree x(p; w) satis es homogeneity of degree if 8 >, x(p; w) = x(p; w) 3
Walras law (non-satiation) x(p; w) satis es Walras law if p x(p; w) = w Notice that the two assumptions are very mild. Homogeneity of degree states that the consumer does not su er from money illusion. If prices and wealth are increased in the same proportion, then choices should be unchanged. This is so because you can easily notice that B p;w = B p;w. Walras law indicates that there is no satiation and that at least one good is desirable. 2. Price e ects and wealth e ects We give now a way to categorize the goods that the consumer can consume. Wealth e ects A good ` is a normal good if @x`(p;w) A good ` is an inferior good if @x`(p;w) < Price e ects A good ` is an ordinary good if @x`(p;w) @p` A good ` is a gi en good if @x`(p;w) > @p` A good ` is a gross substitute for good k if @x`(p;w) @p k > A good ` is a gross complement for good k if @x`(p;w) @p k < If both assumptions are satis ed, we derive a few restrictions that x(p; w) must satisfy. 2.2 Restrictions Restriction imposed by homogeneity of degree We know that x(p; w) x(p; w) = Let us -di erentiate this expression and evaluate the derivative at =. Pick some good `. Then @x`(p; w) @p +@x`(p; w) @p 2 ++@x`(p; w) @p @ @p 2 @ @p L 4 @p L +@x`(p; w) @ =
Thus, @x`(p; w) @p p + @x`(p; w) @p 2 p 2 + + @x`(p; w) @p L p L + @x`(p; w) w = Evaluating the expression at =, we end up with the following expression LX @x`(p; w) p k + @x`(p; w) w = @p k k= This expression is known as Euler s equation. Notice that this expression is for a speci c good ` that we chose. Therefore we must have L such expressions, one for each good. Hence, 8` = ; ; L LX @x`(p; w) p k + @x`(p; w) w = @p k k= We can put the L expression into matrix form to get, D p x(p; w)p + D w x(p; w)w = where and @x (p;w) @x @p (p;w) @p L D p x(p; w) = B C @ A @x L (p;w) @x @p L (p;w) @p {z L } LL D w x(p; w) = B @ @x (p;w) C C A @x L (p;w) {z } L 5
Restrictions imposed by Walras law We know that p x(p; w) = w. (a) Cournot aggregation condition Pick a given price p k for some good k and di erentiate the expression above with respect to p k. p @x (p; w) @p k + + p k @x k (p; w) @p k + x k (p; w) + + p L @x L (p; w) @p k = Hence, LX `= @x`(p; w) p` + x k (p; w) = @p k Notice that this expression is for a speci c p k that we chose. Therefore we must have L such expressions, one for each price. Hence 8k = ; ; L LX `= @x`(p; w) p` + x k (p; w) = @p k In matrix notation, p D p x(p; w) + x(p; w) T = (b) Engel aggregation condition Now let us take the derivative of the budget line with respect to w. p @x (p; w) + + p L @x L (p; w) = Which gives, LX k= p k @x k (p; w) Notice that we have only one such expression. In matrix notation, we obtain p D w x(p; w) = = 6
3 Further restrictions on x(p; w) the weak axiom of revealed preferences (WARP) We are going to introduce a property (WARP) that can be seen as a basic expression of rationality of a consumer or alternatively, an expression for the basic consistency of choices. Consider two di erent price-wealth situations (p; w) and (p ; w ). At (p; w), x(p; w) is the bundle chosen whereas at (p ; w ), x(p ; w ) is chosen. Assume that x(p; w) 6= x(p ; w ) and p x(p ; w ) w What can we say about this? Well, our assumption p x(p ; w ) w tells us that bundle x(p ; w ) is a ordable at price-wealth situation (p; w) but is not chosen bundle x(p; w) is chosen instead. In the language of revealed preferences, this means that x(p; w) is revealed (directly) preferred to x(p ; w ). But then what should we say about p x(p; w)? Since x(p; w) is revealed preferred to x(p ; w ), a rational consumer should choose x(p; w) over x(p ; w ) whenever this is possible. But at (p ; w ) we are told that x(p ; w ) 6= x(p; w) is chosen over x(p; w). Then it must be the case that x(p; w) is not a ordable at (p ; w ). Otherwise, this would be an indication that the consumer is changing his mind he states preferring x(p; w) over x(p ; w ) in some situation and preferring x(p ; w ) over x(p; w) in some situations. We want this type of preference reversals hence inconsistencies in how the consumer is making choices to be avoided. A consumer, if rational, must not exhibit these type of inconsistencies. This is the message contained in the WARP. One of the central question addressed in this part is whether a consumer, who is consistent in how he performs choices across price-income pairs in the sense of the WARP, is a rational agent as one whose behavior can be rationalized as coming from a process of utility maximization subject to budget constraint(s). And if the answer to that question turns out to be negative, then how far is the consumer from being a rational agent. However these questions are a bit premature at this stage. Let us now turn ourselves to a formal de nition of the WARP. We will then study its implications. Weak axiom of revealed preferences A consumer satis es the WARP if for each (p; w) and (p ; w ) such that px(p ; w ) w and x(p; w) 6= x(p ; w ), we have p x(p; w) > w 7
Notice that if x(p; w) satis es the WARP, then x(p; w) satis es homogeneity of degree (you should be able to show that!) Assuming that the WARP is satis ed will imply several important restrictions. Slutsky compensated price change Consider two situations (p; w) and (p ; w ). We say that there is a Slutsky compensated price change when going from (p; w) to (p ; w ) if p x(p; w) = w. That is, bundle x(p; w) is exactly a ordable at (p ; w ). The way to think about a Slutsky compensated price change is as follows. Suppose that price changes in such a way that only the price of good decreases. Then wealth is "taken away" from the consumer from w to w so that x(p; w) is exactly a ordable at (p ; w ). =) purchasing power remains constant when going from (p; w) to (p ; w ). In introductory economics course, you are often told that the demand for a given good ` moves in the opposite direction than p` does. This is usually called the law of demand. You should of course know better! The gi en good case which imply that demand is upward sloping should convince you that this claim is wrong. However, the WARP will imply that the law of demand holds but only for some very speci c price changes, namely Slutsky compensated price changes. We call this the compensated law of demand. Theorem Compensated law of demand. Let (p; w) and (p ; w ) be such that p x(p; w) = w Suppose x(p; w) satis es Walras Law. Then x(p; w) satis es the WARP if and only if (p p) [x(p ; w ) x(p; w)] () Proof We are going to prove both directions of the theorem in turn. Part if part! If the WARP holds, then inequality () holds. Suppose the WARP holds and (p ; w ) is a Slutsky compensated price change. Notice that if x(p; w) = x(p ; w ), then (p p) [] = and we are done. So assume that x(p; w) 6= x(p ; w ). Let us develop the product in the left-hand side of Equation () and obtain [p x(p ; w ) p x(p; w) ] + [ p x(p ; w ) + p x(p; w) {z } {z } ] A B 8
Let s look at A rst. By Walras law, p x(p ; w ) = w. Next, since we have a compensated price change, it is also the case that p x(p; w) = w. Hence A =. What about B? By Walras law, p x(p; w) = w We next have p x(p ; w ). Remember that because we have a compensated price change p x(p; w) = w. Hence x(p; w) is a ordable at (p ; w ) but not chosen x(p ; w ) is revealed preferred to x(p; w). By the WARP, we must have p x(p ; w ) > w, that is consistency of choices imply that x(p ; w ) is no longer a ordable at x(p; w) otherwise this would imply that the consumer is changing his mind. Therefore, summing up the information gathered with A and B, we end up with (p p) [x(p ; w ) x(p; w)] < The compensated law of demand holds. This proves Part. Notice that we have shown that if x(p; w) 6= x(p ; w ), then the inequality in equation () must be strict. Part 2 only if part! If inequality () holds, then the WARP holds. Before proving the claim, notice that x(p; w) satis es the WARP, if and only for all compensated price change x(p; w) satis es the WARP a fact that we will not prove here (you can nd its proof in MWG, bottom of page 3 and top of page 32. You must look at it). So we need to prove that if there is a violation of the WARP, we can construct a violation of the WARP with a compensated price change. Assume that inequality () holds but the WARP does not hold (by contradiction). Then there exists a compensated price change from a given price-wealth situation (p ; w ) to (p; w) such that x(p; w) 6= x(p ; w ) and the WARP is violated. That is there exist (p ; w ), (p; w) such that By Walras law, this yields to p x(p; w) w p x(p ; w ) = w p [x(p; w) x(p ; w )] = and [p x(p ; w )] [p x(p; w) ] {z } {z } =w w 9
Hence, we have a violation of the compensated law of demand. This is in contradiction with the initial assumption that equation () was satis ed while the WARP was violated. Q.E.D. 3. A di erentiable version of the compensated law of demand the Slutsky matrix Let us assume now that the demand x(p; w) is di erentiable, i.e. for each ` = ; ; L, x`(p; w) is a di erentiable function. Remember that equation () told us that for any compensated price change (p p) [x(p ; w ) x(p; w)] Let p = (p p) and x = x(p ; w ) x(p; w). The equation above can be rewritten as p x Therefore, if x(p; w) is di erentiable, we obtain a di erentiable version of the compensated law of demand dp dx (2) That is, prices and demand move in opposite directions for any Slutsky compensated price change. Notice that dx is just the total derivative of x(p; w). That is, in matrix notation, dx = D p x(p; w) dp + D {z } {z} w x(p; w) {z } {z} dw L L LL Hence dx is the sum of two L matrices. Notice that because we are dealing with a Slutsky compensated price change, the variation in wealth is dw = x(p; w) T dp Hence dx = [D p x(p; w) + D w x(p; w)x(p; w) T ]dp
Equation (2) then becomes dp [D p x(p; w) {z } LL + D w x(p; w) x(p; w) T ]dp {z } {z } L Let S(p; w) D p x(p; w)+d w x(p; w)x(p; w) T. We call S(p; w) the Slutsky matrix, or the matrix of substitution e ects. This should be reminiscent of the Slutsky equation you studied in undergrad micro up to the following minor di erence. Here we are dealing with Slutsky compensated price changes. The kind of compensated changes you saw in undergrad were di erent following a decrease in, say, the price p when going from (p; w) to (p ; w ), wealth was adjusted in such a way that the consumer could reach the same utility level as with x(p; w). This does not imply that x(p; w) was still a ordable. This type of change is called Hicks compensated price change. As you can see they are di erent from a Slutsky compensated price change. The former involves allowing the consumer to be at the same utility level than before, whereas the latter involves allowing the consumer to buy the same bundle as before. Note that substitution e ects are in general unobservable. s (p; w) s L (p; w) S(p; w) = B C @ A s L (p; w) s LL (p; w) where 8`; k L s`k = @x`(p; w) + @x`(p; w) x k (p; w) @p k! Substitution e ect s`k tells us how the demand x`(p; w) changes when the price of good k, p k, changes if the consumer s purchasing power is kept the same. Hence only relative prices have changed. The income e ect hidden in a price change i.e. the indirect associated change in purchasing power is netted out so that only the direct e ect associated with the price change remains. We say that good ` is a net substitute for good k if s`k > ; and a net complement if s`k <. A consequence of the di erentiable version of the compensated law of demand is the following.
Theorem 2 Suppose x(p; w) is di erentiable, satis es Walras law and the WARP. Then at any (p; w), S(p; w) satis es v S(p; w)v 8v 2 R L The above Theorem states that S(p; w) is a negative semi-de nite matrix.... Negative semi-de niteness Let A be an L L matrix. A submatrix k k of A formed by eliminating L k columns, say columns i ; ; i L k and the same L k rows i ; ; i L k ; is called a submatrix of principal order k. The determinant of a submatrix k k formed as explained above is called a principal minor of order k of the matrix A. For example let A be a 3 3 matrix a a 2 a 3 A = @ a 2 a 22 a 23 A a 3 a 32 a 33 The principal minor of order are all the elements on the diagonals. The principal minors of order 2 are the determinants of all the 2 2 matrices obtained by deleting the same row and the same column. That is, there are three such principal minors a22 a 23 det a 32 a 33 ; det a a 3 a 3 a 33 ; det a a 2 a 2 a 22 Finally there is only one principal minor of order 3, and it is det A itself. So how do we check that a matrix A is negative semi-de nite? De nition Let A be an LL matrix. Then A is negative semi-de nite if and only if v Av for all v 2 R L Although this de nition may be useful at times to prove that a matrix is not negative semi-de nite, it looks rather impossible to check that the claim is true for all possible vectors v. So do we have alternative methods? We can use the principal minors to determine the negative semi-de niteness of A 2
De nition 2 Let A be an LL matrix. Then A is negative semi-de nite if all the principal minor of order k, with k odd; are less than or equal to ; and all the principal minors of order k; with k even, are greater than or equal to. Notice that in de nition, we have an if and only if statement whereas de nition 2 contains an if statement. This means that de nition 2 provides only a necessary condition but not a su cient condition. In some odd cases which we will not encounter in this course one can nd that de nition 2 is satis ed by A, conclude that A is then negative semi-de nite, although there exists a vector v such that v Av >. The conditions laid down in De nition 2 are both necessary and su cient if A happens to be a symmetric matrix which is not the case for S(p; w). De nition 3 Let A be a symmetric L L matrix. Then A is negative semi-de nite if and only if all the principal minor of order k, with k odd; are less than or equal to ; and all the principal minors of order k; with k even, are greater than or equal to. De nition 4 Let A be an L L matrix not necessarily symmetric. Then A is negative semi-de nite if and only if the symmetric transformed ^A of A, with ^A = 2 (A + AT ) is negative semi-de nite. That is, for ^A, all the principal minor of order k, with k odd; are less than or equal to ; and all the principal minors of order k; with k even, are greater than or equal to. Like I said, in this course we will either make use of the standard de nition v Av or De nition 2. Applying De nition 2, we say that A is negative semi-de nite if a`` 8` = ; 2; 3 a 22 a 33 a 23 a 32 a a 33 a 3 a 3 a a 22 a 2 a 2 det(a) = a (a 22 a 33 a 23 a 32 ) a 2 (a 2 a 33 a 3 a 32 ) + a 3 (a 2 a 23 a 3 a 22 ) 3
... The negative semi-de niteness of S(p; w) has two direct implications. () For each good `, s`;`(p; w), i.e. demand and price move in opposite directions (2) A gi en good is an inferior good. Suppose good ` is a gi en good. Then we know that, s`;`(p; w) = @x`(p; w) + @x`(p; w) x`(p; w) @p` {z } > by assumption Hence for the inequality to hold, we must have Hence good ` is an inferior good. @x`(p; w) x`(p; w) < Let us now explore further the properties of S(p; w). First let us check whether S(p; w) is a symmetric matrix. Recall that S(p; w) is symmetric if for each `; k we have s`;k (p; w) = s k;`(p; w). It turns out that the crucial di erence between the utility-based approach and the choice-based approach is the symmetricity or lack thereof of the Slutsky matrix. If the number of goods is L = 2, then S(p; w) is necessarily symmetric. However, when L > 2; this is no longer true. In the utility-based approach, S(p; w) is always symmetric. 4
Proposition 3 S(p; w) is not in general a symmetric matrix Proof It is su cient to look at the following counter-example to prove the claim. Let L = 3 and x(p; w) be given by the following demand system x (p; w) = x 2 (p; w) = x 3 (p; w) = p 2 w p (p + p 2 + p 3 ) p 3 w p 2 (p + p 2 + p 3 ) p w p 3 (p + p 2 + p 3 ) Let w = p = p 2 = p 3 =. As shown in class, it can be checked that the Slutsky matrix is given by S(p; w) = @ 3 3 3 3 3 3 A For each `; k we have s`;k (p; w) 6= s k;`(p; w). necessarily symmetric. Therefore S(p; w) is not Q.E.D. Another important property of S(p; w) is that it is singular, and this for any number of goods. That is 8p 2 R L ++, p S(p; w) = and S(p; w)p = Equivalent consequences of the singularity of the Slutsky matrix are as follows (i) The rows or the columns of S(p; w) are linearly dependent (ii) S(p; w) has rank less than L (iii) S(p; w) is not invertible, i.e. det(s(p; w)) = (iv) S(p; w) can never be negative de nite The singularity of S(p; w) is a consequence of the properties of homogeneity of degree and Walras law. 5
Proposition 4 Suppose that x(p; w) is di erentiable, homogeneous of degree, and satis es Walras law. Then For all p, p S(p; w) = and S(p; w)p = Proof First part p S(p; w) = For each good `, the following is obtained by applying the matrix operation p S(p; w) to one of the column of S(p; w) remember that p is a L column vector while S(p; w) is a L L matrix; hence p S(p; w) is a L vector, i.e. we have L entries. We have p s + p 2 s 2 + + p L s L () p s 2 + p 2 s 22 + + p L s L2 (2). p s L + p 2 s 2L + + p L s LL (L) Let us just show that the claim is true for (). The reasoning for the remaining (L ) equation is analog. In order to prove the claim, let us develop () by substituting in () the respective Slutsky equations in place of the s`k terms. We get, p @x (p;w) @p + @x (p;w) x (p; w) +p @x2 (p;w) 2 @p + @x 2(p;w) x (p; w) ++ @xl (p;w) @p + @x L(p;w) x (p; w) Let us write the expression in a more compact fashion. We obtain LX `= @x`(p; w) p` + @p LX `= p` @x`(p; w) x (p; w) Notice that P L @x`(p;w) `= p` @p = x (p; w) by the Cournot aggregation condition, whereas P L @x`(p;w) `= p` = by the Engel aggregation condition. Hence we obtain the desired conclusion. Second part S(p; w)p = 6
For each good `, the following is obtained by applying the matrix operation S(p; w)p to one of the column of S(p; w) remember that p is a L column vector while S(p; w) is a L L matrix; hence S(p; w)p is a L vector, i.e. we have L entries. We have s p + s 2 p 2 + + s L p L () s 2 p + s 22 p 2 + + s 2L p L (2). s L + s L2 p 2 + + s LL p L (L) Let us just show that the claim is true for (). The reasoning for the remaining (L ) equation is analog. In order to prove the claim, let us develop () by substituting in () the respective Slutsky equations in place of the s`k terms. @x (p;w) @p + @x (p;w) x (p; w) p + @x (p;w) @p 2 + @x (p;w) x 2 (p; w) p 2 ++ @x (p;w) @p L + @x (p;w) x L (p; w) Let us write the expression in a more compact fashion. We obtain LX @x (p; w) `= `= @p` p` + LX `= This can be rewritten as LX @x (p; w) p` + @x (p; w) @p` Notice that P L `= @x (p;w) p` = @p` @x (p; w) x`(p; w)p` LX p`x`(p; w) `= @x (p;w) w by the Euler s equation applied to good. Note further that P L `= p`x`(p; w) is simply w by Walras law. Hence we obtain the desired conclusion, which completes the proof. Q.E.D. 7
3.2 Equivalence between the WARP and utility maximization? As we have seen, an important di erence between the choice-based approach and the utility-approach is that the WARP is not enough to get the symmetricity of S(p; w). Is this only a purely technical feature or does this have consequences? In particular how is a theory based on the WARP related to a theory based on utility maximization? Are these two equivalent? The answer is no, as shown in the following example. Example Non-equivalence between the WARP and utility maximization (Hicks, 953) Let L = 3 and w = 8 Consider the following three situations p = (2; ; 2) x = (; 2; 2) p 2 = (2; 2; ) x 2 = (2; ; 2) p 3 = (; 2; 2) x 3 = (2; 2; ) You can check that this consumer satis es the WARP (please check!) We make three observations. p x 3 = 8 = w so that x is revealed preferred to x 3 2 p 2 x = 8 = w so that x 2 is revealed preferred to x 3 p 3 x 2 = 8 = w so that x 3 is revealed preferred to x 2 Remember your old undergrad micro course in which you studied preference relations? If you remember, you used the binary relation symbol. For all x; y 2 R L +, x y means that x is at least as good as y; x y means that x is strictly preferred to y; and x y means that x is equivalent to y. Let us rewrite the consequences of (i), (ii) and (iii) in the language of preference relations, i.e. suppose that our consumer is equipped with a preference relation. Then, p x 3 = 8 = w so that x x 3 2 p 2 x = 8 = w so that x 2 x 3 p 3 x 2 = 8 = w so that x 3 x So we have x x 3, x 3 x 2 but x 2 x. Does this ring a bell? Yes! This means that preference relation violates transitivity, one of the basic property satis ed by a rational consumer in the utility-based approach. We 8
have cycles in the ranking of alternatives. This consumer is not rational, although he satis es the WARP. Satisfying the WARP is not enough to get the equivalence between the choice-based approach and the utility-based approach. We need a property that is stronger than the WARP! the strong axiom of revealed preferences (SARP) which will avoid these kind of preference cycles. Because of our time constraint, we will not cover the study of the SARP. 9