Advanced Microeconomics Fall Lecture Note 1 Choice-Based Approach: Price e ects, Wealth e ects and the WARP
|
|
- Emmeline Constance Sims
- 6 years ago
- Views:
Transcription
1 Prof. Olivier Bochet Room A.34 Phone 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 +
2 A point x 2 R L + is called a bundle. Formally, it is a column vector x = 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 = 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
3 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) = 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
4 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 A good ` is an inferior good < Price e ects A good ` is an ordinary A good ` is a gi en good A good ` is a gross substitute for good k > A good ` is a gross complement for good 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 `. +@x`(p; 2 L L +@x`(p; =
5 p 2 p 2 + L p L w) w = Evaluating the expression at =, we end up with the following expression w) p k w) w 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 w) p k w) w 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 (p;w) L D p x(p; w) = B L {z L } LL D w x(p; w) (p;w) C C L (p;w) {z } L 5
6 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; k + + p k (p; k + x k (p; w) + + p L (p; k = Hence, LX w) p` + x k (p; w) 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 w) p` + x k (p; w) 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; w) + + p L (p; w) = Which gives, LX k= p k (p; w) Notice that we have only one such expression. In matrix notation, we obtain p D w x(p; w) = = 6
7 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
8 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
9 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
10 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
11 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 A s L (p; w) s LL (p; w) where 8`; k L s`k w) w) x k (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.
12 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
13 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
14 ... 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) w) w) x`(p; {z } > by assumption Hence for the inequality to hold, we must have Hence good ` is an inferior 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
15 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) 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
16 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 p L s L () p s 2 + p 2 s 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;w) x (p; w) (p;w) 2(p;w) x (p; w) L(p;w) x (p; w) Let us write the expression in a more compact fashion. We obtain LX w) p` LX `= w) x (p; w) Notice that P `= = x (p; w) by the Cournot aggregation condition, whereas P `= p` = by the Engel aggregation condition. Hence we obtain the desired conclusion. Second part S(p; w)p = 6
17 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 s L p L () s 2 p + s 22 p s 2L p L (2). s L + s L2 p 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 (p;w) x (p; w) p 2 (p;w) x 2 (p; w) p 2 L (p;w) x L (p; w) Let us write the expression in a more compact fashion. We obtain (p; w) `= p` + LX `= This can be rewritten as (p; w) p` (p; Notice that P L (p;w) (p; w) x`(p; w)p` LX p`x`(p; w) (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
18 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
19 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
Microeconomics, Block I Part 1
Microeconomics, Block I Part 1 Piero Gottardi EUI Sept. 26, 2016 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, 2016 1 / 53 Choice Theory Set of alternatives: X, with generic elements x,
More informationLast Revised: :19: (Fri, 12 Jan 2007)(Revision:
0-0 1 Demand Lecture Last Revised: 2007-01-12 16:19:03-0800 (Fri, 12 Jan 2007)(Revision: 67) a demand correspondence is a special kind of choice correspondence where the set of alternatives is X = { x
More informationMicroeconomics II Lecture 4. Marshallian and Hicksian demands for goods with an endowment (Labour supply)
Leonardo Felli 30 October, 2002 Microeconomics II Lecture 4 Marshallian and Hicksian demands for goods with an endowment (Labour supply) Define M = m + p ω to be the endowment of the consumer. The Marshallian
More informationRevealed Preferences and Utility Functions
Revealed Preferences and Utility Functions Lecture 2, 1 September Econ 2100 Fall 2017 Outline 1 Weak Axiom of Revealed Preference 2 Equivalence between Axioms and Rationalizable Choices. 3 An Application:
More informationLecture 7: General Equilibrium - Existence, Uniqueness, Stability
Lecture 7: General Equilibrium - Existence, Uniqueness, Stability In this lecture: Preferences are assumed to be rational, continuous, strictly convex, and strongly monotone. 1. Excess demand function
More informationECON501 - Vector Di erentiation Simon Grant
ECON01 - Vector Di erentiation Simon Grant October 00 Abstract Notes on vector di erentiation and some simple economic applications and examples 1 Functions of One Variable g : R! R derivative (slope)
More informationRecitation 2-09/01/2017 (Solution)
Recitation 2-09/01/2017 (Solution) 1. Checking properties of the Cobb-Douglas utility function. Consider the utility function u(x) Y n i1 x i i ; where x denotes a vector of n di erent goods x 2 R n +,
More informationEconS Micro Theory I Recitation #4b - Demand theory (Applications) 1
EconS 50 - Micro Theory I Recitation #4b - Demand theory (Applications). Exercise 3.I.7 MWG: There are three commodities (i.e., L=3) of which the third is a numeraire (let p 3 = ) the Walrasian demand
More informationWeek 9: Topics in Consumer Theory (Jehle and Reny, Chapter 2)
Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter 2) Tsun-Feng Chiang *School of Economics, Henan University, Kaifeng, China November 15, 2015 Microeconomic Theory Week 9: Topics in Consumer Theory
More informationECON0702: Mathematical Methods in Economics
ECON0702: Mathematical Methods in Economics Yulei Luo SEF of HKU January 12, 2009 Luo, Y. (SEF of HKU) MME January 12, 2009 1 / 35 Course Outline Economics: The study of the choices people (consumers,
More informationMicroeconomic Theory-I Washington State University Midterm Exam #1 - Answer key. Fall 2016
Microeconomic Theory-I Washington State University Midterm Exam # - Answer key Fall 06. [Checking properties of preference relations]. Consider the following preference relation de ned in the positive
More informationThe Kuhn-Tucker Problem
Natalia Lazzati Mathematics for Economics (Part I) Note 8: Nonlinear Programming - The Kuhn-Tucker Problem Note 8 is based on de la Fuente (2000, Ch. 7) and Simon and Blume (1994, Ch. 18 and 19). The Kuhn-Tucker
More informationUtility Maximization Problem
Demand Theory Utility Maximization Problem Consumer maximizes his utility level by selecting a bundle x (where x can be a vector) subject to his budget constraint: max x 0 u(x) s. t. p x w Weierstrass
More informationNotes I Classical Demand Theory: Review of Important Concepts
Notes I Classical Demand Theory: Review of Important Concepts The notes for our course are based on: Mas-Colell, A., M.D. Whinston and J.R. Green (1995), Microeconomic Theory, New York and Oxford: Oxford
More informationCONSUMER DEMAND. Consumer Demand
CONSUMER DEMAND KENNETH R. DRIESSEL Consumer Demand The most basic unit in microeconomics is the consumer. In this section we discuss the consumer optimization problem: The consumer has limited wealth
More informationAdvanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2
Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 Prof. Dr. Oliver Gürtler Winter Term 2012/2013 1 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J 1. Introduction
More informationConsumer Theory. Ichiro Obara. October 8, 2012 UCLA. Obara (UCLA) Consumer Theory October 8, / 51
Consumer Theory Ichiro Obara UCLA October 8, 2012 Obara (UCLA) Consumer Theory October 8, 2012 1 / 51 Utility Maximization Utility Maximization Obara (UCLA) Consumer Theory October 8, 2012 2 / 51 Utility
More informationGi en Demand for Several Goods
Gi en Demand for Several Goods Peter Norman Sørensen January 28, 2011 Abstract The utility maimizing consumer s demand function may simultaneously possess the Gi en property for any number of goods strictly
More informationLocal disaggregation of demand and excess demand functions: a new question
Local disaggregation of demand and excess demand functions: a new question Pierre-Andre Chiappori Ivar Ekeland y Martin Browning z January 1999 Abstract The literature on the characterization of aggregate
More information1 Uncertainty and Insurance
Uncertainty and Insurance Reading: Some fundamental basics are in Varians intermediate micro textbook (Chapter 2). A good (advanced, but still rather accessible) treatment is in Kreps A Course in Microeconomic
More informationChapter 4. Maximum Theorem, Implicit Function Theorem and Envelope Theorem
Chapter 4. Maximum Theorem, Implicit Function Theorem and Envelope Theorem This chapter will cover three key theorems: the maximum theorem (or the theorem of maximum), the implicit function theorem, and
More informationNonlinear Programming (NLP)
Natalia Lazzati Mathematics for Economics (Part I) Note 6: Nonlinear Programming - Unconstrained Optimization Note 6 is based on de la Fuente (2000, Ch. 7), Madden (1986, Ch. 3 and 5) and Simon and Blume
More informationEconomics 121b: Intermediate Microeconomics Midterm Suggested Solutions 2/8/ (a) The equation of the indifference curve is given by,
Dirk Bergemann Department of Economics Yale University Economics 121b: Intermediate Microeconomics Midterm Suggested Solutions 2/8/12 1. (a) The equation of the indifference curve is given by, (x 1 + 2)
More informationEcon Spring Review Set 1 - Answers ELEMENTS OF LOGIC. NECESSARY AND SUFFICIENT. SET THE-
Econ 4808 - Spring 2008 Review Set 1 - Answers ORY ELEMENTS OF LOGIC. NECESSARY AND SUFFICIENT. SET THE- 1. De ne a thing or action in words. Refer to this thing or action as A. Then de ne a condition
More informationHicksian Demand and Expenditure Function Duality, Slutsky Equation
Hicksian Demand and Expenditure Function Duality, Slutsky Equation Econ 2100 Fall 2017 Lecture 6, September 14 Outline 1 Applications of Envelope Theorem 2 Hicksian Demand 3 Duality 4 Connections between
More informationConsumer theory Topics in consumer theory. Microeconomics. Joana Pais. Fall Joana Pais
Microeconomics Fall 2016 Indirect utility and expenditure Properties of consumer demand The indirect utility function The relationship among prices, incomes, and the maximised value of utility can be summarised
More informationStructural Properties of Utility Functions Walrasian Demand
Structural Properties of Utility Functions Walrasian Demand Econ 2100 Fall 2017 Lecture 4, September 7 Outline 1 Structural Properties of Utility Functions 1 Local Non Satiation 2 Convexity 3 Quasi-linearity
More informationUtility Maximization Problem. Advanced Microeconomic Theory 2
Demand Theory Utility Maximization Problem Advanced Microeconomic Theory 2 Utility Maximization Problem Consumer maximizes his utility level by selecting a bundle x (where x can be a vector) subject to
More informationAdvanced Microeconomics
Advanced Microeconomics Partial and General Equilibrium Giorgio Fagiolo giorgio.fagiolo@sssup.it http://www.lem.sssup.it/fagiolo/welcome.html LEM, Sant Anna School of Advanced Studies, Pisa (Italy) Part
More informationi) This is simply an application of Berge s Maximum Theorem, but it is actually not too difficult to prove the result directly.
Bocconi University PhD in Economics - Microeconomics I Prof. M. Messner Problem Set 3 - Solution Problem 1: i) This is simply an application of Berge s Maximum Theorem, but it is actually not too difficult
More informationIntro to Economic analysis
Intro to Economic analysis Alberto Bisin - NYU 1 Rational Choice The central gure of economics theory is the individual decision-maker (DM). The typical example of a DM is the consumer. We shall assume
More informationMICROECONOMIC THEORY I PROBLEM SET 1
MICROECONOMIC THEORY I PROBLEM SET 1 MARCIN PĘSKI Properties of rational preferences. MWG 1.B1 and 1.B.2. Solutions: Tutorial Utility and preferences. MWG 1.B.4. Solutions: Tutorial Choice structure. MWG
More informationSeptember Math Course: First Order Derivative
September Math Course: First Order Derivative Arina Nikandrova Functions Function y = f (x), where x is either be a scalar or a vector of several variables (x,..., x n ), can be thought of as a rule which
More informationMidterm #1 EconS 527 Wednesday, February 21st, 2018
NAME: Midterm #1 EconS 527 Wednesday, February 21st, 2018 Instructions. Show all your work clearly and make sure you justify all your answers. 1. Question 1 [10 Points]. Discuss and provide examples of
More informationEcon 121b: Intermediate Microeconomics
Econ 121b: Intermediate Microeconomics Dirk Bergemann, Spring 2012 Week of 1/29-2/4 1 Lecture 7: Expenditure Minimization Instead of maximizing utility subject to a given income we can also minimize expenditure
More informationUniqueness, Stability, and Gross Substitutes
Uniqueness, Stability, and Gross Substitutes Econ 2100 Fall 2017 Lecture 21, November 14 Outline 1 Uniquenness (in pictures) 2 Stability 3 Gross Substitute Property Uniqueness and Stability We have dealt
More informationSecond Welfare Theorem
Second Welfare Theorem Econ 2100 Fall 2015 Lecture 18, November 2 Outline 1 Second Welfare Theorem From Last Class We want to state a prove a theorem that says that any Pareto optimal allocation is (part
More informationChapter 1. Consumer Choice under Certainty. 1.1 Budget, Prices, and Demand
Chapter 1 Consumer Choice under Certainty 1.1 Budget, Prices, and Demand Consider an individual consumer (household). Her problem: Choose a bundle of goods x from a given consumption set X R H under a
More informationMicroeconomics, Block I Part 2
Microeconomics, Block I Part 2 Piero Gottardi EUI Sept. 20, 2015 Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, 2015 1 / 48 Pure Exchange Economy H consumers with: preferences described
More informationAdvanced Microeconomic Theory. Chapter 2: Demand Theory
Advanced Microeconomic Theory Chapter 2: Demand Theory Outline Utility maximization problem (UMP) Walrasian demand and indirect utility function WARP and Walrasian demand Income and substitution effects
More informationBEEM103 UNIVERSITY OF EXETER. BUSINESS School. January 2009 Mock Exam, Part A. OPTIMIZATION TECHNIQUES FOR ECONOMISTS solutions
BEEM03 UNIVERSITY OF EXETER BUSINESS School January 009 Mock Exam, Part A OPTIMIZATION TECHNIQUES FOR ECONOMISTS solutions Duration : TWO HOURS The paper has 3 parts. Your marks on the rst part will be
More informationEcon Review Set 2 - Answers
Econ 4808 Review Set 2 - Answers EQUILIBRIUM ANALYSIS 1. De ne the concept of equilibrium within the con nes of an economic model. Provide an example of an economic equilibrium. Economic models contain
More informationWeek 6: Consumer Theory Part 1 (Jehle and Reny, Chapter 1)
Week 6: Consumer Theory Part 1 (Jehle and Reny, Chapter 1) Tsun-Feng Chiang* *School of Economics, Henan University, Kaifeng, China November 2, 2014 1 / 28 Primitive Notions 1.1 Primitive Notions Consumer
More information1 Uncertainty. These notes correspond to chapter 2 of Jehle and Reny.
These notes correspond to chapter of Jehle and Reny. Uncertainty Until now we have considered our consumer s making decisions in a world with perfect certainty. However, we can extend the consumer theory
More informationGeneral Equilibrium and Welfare
and Welfare Lectures 2 and 3, ECON 4240 Spring 2017 University of Oslo 24.01.2017 and 31.01.2017 1/37 Outline General equilibrium: look at many markets at the same time. Here all prices determined in the
More informationMicroeconomic Theory I Midterm October 2017
Microeconomic Theory I Midterm October 2017 Marcin P ski October 26, 2017 Each question has the same value. You need to provide arguments for each answer. If you cannot solve one part of the problem, don't
More informationECON2285: Mathematical Economics
ECON2285: Mathematical Economics Yulei Luo FBE, HKU September 2, 2018 Luo, Y. (FBE, HKU) ME September 2, 2018 1 / 35 Course Outline Economics: The study of the choices people (consumers, firm managers,
More informationLecture 3 - Axioms of Consumer Preference and the Theory of Choice
Lecture 3 - Axioms of Consumer Preference and the Theory of Choice David Autor 14.03 Fall 2004 Agenda: 1. Consumer preference theory (a) Notion of utility function (b) Axioms of consumer preference (c)
More informationRecitation #2 (August 31st, 2018)
Recitation #2 (August 1st, 2018) 1. [Checking properties of the Cobb-Douglas utility function.] Consider the utility function u(x) = n i=1 xα i i, where x denotes a vector of n different goods x R n +,
More informationRice University. Answer Key to Mid-Semester Examination Fall ECON 501: Advanced Microeconomic Theory. Part A
Rice University Answer Key to Mid-Semester Examination Fall 006 ECON 50: Advanced Microeconomic Theory Part A. Consider the following expenditure function. e (p ; p ; p 3 ; u) = (p + p ) u + p 3 State
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationAppendix for "O shoring in a Ricardian World"
Appendix for "O shoring in a Ricardian World" This Appendix presents the proofs of Propositions - 6 and the derivations of the results in Section IV. Proof of Proposition We want to show that Tm L m T
More informationEconS Microeconomic Theory II Homework #9 - Answer key
EconS 503 - Microeconomic Theory II Homework #9 - Answer key 1. WEAs with market power. Consider an exchange economy with two consumers, A and B, whose utility functions are u A (x A 1 ; x A 2 ) = x A
More informationf( x) f( y). Functions which are not one-to-one are often called many-to-one. Take the domain and the range to both be all the real numbers:
I. UNIVARIATE CALCULUS Given two sets X and Y, a function is a rule that associates each member of X with exactly one member of Y. That is, some x goes in, and some y comes out. These notations are used
More informationMicroeconomics. Joana Pais. Fall Joana Pais
Microeconomics Fall 2016 Primitive notions There are four building blocks in any model of consumer choice. They are the consumption set, the feasible set, the preference relation, and the behavioural assumption.
More informationLecture 8: Basic convex analysis
Lecture 8: Basic convex analysis 1 Convex sets Both convex sets and functions have general importance in economic theory, not only in optimization. Given two points x; y 2 R n and 2 [0; 1]; the weighted
More informationSolving Extensive Form Games
Chapter 8 Solving Extensive Form Games 8.1 The Extensive Form of a Game The extensive form of a game contains the following information: (1) the set of players (2) the order of moves (that is, who moves
More informationFirst Welfare Theorem
First Welfare Theorem Econ 2100 Fall 2017 Lecture 17, October 31 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Past Definitions A feasible allocation (ˆx, ŷ) is Pareto optimal
More informationx 1 1 and p 1 1 Two points if you just talk about monotonicity (u (c) > 0).
. (a) (8 points) What does it mean for observations x and p... x T and p T to be rationalized by a monotone utility function? Notice that this is a one good economy. For all t, p t x t function. p t x
More informationMicroeconomics I Fall 2007 Prof. I. Hafalir
Microeconomics I Fall 2007 Prof. I. Hafalir Chris Almost Contents Contents 1 1 Demand Theory 2 1.1 Preference relations............................. 2 1.2 Utility functions................................
More informationOptimal taxation with monopolistic competition
Optimal taxation with monopolistic competition Leslie J. Reinhorn Economics Department University of Durham 23-26 Old Elvet Durham DH1 3HY United Kingdom phone +44 191 334 6365 fax +44 191 334 6341 reinhorn@hotmail.com
More informationEconomics 101. Lecture 2 - The Walrasian Model and Consumer Choice
Economics 101 Lecture 2 - The Walrasian Model and Consumer Choice 1 Uncle Léon The canonical model of exchange in economics is sometimes referred to as the Walrasian Model, after the early economist Léon
More informationIntroduction: structural econometrics. Jean-Marc Robin
Introduction: structural econometrics Jean-Marc Robin Abstract 1. Descriptive vs structural models 2. Correlation is not causality a. Simultaneity b. Heterogeneity c. Selectivity Descriptive models Consider
More informationSometimes the domains X and Z will be the same, so this might be written:
II. MULTIVARIATE CALCULUS The first lecture covered functions where a single input goes in, and a single output comes out. Most economic applications aren t so simple. In most cases, a number of variables
More informationNotes on Consumer Theory
Notes on Consumer Theory Alejandro Saporiti Alejandro Saporiti (Copyright) Consumer Theory 1 / 65 Consumer theory Reference: Jehle and Reny, Advanced Microeconomic Theory, 3rd ed., Pearson 2011: Ch. 1.
More informationPreferences and Utility
Preferences and Utility How can we formally describe an individual s preference for different amounts of a good? How can we represent his preference for a particular list of goods (a bundle) over another?
More informationCHAPTER 4: HIGHER ORDER DERIVATIVES. Likewise, we may define the higher order derivatives. f(x, y, z) = xy 2 + e zx. y = 2xy.
April 15, 2009 CHAPTER 4: HIGHER ORDER DERIVATIVES In this chapter D denotes an open subset of R n. 1. Introduction Definition 1.1. Given a function f : D R we define the second partial derivatives as
More informationSome Notes on Adverse Selection
Some Notes on Adverse Selection John Morgan Haas School of Business and Department of Economics University of California, Berkeley Overview This set of lecture notes covers a general model of adverse selection
More informationIt is convenient to introduce some notation for this type of problems. I will write this as. max u (x 1 ; x 2 ) subj. to. p 1 x 1 + p 2 x 2 m ;
4 Calculus Review 4.1 The Utility Maimization Problem As a motivating eample, consider the problem facing a consumer that needs to allocate a given budget over two commodities sold at (linear) prices p
More informationThe Fundamental Welfare Theorems
The Fundamental Welfare Theorems The so-called Fundamental Welfare Theorems of Economics tell us about the relation between market equilibrium and Pareto efficiency. The First Welfare Theorem: Every Walrasian
More informationGARP and Afriat s Theorem Production
GARP and Afriat s Theorem Production Econ 2100 Fall 2017 Lecture 8, September 21 Outline 1 Generalized Axiom of Revealed Preferences 2 Afriat s Theorem 3 Production Sets and Production Functions 4 Profits
More informationChapter 1 - Preference and choice
http://selod.ensae.net/m1 Paris School of Economics (selod@ens.fr) September 27, 2007 Notations Consider an individual (agent) facing a choice set X. Definition (Choice set, "Consumption set") X is a set
More informationEconomics 401 Sample questions 2
Economics 401 Sample questions 1. What does it mean to say that preferences fit the Gorman polar form? Do quasilinear preferences fit the Gorman form? Do aggregate demands based on the Gorman form have
More informationEC487 Advanced Microeconomics, Part I: Lecture 2
EC487 Advanced Microeconomics, Part I: Lecture 2 Leonardo Felli 32L.LG.04 6 October, 2017 Properties of the Profit Function Recall the following property of the profit function π(p, w) = max x p f (x)
More informationMicroeconomic Theory: Lecture 2 Choice Theory and Consumer Demand
Microeconomic Theory: Lecture 2 Choice Theory and Consumer Demand Summer Semester, 2014 De nitions and Axioms Binary Relations I Examples: taller than, friend of, loves, hates, etc. I Abstract formulation:
More information1 Objective. 2 Constrained optimization. 2.1 Utility maximization. Dieter Balkenborg Department of Economics
BEE020 { Basic Mathematical Economics Week 2, Lecture Thursday 2.0.0 Constrained optimization Dieter Balkenborg Department of Economics University of Exeter Objective We give the \ rst order conditions"
More informationMathematical models in economy. Short descriptions
Chapter 1 Mathematical models in economy. Short descriptions 1.1 Arrow-Debreu model of an economy via Walras equilibrium problem. Let us consider first the so-called Arrow-Debreu model. The presentation
More informationWeek 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 1 / Reny, 32
Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer Theory (Jehle and Reny, Chapter 1) Tsun-Feng Chiang* *School of Economics, Henan University, Kaifeng, China November 1, 2015 Week 7: The Consumer
More informationTutorial 3: Optimisation
Tutorial : Optimisation ECO411F 011 1. Find and classify the extrema of the cubic cost function C = C (Q) = Q 5Q +.. Find and classify the extreme values of the following functions (a) y = x 1 + x x 1x
More informationOn the microeconomic foundations of linear demand for di erentiated products
On the microeconomic foundations of linear demand for di erentiated products Rabah Amir y, Philip Erickson z, and Jim Jin x Nov 9, 205 Abstract This paper provides a thorough exploration of the microeconomic
More informationRevealed Preference 2011
Revealed Preference 2011 Motivation: 1. up until now we have started with preference and then described behaviour 2. revealed preference works backwards - start with behaviour and describe preferences
More informationGeneralized Convexity in Economics
Generalized Convexity in Economics J.E. Martínez-Legaz Universitat Autònoma de Barcelona, Spain 2nd Summer School: Generalized Convexity Analysis Introduction to Theory and Applications JULY 19, 2008 KAOHSIUNG
More informationIndex. Cambridge University Press An Introduction to Mathematics for Economics Akihito Asano. Index.
, see Q.E.D. ln, see natural logarithmic function e, see Euler s e i, see imaginary number log 10, see common logarithm ceteris paribus, 4 quod erat demonstrandum, see Q.E.D. reductio ad absurdum, see
More informationSeminars on Mathematics for Economics and Finance Topic 5: Optimization Kuhn-Tucker conditions for problems with inequality constraints 1
Seminars on Mathematics for Economics and Finance Topic 5: Optimization Kuhn-Tucker conditions for problems with inequality constraints 1 Session: 15 Aug 2015 (Mon), 10:00am 1:00pm I. Optimization with
More informationIntroduction to General Equilibrium
Introduction to General Equilibrium Juan Manuel Puerta November 6, 2009 Introduction So far we discussed markets in isolation. We studied the quantities and welfare that results under different assumptions
More informationMATHEMATICAL PROGRAMMING I
MATHEMATICAL PROGRAMMING I Books There is no single course text, but there are many useful books, some more mathematical, others written at a more applied level. A selection is as follows: Bazaraa, Jarvis
More informationLecture 1- The constrained optimization problem
Lecture 1- The constrained optimization problem The role of optimization in economic theory is important because we assume that individuals are rational. Why constrained optimization? the problem of scarcity.
More informationQuestion 1. (p p) (x(p, w ) x(p, w)) 0. with strict inequality if x(p, w) x(p, w ).
University of California, Davis Date: August 24, 2017 Department of Economics Time: 5 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE Please answer any three
More informationEconomics 201B Second Half. Lecture 10, 4/15/10. F (ˆp, ω) =ẑ(ˆp) when the endowment is ω. : F (ˆp, ω) =0}
Economics 201B Second Half Lecture 10, 4/15/10 Debreu s Theorem on Determinacy of Equilibrium Definition 1 Let F : R L 1 + R LI + R L 1 be defined by F (ˆp, ω) =ẑ(ˆp) when the endowment is ω The Equilibrium
More informationMicroeconomic Theory I Midterm
Microeconomic Theory I Midterm November 3, 2016 Name:... Student number:... Q1 Points Q2 Points Q3 Points Q4 Points 1a 2a 3a 4a 1b 2b 3b 4b 1c 2c 4c 2d 4d Each question has the same value. You need to
More informationIn the Ramsey model we maximized the utility U = u[c(t)]e nt e t dt. Now
PERMANENT INCOME AND OPTIMAL CONSUMPTION On the previous notes we saw how permanent income hypothesis can solve the Consumption Puzzle. Now we use this hypothesis, together with assumption of rational
More informationBoundary Behavior of Excess Demand Functions without the Strong Monotonicity Assumption
Boundary Behavior of Excess Demand Functions without the Strong Monotonicity Assumption Chiaki Hara April 5, 2004 Abstract We give a theorem on the existence of an equilibrium price vector for an excess
More informationThe Ohio State University Department of Economics. Homework Set Questions and Answers
The Ohio State University Department of Economics Econ. 805 Winter 00 Prof. James Peck Homework Set Questions and Answers. Consider the following pure exchange economy with two consumers and two goods.
More informationNotes on the Mussa-Rosen duopoly model
Notes on the Mussa-Rosen duopoly model Stephen Martin Faculty of Economics & Econometrics University of msterdam Roetersstraat 08 W msterdam The Netherlands March 000 Contents Demand. Firm...............................
More informationLecture 6: Contraction mapping, inverse and implicit function theorems
Lecture 6: Contraction mapping, inverse and implicit function theorems 1 The contraction mapping theorem De nition 11 Let X be a metric space, with metric d If f : X! X and if there is a number 2 (0; 1)
More informationIntroduction to General Equilibrium: Framework.
Introduction to General Equilibrium: Framework. Economy: I consumers, i = 1,...I. J firms, j = 1,...J. L goods, l = 1,...L Initial Endowment of good l in the economy: ω l 0, l = 1,...L. Consumer i : preferences
More informationRisk Aversion over Incomes and Risk Aversion over Commodities. By Juan E. Martinez-Legaz and John K.-H. Quah 1
Risk Aversion over Incomes and Risk Aversion over Commodities By Juan E. Martinez-Legaz and John K.-H. Quah 1 Abstract: This note determines the precise connection between an agent s attitude towards income
More informationAddendum to: International Trade, Technology, and the Skill Premium
Addendum to: International Trade, Technology, and the Skill remium Ariel Burstein UCLA and NBER Jonathan Vogel Columbia and NBER April 22 Abstract In this Addendum we set up a perfectly competitive version
More informationProblem set 2 solutions Prof. Justin Marion Econ 100M Winter 2012
Problem set 2 solutions Prof. Justin Marion Econ 100M Winter 2012 1. I+S effects Recognize that the utility function U =min{2x 1,4x 2 } represents perfect complements, and that the goods will be consumed
More informationAlp Simsek (MIT) Recitation Notes: 1. Gorman s Aggregation Th eorem2. Normative Representative November 9, Household Theorem / 16
14.452 Recitation Notes: 1. Gorman s Aggregation Theorem 2. Normative Representative Household Theorem 3. Representative Firm Theorem (Recitation 2 on November 6, 2009) (Reference: "Introduction to Modern
More information