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# Economic Theory ECON 101A

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This 75 page Class Notes was uploaded by Dr. Janiya Bernier on Thursday October 22, 2015. The Class Notes belongs to ECON 101A at University of California - Berkeley taught by S. Dellavigna in Fall. Since its upload, it has received 42 views. For similar materials see /class/226700/econ-101a-university-of-california-berkeley in Economcs at University of California - Berkeley.

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Date Created: 10/22/15

Economics 101A Lecture 3 revised Stefano DellaVigna September 2 2003 Outline 1 Envelope Theorem 2 Convexity and concavity 3 Constrained Maximization 1 Envelope Theorem 0 You now know how 901quot varies if p1 varies o How does the function h vary at the optimum as p1 varies o Differentiate hxip1p2x p1p2p1p2 with respect to p1 dhTP1p27 p1p27p17p2 dPl 3hX7P 3TX7P 3961 3P1 3hX7P 39030551 T 8x2 am 8h X 71 3P1 0 Can we say something about the first two terms They are zero o Envelope Theorem for unconstrained maximization Assume that you maximize function fx p with re spect to 90 Consider then the function f at the op timum that is fxpp The total differential of this function with respect to 19 equals the partial derivative with respect to pi dfXp p 8fXp p dpq 3197 39 0 You can disregard the indirect effects Graphical in tuition 2 Convexity and concavity Function f from C C R to R is concave if H7596 1 0y 2 tf 1 tfy for all 903 6 C and for all t 6 01 0 Notice C must be convex set ie if 96 E C and y E C then tx1 ty EC forte 01 0 Function f from C C R to R is strictly concave if H7596 1 074 gt tf 1 tfy for all 903 6 C and for all t 6 01 0 Function f from R to R is convex if f is concave Alternative characterization of convexity A function f twice differentiable is concave if and only if for all 90 the subdeterminants of the Hessian matrix have the property H1 S O H2 Z 0 H3 S O and so on For the univariate case this reduces to f S O For the bivariate case this reduces to fag 96 S O and 2 3 fyy lt 2 0 A twice differentiable function is strictly concave if the same property holds with strict inequalities 0 Examples 1 For which values of a b and c is f ax3 13902 cm d is the function concave over R Strictly concave Convex 2 Is fxy x2 y2 concave o For Example 2 compute the Hessian matrix fa 7fy 3330 7 79 fl 9513 7 gay Hessian matrix H I H lt 773733 ITy y 0 Compute H1l 3233 and lH2l mm 99 lt gt2 339 Why are convexity and concavity important Theorem Consider a twice differentiable concave convex function over C C R If the point X0 satisfies the fist order conditions it is a global max imum minimum For the proof we need to check that the second order conditions are satisfied These conditions are satisfied by definition of con cavity We have only proved that it is a local maximum 3 Constrained maximization o Nicholson Ch 2 pp 39 46 0 So far unconstrained maximization on R or open subsets o What if there are constraints to be satisfied 0 Example 1 maxyac y subject to 3x y 5 0 Substitute it in maxing a gtlt 5 396 0 Solution 90 0 Example 2 maxing my subject to ac expyy expx 5 0 Solution Graphical intuition on general solution Example 3 maxing fxy a gtlt y st hxy 2 y2 1 0 Draw 0 hxy 2 y2 1 Drawxgt1ltyKwith KgtOVaryK Where is optimum Where dydm along curve my 2 K equals dydm along curve 902 y2 1 0 Write down these slopes o Idea Use implicit function theorem 0 Heuristic solution of system rggyx fy st hxy 0 0 Assume continuity and differentiability of h hy7 0orh7 0 o Implicit function Theorem Express y as a function of 90 or 90 as function of y 0 Write system as maxg fxgx fmgx Wm 51 o What is 839 63 0 Substitute in and get O or f 9 hamp79 fg 9 h 79 o Lagrange Multiplier Theorem necessary condi tion Consider a problem of the type mlrpfgn f 961 962 m an p hl 1 2 an p 0 h2 17 27 397 71 0 hm 1 2 an p 0 with n gt m Let X Xp be a local solution to this problem 0 Assume f and h differentiable at 90 the following Jacobian matrix at X has maximal rank ah ah Tm X a711 X J ah ah Tm X a X 0 Then there exists a vector A A1 gtm such that X A maximize the Lagrangean function Lx A fx p ikg h x P j o Casen2m1 0 First order conditions are afXpgt3hXp0 8967 8907 fori12 o Rewrite as fml hml 1 12 h mQ Constrained Maximization Sufficient condition for the case n 2m 1 If X satisfies the Lagrangean condition and the de terminant of the bordered Hessian ah 8h 0 X X 8h 8 L a L ah 82L 82L 8 2X 83318332 X 83028332 X is positive then X is a constrained maximum If it is negative then x is a constrained minimum Why This isjust the Hessian of the Lagrangean L with respect to A 1 and 2 Example 4 maxm x2 yy2 st x2y2 p 0 maxmy 2 96y 242 M962 242 p Foc with respect to ac Foc with respect to y Foc with respect to A Candidates to solution Maxima and minima Economics 101A Lecture 24 Stefano DellaVigna April 24 2008 Outline 1 Dynamic Games 3 Oligopoly Stackelberg 3 General Equilibrium Introduction 4 Edgeworth Box Pure Exchange 5 Barter 1 Dynamic Games II 0 Can use this to study finiter repeated games 0 Suppose we play the prisoner s dilemma game ten times 1 2 D ND D 4 4 1 5 ND 5 1 2 2 o What is the subgame perfect equilibrium o The result differs if infinite repetition with a proba bility of terminating 0 Can have cooperation 0 Strategy of repeated game Cooperate ND as long as opponent always co operate Defect D forever after first defection 0 Theory of repeated games Econ 104 2 Oligopoly Stackelberg o Nicholson Ch 15 pp 543 545 better than Ch 14 pp 423 424 9th 0 Setting as in problem set 0 2 Firms 0 Cost Cy 2 cy with c gt O 0 Demand pY a bY with a gt c gt O and bgt0 0 Difference Firm 1 makes the quantity decision first 0 Use subgame perfect equilibrium Solution Solve first for Firm 2 decision as function of Firm 1 decision rnygx a by2 byiquot y2 cy2 Foca 2by 2quot by lk c O Firm 2 best response function a c 341 2b 239 y2 o Firm 1 takes this response into account in the max im ization my xw by1 bka y1y1 Cyl or b W D max a c yl 91 2b 2 91 91 oFoc a 2by1 by1 CO or a c and 0 Total production c a Y5y1ltygt2k3T Price equals blt3a cgt 1 3 a a c p 4 b 4 4 0 Compare to monopoly a c yM 2b and ac PM 2 0 Compare to Cournot ygtk gtlt 2265C D y1y2 3b and 1 2 a c 131 3 3 Compare with Cournot outcome Firm 2 best response function 24 2 2b Firm 1 best response function a c 2b y1 Intersection gives Cou rnot a c yl 2 y 2 2 Stackelberg Equilibrium is point on Best Response of Firm 2 that maximizes profits of Firm 1 Plot iso profit curve of Firm 1 l a C 741 by1y2 574 Solve for yg along iso profit a c 1 y2 b yl byl Iso profit curve is flat for dyg dyl 5041 or 91 Figure 3 General Equilibrium Introduction 0 So far we looked at consumers Demand for goods Choice of leisure and work Choice of risky activities 0 We also looked at producers Production in perfectly competitive firm Production in monopoly Production in oligopoly 0 We also combined consumers and producers Supply Demand Market equilibrium 0 Partial equilibrium one good at a time 0 General equilibrium Demand and supply for all goods supply of young workerT wage of experi enced workers minimum wageT effect on higher earners steel tariffT effect on car price 4 Edgeworth Box Pure Exchange 0 Nicholson Ch 13 pp 441 444 476 478 Ch 12 pp 335 338 369 370 9th 0 2 consumers in economy 73 12 0 2 goods 1 2 Endowment of consumer 73 good j w 1 2 1 2 0 Total endowment wl 0J2 wl wl 002 002 o No production here With production as in book wl 0J2 are optimally produced o Edgeworth box 0 Draw preferences of agent 1 0 Draw preferences of agent 2 0 Consumption of consumer 73 good j o Feasible consumption gmmwi o If preferences monotonic w for all 73 0 Can map consumption eves into box 5 Barter 0 Consumers can trade goods 1 and 2 o Allocation can be outcome of barter if 0 Individual rationality Z uiwiw for all 73 A A A A o Pareto EffICIency There IS no allocation 90 90 such that I 164503563 2 963 963 for all 7 with strict inequality for at least one agent o Barter outcomes in Edgeworth box 0 Endowments wl 0J2 0 Area that satisfies individual rationality condition 0 Points that satisfy pareto efficiency 0 Pareto set Set of points where indifference curves are tangent Contract curve Subset of Pareto set inside the individually rational area Contract curve Set of barter equilibria Multiple equilibria Depends on bargaining power Bargaining is time and information intensive proce dure What if there are prices instead 6 Next lecture o Walrasian Equilibrium Economics 101A Lecture 24 Stefano DellaVigna Novem ber 28 2006 Outline 1 Dynamic Games 2 Oligopoly Stackelberg 1 Dynamic Games 0 Nicholson Ch 15 pp 449 454OLD Ch 10 pp 256 259 0 Dynamic games one player plays after the other 0 Decision trees Decision nodes Strategy is a plan of action at each decision node 0 Example battle of the sexes game She He Ballet Football Ballet 21 00 Football 00 12 0 Dynamic version she plays first o Subgame perfect equilibrium At each node of the tree the player chooses the strategy with the highest payoff given the other players strategy 0 Backward induction Find optimal action in last pe riod and then work backward 0 Solution 0 Example 2 Entry Game 1 2 Enter Do not Enter Enter 1 1 100 Do not Enter 05 00 0 Exercise Dynamic version 0 Coordination games solved if one player plays first 0 Can use this to study finiter repeated games 0 Suppose we play the prisoner s dilemma game ten times 1 2 D ND D 4 4 1 5 ND 5 1 2 2 o What is the subgame perfect equilibrium o The result differs if infinite repetition with a proba bility of terminating 0 Can have cooperation 0 Strategy of repeated game Cooperate ND as long as opponent always co operate Defect D forever after first defection 0 Theory of repeated games Econ 104 2 Oligopoly Stackelberg 0 Setting as in problem set 0 2 Firms Cost Cy 2 cy with c gt 0 Demand pY a bY with a gt c gt O and bgt0 0 Difference Firm 1 makes the quantity decision first 0 Use subgame perfect equilibrium Solution Solve first for Firm 2 decision as function of Firm 1 decision rnygx a by2 byiquot y2 cy2 Foca 2by 2quot by lk c O Firm 2 best response function a c 341 2b 239 y2 o Firm 1 takes this response into account in the max im ization my xw by1 bka y1y1 Cyl or b W D max a c yl 91 2b 2 91 91 oFoc a 2by1 by1 CO or a c and 0 Total production c a Y5y1ltygt2k3T Price equals blt3a cgt 1 3 a a c p 4 b 4 4 0 Compare to monopoly a c yM 2b and ac PM 2 0 Compare to Cournot ygtk gtlt 2265C D y1y2 3b and 1 2 a c 131 3 3 Compare with Cournot outcome Firm 2 best response function 24 2 2b Firm 1 best response function a c 2b y1 Intersection gives Cou rnot a c yl 2 y 2 2 Stackelberg Equilibrium is point on Best Response of Firm 2 that maximizes profits of Firm 1 Plot iso profit curve of Firm 1 l a C 741 by1y2 574 Solve for yg along iso profit a c 1 y2 b yl byl Iso profit curve is flat for dyg dyl 5041 or 91 Figure 3 General Equilibrium Introduction 0 So far we looked at consumers Demand for goods Choice of leisure and work Choice of risky activities 0 We also looked at producers Production in perfectly competitive firm Production in monopoly Production in oligopoly 0 We also combined consumers and producers Supply Demand Market equilibrium 0 Partial equilibrium one good at a time 0 General equilibrium Demand and supply for all goods supply of young workerT wage of experi enced workers minimum wageT effect on higher earners steel tariffT effect on car price 4 Edgeworth Box Pure Exchange Nicholson Ch 12 pp 335 338 369 370 OLD Ch 16 pp 422 425 0 2 consumers in economy 73 12 0 2 goods 1 2 Endowment of consumer 73 good j w 1 2 1 2 0 Total endowment wl 0J2 wl wl 002 002 o No production here With production as in book wl 0J2 are optimally produced o Edgeworth box 0 Draw preferences of agent 1 0 Draw preferences of agent 2 0 Consumption of consumer 73 good j o Feasible consumption gmmwi o If preferences monotonic w for all 73 0 Can map consumption eves into box 5 Next lecture o Barter 0 General equilibrium Economics 101A Lecture 3 Stefano DellaVigna September 5 2006 Outline 1 Envelope Theorem 2 Convexity and concavity 3 Constrained Maximization 4 Envelope Theorem 1 Envelope Theorem Ch 2 pp 33 37 OLD 34 39 You now know how 901quot varies if p1 varies How does h X vary as p1 varies Differentiate hTP17P27 P17P27P17P2 With respect to p1 dhip1p27 p1p27p1p2 dPl 8W p anew p 3961 3191 I 8hx p 8 X p T 8x2 am 8h X p 3191 o The first two terms are zero o Envelope Theorem for unconstrained maximization Assume that you maximize function fx p with re spect to 90 Consider then the function f at the op timum that is fxpp The total differential of this function with respect to 19 equals the partial derivative with respect to pi dfXp p 8fXp p dpq 3197 39 0 You can disregard the indirect effects Graphical in tuition 2 Convexity and concavity Function f from C C R to R is concave if H7596 1 0y 2 tf 1 tfy for all 903 6 C and for all t 6 01 0 Notice C must be convex set ie if 96 E C and y E C then tx1 ty EC forte 01 0 Function f from C C R to R is strictly concave if H7596 1 074 gt tf 1 tfy for all 903 6 C and for all t 6 01 0 Function f from R to R is convex if f is concave Alternative characterization of convexity A function f twice differentiable is concave if and only if for all 90 the subdeterminants of the Hessian matrix have the property H1 S O H2 Z 0 H3 S O and so on For the univariate case this reduces to f S O For the bivariate case this reduces to fag 96 S O and 2 3 fyy lt 2 0 A twice differentiable function is strictly concave if the same property holds with strict inequalities 0 Examples 1 For which values of a b and c is f ax3 13902 cm d is the function concave over R Strictly concave Convex 2 Is fxy x2 y2 concave o For Example 2 compute the Hessian matrix fa 7fy 3330 7 79 fl 9513 7 gay Hessian matrix H I H lt 773733 ITy y 0 Compute H1l 3233 and lH2l mm 99 lt gt2 339 Why are convexity and concavity important Theorem Consider a twice differentiable concave convex function over C C R If the point X0 satisfies the fist order conditions it is a global max imum minimum For the proof we need to check that the second order conditions are satisfied These conditions are satisfied by definition of con cavity We have only proved that it is a local maximum 3 Constrained maximization 0 Ch 2 pp 38 44 OLD 39 46 0 So far unconstrained maximization on R or open subsets o What if there are constraints to be satisfied 0 Example 1 maxyac y subject to 3x y 5 0 Substitute it in maxing a gtlt 5 396 0 Solution 90 0 Example 2 maxing my subject to ac expyy expx 5 0 Solution Graphical intuition on general solution Example 3 maxing fxy a gtlt y st hxy 2 y2 1 0 Draw 0 hxy 2 y2 1 Drawxgt1ltyKwith KgtOVaryK Where is optimum Where dydm along curve my 2 K equals dydm along curve 902 y2 1 0 Write down these slopes o Idea Use implicit function theorem 0 Heuristic solution of system rggyx fy st hxy 0 0 Assume continuity and differentiability of h hy7 0orh7 0 o Implicit function Theorem Express y as a function of 90 or 90 as function of y 0 Write system as maxg fxgx fmgx Wm 51 o What is 839 63 0 Substitute in and get O or f 9 hamp79 fg 9 h 79 o Lagrange Multiplier Theorem necessary condi tion Consider a problem of the type mlrpfgn f 961 962 m an p hl 1 2 an p 0 h2 17 27 397 71 0 hm 1 2 an p 0 with n gt m Let X Xp be a local solution to this problem 0 Assume f and h differentiable at 90 the following Jacobian matrix at X has maximal rank ah ah Tm X a711 X J ah ah Tm X a X 0 Then there exists a vector A A1 gtm such that X A maximize the Lagrangean function Lx A fx p ikg h x P j o Casen2m1 0 First order conditions are afXpgt3hXp0 8967 8907 fori12 o Rewrite as fml hml 1 12 h mQ Constrained Maximization Sufficient condition for the case n 2m 1 If X satisfies the Lagrangean condition and the de terminant of the bordered Hessian ah 8h 0 X X 8h 8 L a L ah 82L 82L 8 2X 83318332 X 83028332 X is positive then X is a constrained maximum If it is negative then x is a constrained minimum Why This isjust the Hessian of the Lagrangean L with respect to A 1 and 2 Example 4 maxm x2 yy2 st x2y2 p 0 maxmy 2 96y 242 M962 242 p Foc with respect to ac Foc with respect to y Foc with respect to A Candidates to solution Maxima and minima 4 Envelope Theorem II o Envelope Theorem Ch 2 pp 44 OLD 46 47 0 Envelope Theorem for Constrained Maximiza tion In problem above consider Fp E fxp p We are interested in dFpdp We can neglect in direct effects 3fXpp m A8hjXPJP dpi 8197 320 7 3197 Example 4 continued maxing 2 my y2 st df 9619 yp dp Envelope Theorem 5 Next Class 0 Next class Preferences Utility Maximization where we get to apply max imization techniques the first time

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