Microeconomics, Block I Part 1

Transcription

1 Microeconomics, Block I Part 1 Piero Gottardi EUI Sept. 26, 2016 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

2 Choice Theory Set of alternatives: X, with generic elements x, y (Weak) Preference Relation: %, binary relation on X x % y: x is weakly preferred to y Rational Choice: for any set B X of available alternatives, choose any element weakly preferred to any other available alternative. That is, any element in C (B; %) = fx 2 B : x % y for all y 2 Bg Relationship between preferences and choices: - % implies choices, but also - choices (potentially observable) allow us to make inferences over individual preferences (unobservable) Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

3 From Preferences to Choices and viceversa % is rational if it is: 1 complete: for each pair x, y 2 X, either x % y and/or y % x holds. 2 transitive: for each triple x, y, z 2 X, x % y and y % z ) x % z Claim For any X nite, C (B; %) is ensured to be nonempty for all B X if % is rational. In contrast, we can nd preferences % violating completeness or transitivity such that C (B; %) is empty. [proof? ) HW 1] If x is chosen when set of available alternatives is B, we say that, for any y 2 B, x % R y, or x is revealed weakly preferred to y % R is a preference relationship consistent with the observation of the choice made (and the rationality of individual) Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

4 Utility Function A utility function U : X! R represents a preference relation % if, for all x, y 2 X : x % y, U(x) U(y) Claim A preference relation can be represented by a utility function only if it is rational [proof?] Note: U(.) representing % is not unique Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

5 Sequences of choices - Consider now a collection of sets of feasible alternatives: B = fb i X, i 2 I g Choice is now a correspondence, C (; %), de ned for B i 2 B: choice rule such that C (B i ; %) B i for all B i 2 B What are the general properties of choice rules? WARP (Weak Axiom of Revealed Preference) is satis ed by C (; %) if: whenever we have x, y 2 B i and x 2 C (B i ; %), for some i 2 I, then for any B j such that x, y 2 B j and y 2 C (B j ; %) we must have that also x 2 C (B j ; %) Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

6 Sequences of choices cts. Proposition 1 If % is rational, then C (; %) must satisfy WARP. Proof. x, y 2 B i and x 2 C (B i ; %) ) x % y y 2 C (B j ; %) ) y % z for all z 2 B j. By transitivity, x % z for all z 2 B j and hence whenever x 2 B j we must have x 2 C (B j ; %). Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

7 Sequences of choices cts. - Can now make more speci c inferences about individual preferences from observation of their choices. - Inverse result of Prop. 1 holds: Proposition 2 If a choice rule C () on B satis es WARP, there always exists a preference relation on X such that C (B i ) = C (B i ; %) for all B i 2 B. Proof. Consider revealed preference relation % R de ned by C (): x % R y whenever 9B i 2 B : x, y 2 B i and x 2 C (B i ; %). It is then immediate to verify that C () = C (; % R ). - Note: % R may not be unique, nor rational, unless B is su ciently rich Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

8 Consumer Set of alternatives: consumption set, set of admissible levels of consumption of the L existing commodities. Will assume, as standard: X = R L + with generic element x X is then a convex set, bounded below. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

9 Consumer Set of alternatives: consumption set, set of admissible levels of consumption of the L existing commodities. Will assume, as standard: X = R L + with generic element x X is then a convex set, bounded below. set of feasible alternatives, attainable by trading in competitive markets with a given income m: budget set Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

10 Markets Competitive markets: consumer takes market prices as given, independent of his decisions (price taker) In addition we consider the case where: - prices are linear: unit price p l of each commodity l 2 f1,.., Lg is xed, independent of level of individual trades (and the same for all agents). - markets are complete: for each commodity l in X there is a market where commodity can be traded - free disposal ) p 0 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

11 Budget set: B(p, m) = ( x 2 X : p x = Set of admissible alternatives is now an in nite set. It is convex, compact for all p 0 How does it change when p changes? and m? Note that B(p, m) = B(αp, αm) for all α > 0 ) L p l x l m l=1 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

12 Consumer s problem: choose level of trades (consumption) so as to maximize utility: max U(x) x s.t. x 2 B(p, m) Does a solution always exist? Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

13 Consumer s problem: choose level of trades (consumption) so as to maximize utility: max U(x) x s.t. x 2 B(p, m) Does a solution always exist? Yes when: - p 0 =) B(p, m) is compact and - U(.) is continuous (maintained assumption) Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

14 Continuity of U(.): Claim: There exists a continuous utility function representing % if: % is rational and continuous (upper contour set P(x) = fy 2 X : y % xg and lower contour set L(x) = fy 2 X : x % yg are closed, for all x 2 X ) Continuity is violated, for instance, by the following lexicographic preferences (for L = 2): x % y whenever either (i) x 1 > y 1 or (ii) x 1 = y 1 and x 2 y 2 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

15 Consumer s demand, general properties solution of consumer s problem for every p, m (choice rule) ) consumer s demand correspondence x(p, m); is nonempty for all p 0, m 0 Will study properties of individual consumer s demand (restrictions on behavior implied by rationality of choice and stability of preferences). - x(p, m) is homogenous of degree zero in p, m: x(p, m) = x(αp, αm) for all p 0, α > 0 (follows from property of budget set established above) - WARP: let x 0 2 x(p 0, m 0 ) whenever p x 0 m and x 0 /2 x(p, m), we must have p 0 x > m 0 8x 2 x(p, m) Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

16 Consumer s demand, general properties, cts. Why? - x(p, m) is a upper hemicontinuous (correspondence, as solution may not be unique) in p, m, for all p 0, m > 0 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

17 Consumer s demand, general properties, cts. Why? - x(p, m) is a upper hemicontinuous (correspondence, as solution may not be unique) in p, m, for all p 0, m > 0 Follows from the Theorem of the Maximum (by the continuity of U(.) and of the budget set correspondence B(p, m)). Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

18 Assumptions on Preferences and Properties of Demand For other properties, may impose (some) additional assumptions: A.1 local non satiation: % is such that, for all x 2 X there exists y 2 N(x) \ X, an arbitrarily small neighborhood in X of x, such that y x (that is, x y). implies that indi erence curves are not thick. corresponding property of U(.)? Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

19 Assumptions on Preferences and Properties of Demand For other properties, may impose (some) additional assumptions: A.1 local non satiation: % is such that, for all x 2 X there exists y 2 N(x) \ X, an arbitrarily small neighborhood in X of x, such that y x (that is, x y). implies that indi erence curves are not thick. corresponding property of U(.)? or the stronger: A.1 monotonicity: y x ) y x [strict if y > x ) y x] corresponds to monotonicity of U(.) Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

20 Assumptions on Preferences and Properties of Demand For other properties, may impose (some) additional assumptions: A.1 local non satiation: % is such that, for all x 2 X there exists y 2 N(x) \ X, an arbitrarily small neighborhood in X of x, such that y x (that is, x y). implies that indi erence curves are not thick. corresponding property of U(.)? or the stronger: A.1 monotonicity: y x ) y x [strict if y > x ) y x] corresponds to monotonicity of U(.) - Claim A.1: Under A.1, x(p, m) satis es the budget identity: p x(p, m) = m for all p, m [Proof?] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

21 Assumptions on Preferences and Properties of Demand cts. A.2 convexity: % is such that, for all x 2 X, the upper contour set P(x) is convex Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

22 Assumptions on Preferences and Properties of Demand cts. A.2 convexity: % is such that, for all x 2 X, the upper contour set P(x) is convex corresponds to quasi-concavity of U(.) Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

23 Assumptions on Preferences and Properties of Demand cts. A.2 convexity: % is such that, for all x 2 X, the upper contour set P(x) is convex corresponds to quasi-concavity of U(.) or the stronger: A.2 strict convexity: if, in addition, y x ) αx + (1 α)y x for all x, y 2 X, α > 0 (that is, indi erence curves have no at parts ; corresponds to strict quasi-concavity of U(.)) Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

24 Assumptions on Preferences and Properties of Demand cts. A.2 convexity: % is such that, for all x 2 X, the upper contour set P(x) is convex corresponds to quasi-concavity of U(.) or the stronger: A.2 strict convexity: if, in addition, y x ) αx + (1 α)y x for all x, y 2 X, α > 0 (that is, indi erence curves have no at parts ; corresponds to strict quasi-concavity of U(.)) - Claim A.2: Under A.2, x(p, m) is convex valued Under A.2 x(p, m) is single valued (a function) [Proof?] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

25 Assumptions on Preferences and Properties of Demand cts. A.3 homothetic: preferences are invariant with respect to expansions along rays from the origin, x y ) αx αy for all x, y 2 X, α > 0 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

26 Assumptions on Preferences and Properties of Demand cts. A.3 homothetic: preferences are invariant with respect to expansions along rays from the origin, x y ) αx αy for all x, y 2 X, α > 0 corresponds to U(.) = g(u(.)), where g(.) is such that g 0 > 0 and u(.) is homogenous of degree 1 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

27 Assumptions on Preferences and Properties of Demand cts. A.3 homothetic: preferences are invariant with respect to expansions along rays from the origin, x y ) αx αy for all x, y 2 X, α > 0 corresponds to U(.) = g(u(.)), where g(.) is such that g 0 > 0 and u(.) is homogenous of degree 1 - Claim A.3: Under A.3, x(p, αm) = αx(p, m) for all m, p 0, α > 0 [Proof? in HW1] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

28 Standard utility functions: Cobb Douglas u(x 1, x 2 ) = x α 1 x 1 CES: u(x 1, x 2 ) = 1 σ x 1 σ + 1 σ x 2 σ, quasi linear: u(x 1, x 2 ) = x 1 + v(x 2 ), with v 0 > 0, v 00 < 0 2 α Do they satisfy A.1, A.2 and A.3? under what conditions? Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

29 FOCs Whenever U(.) is continuously di erentiable, any solution of the consumer s problem satis es the following system of Kuhn Tucker necessary conditions, stated here for the case of interior solutions, assuming A.1 (lns) holds: or, more compactly, and U x l = λp l, l = 1,.., L DU λp = 0 m p x = 0. for some λ > 0 (Lagrange multiplier). If A.2 0 (U(.) strictly quasi concave) also holds, these conditions are also su cient and solution is unique. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

30 Comparative statics Claim: Assume U(.) is twice continuously di erentiable and satis es A.1, A.2 0 : x(p, m) is then a continuously di erentiable function at all p, m such that x(p, m) 0. Under above conditions, properties of x(p, m) can be obtained with IFT from previous FOCs: D 2 U p p T 0 dx dλ = λi x T 0 dp + 1 dm Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

31 Comparative statics cts Claim: Under above assumptions, D 2 U p p T 0 is invertible (when evaluated at (x, λ, p) satisfying FOCs). Proof: Suppose not; i.e. 9(y T, z) 2 R L+1, (y T, z) 6= 0 : D 2 U p y D p T = 2 Uy pz = 0 0 z p y Premultiplying D 2 Uy pz by y T yields: y T D 2 Uy y pz < 0 if y p = 0 (by A.2 0 ), a contradiction. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

32 Comparative statics cts. Setting: we get: D 2 1 U p p T 0 where the following properties hold: D m x = v S v T v t D p x = λs vx T = λs D m x x T v p = 1 S is symmetric, negative semi-de nite and such that p T S = 0. [Proof? in HW2] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

33 Comparative statics cts. Will show the properties found above of D p x, D m x depend on the properties of demand we had already seen: The homogeneity property of x(p, m) implies: D p x p + D m x m = 0 Claim A.1 (budget identity property).then implies: p T D p x + x T = 0 T p D m x = 1 [Check these properties are satis ed by expressions found in previous slide] WARP =)? [more later] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

34 Comparative statics (homothetic preferences) If A.3 also holds, from Claim A.3 we get: D m x = x(p, 1) >> 0 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

35 Indirect Utility Properties: V (p, m) = U(x(p, m)) V (p, m)/ m = λ > 0: consumer s marginal utility of wealth equals the shadow value of relaxing the constraint V (p, m)/ p l 0 for all l, p [argument:?] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

36 Indirect Utility Properties: V (p, m) = U(x(p, m)) V (p, m)/ m = λ > 0: consumer s marginal utility of wealth equals the shadow value of relaxing the constraint V (p, m)/ p l 0 for all l, p [argument:?] V (p, m) is quasi-convex in p: the lower contour set fp : V (p, m) V g is convex [argument: Take any pair p 0, p 00 such that V (p 0, m), V (p 00, m) V, and consider ˆp = αp 0 + (1 α)p 00 for α 2 [0, 1]. Note that for all x such that ˆp x m we must have either p 0 x m and/or p 00 x m; thus U(x) V.] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

37 Indirect Utility Properties: V (p, m) = U(x(p, m)) V (p, m)/ m = λ > 0: consumer s marginal utility of wealth equals the shadow value of relaxing the constraint V (p, m)/ p l 0 for all l, p [argument:?] V (p, m) is quasi-convex in p: the lower contour set fp : V (p, m) V g is convex [argument: Take any pair p 0, p 00 such that V (p 0, m), V (p 00, m) V, and consider ˆp = αp 0 + (1 α)p 00 for α 2 [0, 1]. Note that for all x such that ˆp x m we must have either p 0 x m and/or p 00 x m; thus U(x) V.] Another, more obvious property is that V (p, m) is homogenous of degree zero in p, m Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

38 Duality Consider the problem: min p x x s.t. U(x) u A solution again exists for all u U(0) and p 0 (and may now exist even when p 0 under strict monotonicity). Let us denote its solution for all p, u by x(p, u) (sometimes referred to as compensated - or Hicksian - demand) and E (p, u) = p x(p, u) de ne the expenditure function. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

39 Duality cts. Claim Under A.1, we can show that, for all p 0, m > 0, u > U(0), the following identities hold: x(p, m) = x(p, u ) for u = V (p, m) and E (p, u ) = m (1) x(p, u) = x(p, m ) for m = E (p, u) and V (p, m ) = u [argument: if x 2 x(p, u ), then u(x 0 ) u(x) ) p x 0 p x; if x 2 x(p, m ), then u(x 0 ) > u(x) ) p x 0 > p x. Use then A.1.] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

40 Slutsky Equation Using previous relationship, from FOCs of the compensated demand problem p + µdu = 0 we get (for ū = u = V (p, m)): U(x) ū = 0 D p x(p, u) = λs hence symmetric, negative semi-de nite. [proof? see next page] Hence, for u = V (p, m): D p x(p, m) = D p x(p, u) D m x x T Note: this property can also be obtained by di erentiating the second identity in (1) above, after substituting E (p, u) for m. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

41 Slutsky Equation, cts. Argument for claim D p x(p, u) = λs : Applying again the IFT to the system of FOC s in previous page, evaluated at ū = u = V (p, m): µd 2 U DU DU T 0 dx dµ = I 0 T 0 dp + 1 dū Note next, by comparing FOC s of the primal and dual problem when the solution for x is the same and so is p, that µ = 1/λ. It is then possible to verify [e.g. by using formulae for partitioned inverse] that for some b, z. µd 2 1 U DU λs DU T = 0 b T b z Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

42 Expenditure function Properties of the expenditure function: E (p, u)/ u > 0 (under lns) E (p, u)/ p l 0 for all l = 1,.., L E (p, u) is concave in p [argument: For any pair p 0, p 00, consider ˆp = αp 0 + (1 α)p 00 for α 2 [0, 1]. Then, for any u, E (ˆp, u) = ˆp x(ˆp, u) = αp 0 x(ˆp, u) + (1 α)p 00 x(ˆp, u), which is in turn αe (p 0, u) + (1 α)e (p 00, u).] Also, E (p, u) is homogeneous of degree one in p Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

43 From expenditure and indirect utility functions to demand By the envelope theorem, compensated demand can be obtained from expenditure function: x(p, u) = D p E (p, u) hence the properties of x(p, u) can also be obtained from those of E (p, u) [e.g., that D p x(p, u) is negative semi-de nite] Similarly, di erentiating the equation de ning the indirect utility, V (p, m) = U(x(p, m)), wrt p and using FOCs of the consumer s problem and the property shown above (p.23) p T D p x = x T, yields [Roy s identity]: D p V (p, m) = λx(p, m) = V (p, m) x(p, m) m Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

44 WARP again WARP for consumer s choice problem implies: if p x(p 0, m 0 ) = m and x(p 0, m 0 ) 6= x(p, m), then p 0 x(p, m) > m 0 Can be restated as: for any compensated price change (p 0, m 0 )! (p, m = p x(p 0, m 0 )) we must have (p p 0 ) (x(p, m) x(p 0, m 0 )) = p x 0 (weak inequality because x may be 0) Claim: Take an arbitrary function x(p, m) that is homogenous of degree zero, di erentiable and satis es WARP as well as the budget identity. Then the matrix of compensated price e ects D p x(p, m) + D m x x T is negative semide nite. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

45 WARP again [argument: di erential version of compensated price change: dp, dm : dm = x dp Induced demand change (change in compensated demand) is dx = D p x dp + D m x(x dp) and we must have, by WARP dp dx 0, and this is true for any dp] Note that now WARP is not enough to ensure (such matrix is also symmetric and hence) that we can nd preferences rationalizing demand x(p, m) Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

46 Producer Theory Focus on production activity of rms operating in competitive markets production plan: y 2 R L, net output of the L goods y i < 0 : input y i > 0 : output Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

47 Producer Theory Focus on production activity of rms operating in competitive markets production plan: y 2 R L, net output of the L goods y i < 0 : input y i > 0 : output Set of alternatives: production set, set of (technologically feasible) production plans Y R L assumed to satisfy the following standard properties: - nonempty - closed - Y \ R L + = f0g: no free lunch and possibility of inaction - free disposal: y 2 Y and y 0 y ) y 0 2 Y Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

48 Production function Whenever the commodities which are outputs are xed, O f1,.., Lg (and hence also the complementary set of those which are inputs), the outer boundary of Y can typically be represented by a (continuous) function: the production function, describing the maximal output level attainable for any level of inputs, written here for the case where O = f1g : y 1 = f (z) i (i) (y 1, z) 2 Y 0 > y 1 : (y1 0, z) 2 Y f : R L 1 +! R + (weakly) monotonically increasing by the free disposal property Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

49 Firm s problem: choose production plan so as to maximize rm s pro ts (why? more later): max π = p y s.t. y 2 Y [or max π = p 1 f (z) w z, for w (p 2,.., p L )] Existence of a solution requires now additional conditions (feasible set may not be compact _ analogies with expenditure minimization problem?). Solution for every p 0 : rm s net supply correspondence y(p) Value of the solution: pro t function π(p) = p y(p) Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

50 Properties of Net Supply correspondence and Pro t function y(p) - if nonempty - is homogeneous of degree 0 in p π(p) is homogeneous of degree 1 in p π(p) is a convex function (contrast with E (p, u), and V (p, m)] [argument: take any pair p 0, p 00 and consider ˆp = αp 0 + (1 α)p 00 for α 2 (0, 1). Note that π(ˆp) = y(ˆp) (αp 0 + (1 α)p 00 ) αy(p 0 ) p 0 + (1 α)y(p 00 ) p 00 = απ(p 0 ) + (1 α)π(p 00 ).] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

51 Assumptions on Technology and other Properties of Supply B.1 Convexity: Y is convex. This implies: (i) not only that, analogously to consumer, more balanced combinations of inputs are more productive; (ii) but also that returns to scale are non increasing: reducing scale does not decrease productivity, y 2 Y ) αy 2 Y for all α 2 [0, 1] Corresponding property of f (.): Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

52 Assumptions on Technology and other Properties of Supply B.1 Convexity: Y is convex. This implies: (i) not only that, analogously to consumer, more balanced combinations of inputs are more productive; (ii) but also that returns to scale are non increasing: reducing scale does not decrease productivity, y 2 Y ) αy 2 Y for all α 2 [0, 1] Corresponding property of f (.): f (.) is concave. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

53 Assumptions on Technology and other Properties of Supply B.1 Convexity: Y is convex. This implies: (i) not only that, analogously to consumer, more balanced combinations of inputs are more productive; (ii) but also that returns to scale are non increasing: reducing scale does not decrease productivity, y 2 Y ) αy 2 Y for all α 2 [0, 1] Corresponding property of f (.): f (.) is concave. Claim B.1: Under B.1, y(p) is convex-valued [why?] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

54 Assumptions on Technology and other Properties of Supply Strengthening B.1: B.1 Strict Convexity: y, y 0 2 Y ) αy + (1 α)y 0 2 inty Corresponds to strict concavity of f (.). Implies that: y(p) is single valued for all p 0 Alternatively: Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

55 Assumptions on Technology and other Properties of Supply Strengthening B.1: B.1 Strict Convexity: y, y 0 2 Y ) αy + (1 α)y 0 2 inty Corresponds to strict concavity of f (.). Implies that: y(p) is single valued for all p 0 Alternatively: B.1 Convex Cone: y 2 Y ) αy 2 Y for all α > 0 returns to scale are constant (productivity invariant to scale) Corresponds to homogeneity of degree 1 of f (.). Implies that: - y(p) is not single valued: y 2 y(p) ) αy 2 y(p) for all α > 0 - π(p) = 0 for all p : y(p) is nonempty. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

56 FOCs Whenever f (.) is di erentiable, any solution of the rm s problem satisfy the following system of rst order conditions (stated here for the case of interior solutions - (locally) unconstrained problem): p 1 Df = w also su cient condition if f (.) is concave. If f (.) is strictly concave, y(p) is a continuous function. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

57 Comparative statics Assume f (.) is twice continuously di erentiable and strictly concave: again applying IFT to FOCs in previous slide, we get: D w z = 1 p 1 D 2 f 1 symmetric, negative de nite From pro t to net supply functions: y(p) = D p π (by the envelope theorem), so that D p y = Dp 2 π is: symmetric, positive semi-de nite (by the convexity of π (.)) - recall z = (y 2,.., y L )! - and such that D p y p = 0 (by the homogeneity of y(p)). Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

58 Cost Minimization The input level z which solves the rm s choice problem also solves the following problem: min C = w z s.t. f (z) y 1 This is perfectly analogous to expenditure minimization problem of consumer. Hence we know that: C (w, y 1 ) is concave in w and such that C / y 1 > 0 (if f (.) is strictly increasing) and C / w l 0, l = 2,.., L z(w, y 1 ) = D w C exhibits same properties of compensated demand function (see previous section: D w z(w, y 1 ) negative semi-de nite,..). Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

59 Why Pro t Maximization? Firm is owned by a (set I of) consumers Ownership of a fraction of the rm for consumer i means the right to receive a fraction θ i 2 [0, 1] of the rm s pro ts. When markets are competitive, the choice of alternative production plans by the rm has the following e ect on the consumer s choice problem: max U(x) x s.t. p x m + θ i (p y) Thus rm s choices only a ect consumer by modifying his income. Hence the consumer favors the choice of the production plan that maximizes the rm s pro ts p y Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

60 Aggregate Demand Aggregate demand with H consumers: x(p, m h ) H H h=1 = x h (p, m h ) h=1 What can we say about its properties? It is clearly: continuous (if so are individual demands), m h ) H h=1 homogeneous of degree 0 in (p, satis es Walras law: p x(p, m h ) H h=1, m h for all p, (m h ) H h=1 h its Jacobian has less structure the larger is H [why?]: D p x(p, m h ) H h=1 = λ h S h v h x h T h Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

61 Aggregation: no distribution e ects 1 When does aggregate demand depend only on aggregate income m h m h, not on its distribution? Aggregate demand is invariant: (i) wrt any in nitesimal change in the distribution of income dm h H h=1 : h dm h = 0 : if, and only if D m x h (p, m h ) = D m x h0 (p, m h0 ) - that is v h = v h0 - for any pair h, h 0 (ii) wrt any (also discrete) change in the distribution of income m h H h=1 : h m h = 0 : when, in addition, D m x h (p, m h ) is independent of m h, for all h. That is, consumers have identical homothetic preferences so that x h (p, m h ) = x(p, 1)m h for all h. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

62 Representative consumer 2. When does aggregate demand exhibit the same properties of individual demand? We say a representative consumer exists when there is a utility function U : R L +! R such that: x(p, m h ) H h=1 = x(p, m; U) and x(p, m; U) 2 arg max U(x) s.t. p x m Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

63 Representative consumer cts. It exists: (i) always at a given point ( p, m h ) H h=1 (that is, at p, ( m h ) H h=1, 9U: x( p, m h ) H h=1 = x( p, m; U)). Let: U(x, ξ h H max ) = h ξ h U h (x h ) h=1 s.t h x h (2) x It is easy to verify that if we set ξ h = λ/ λ h for all h, for some λ > 0, we have x( p, m h ) H h=1 2 arg maxpx m U(x, ξ h ). h [argument: Let x h = x h ( p, m h ), x = h x h. We need to show that DU( x) = λ p for some λ > 0, knowing that DU h ( x h ) = λ h p for all h. Note that, at a solution of above problem (2), for all x we have DU(x) = ξ h DU h (x h ) for all h. [why?] Premultiplying DU h ( x h ) = λ h p by ξ h yields: ξ h DU h ( x h ) = ξ h λ h p for all h. Using ξ h = λ/ λ h, we obtain that ξ h DU h ( x h ) is h-invariant, and hence equal to DU( x), and this is in turn equal to λ p.] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

64 Representative consumer cts. (ii) locally (that is, at a given point (p, m h ) H h=1 and wrt in nitesimal changes in p): at (p, m h ) H h=1 9U : x(p, m h ) H h=1 = x(p, m; U)) and D p x(p, m h ) H h=1 = Dp x(p, m; U)). The latter property requires: D p x(p, m h ) H h=1 = λs vx T for some λ > 0, S symmetric, negative semide nite, of rank L 1, such that p T S = 0, and v such that p v = 1. This holds if v h = v for all h (analogies with 1(i)). Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

65 Representative consumer cts. (iii) globally: 9U : x(p, m h ) H h=1 = x(p, m; U) for all p, m h ) H h=1. Property holds if consumers have identical homothetic preferences. In this case U(x) = U h (x) for all h and m = h m h. And if all consumers have quasilinear preferences? Representative consumer may exist, under weaker conditions on preferences, if we x the distribution of income. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

66 Law of Demand and WARP In general, as shown above (1.), aggregate demand depends on income distribution. We can still write it as a function of aggregate income m if we x the distribution of income: m h = α h m for some given α h 0, for all h, α h = 1. h Does aggregate demand x(p, m) satisfy WARP? In general no. Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

67 Law of Demand and WARP In general, as shown above (1.), aggregate demand depends on income distribution. We can still write it as a function of aggregate income m if we x the distribution of income: m h = α h m for some given α h 0, for all h, α h = 1. h Does aggregate demand x(p, m) satisfy WARP? In general no. Take (p, m) and (p 0, m 0 ) such that x (p 0, m 0 ) 6= x (p, m) and: p x p 0, m 0 m. It is immediate to see that the following inequality can still hold: p 0 x (p, m) m 0 [because p x (p 0, m 0 ) m does not imply p x h p 0, m h0 m h for all h; similarly for p 0 x (p, m) m 0.] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

68 Law of Demand and WARP cts. A su cient condition for aggregate demand to satisfy WARP is that individual (uncompensated) demand satis es the Law of Demand (LOD): (p 0 p) x(p 0, m) x(p, m) < 0 for all p, p 0, m: x(p 0, m) 6=x(p, m) [This implies that D p x h is negative semide nite for all h, always true for compensated demand (that is, substitution e ects prevail over income e ects).] This property carries over to aggregate demand and we show it implies WARP. [Let (p, m), (p 0, m 0 ): x(p, m) 6= x(p 0, m 0 ), p x(p 0, m 0 ) m. Take p 00 (m/m 0 )p 0,; x(p 00, m) = x(p 0, m 0 ) by the homogeneity of degree zero of demand. By LOD (p 00 p) [x(p 00, m) x(p, m)] < 0. Using Walras law and the property p x(p 00, m) m we then get p 00 x(p, m) > m,.p 0 x(p, m) > m 0. QED] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

69 Law of Demand and WARP cts. A su cient condition for individual demand to satisfy the Law of Demand is that preferences of every agent are homothetic (though possibly di erent among them). [proof?] Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

70 Aggregation and Production Suppose there are F rms, with production set Y f, f = 1,.., F. When does a representative producer exist? That is, when can we nd a production set Y such that y(p) 2 arg max p y = p y f (p)? y 2Y f Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53

71 Aggregation and Production Suppose there are F rms, with production set Y f, f = 1,.., F. When does a representative producer exist? That is, when can we nd a production set Y such that y(p) 2 arg max p y = p y f (p)? y 2Y Always, no restriction needed in this case: [proof? see HW3] Note also that f set Y = Y f f D w z f = 1 p 1 D 2 f f 1 f which is also always symmetric, negative de nite (LOD). Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, / 53 f