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lecture19_slides - Econ 121 Intermediate Microeconomics...

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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 19 Outline of the course I. Introduction II. Individual choice III. Competitive markets IV. Market failure Outline of the course I. Introduction II. Individual choice III. Competitive markets     Partial equilibrium (Ch. 16) Exchange economies (Ch. 31) Production (Ch. 18–23) Efficiency and equilibrium with production (Ch. 32) IV. Market failure A simple economy with production  2 consumers, A and B  2 consumption goods, 1 and 2, which A and B care about J 1 firms, labeled j D 1; : : : ; J  N input goods, n D 1; : : : ; N , which the firms can use to produce consumption goods 1 and 2  Consumption goods may or may not be in positive net supply: !1 N 0; !2 N 0:  Inputs goods are in positive net supply: N N N I1 > 0; I2 > 0; : : : ; IN > 0 A simple economy with production  Preferences B B A A uA.x1 ; x2 / and uB .x1 ; x2 /  Technology constraints of firm j : j j j;1 j;1 j j j;2 j;2 y1 D f1 .I1 ; : : : ; IN / y2 D f2 .I1 ; : : : ; IN / where j yi j;i In D # of units of consumption good i produced by firm j D # of units of input n used by firm j in the production of cons. good i Thus, firms are heterogeneous in their technologies. Allocations An allocation is a list .x A; x B ; I 1;1 ; : : : ; I J;1 ; I 1;2; : : : ; I J;2; y 1; : : : ; y J / comprising:  a consumption bundle for each consumer: A A x A D .x1 ; x2 /; B B x B D .x1 ; x2 /  an output bundle for each firm j D 1; : : : ; J : j j y j D .y1 ; y2 /  an input bundle for each firm j D 1; : : : ; J and each consumption good i D 1; 2: j;i j;i I j;i D .I1 ; : : : ; IN / Resource constraints Consumption good 1: A B x1 C x1 !1 C N J X y1 J X y2 j j D1 Consumption good 2: B A x2 C x2 !2 C N j D1 Input n D 1; : : : ; N : J X j;2 j;1 .In C In / j D1 Thus, we have 2 C N resource constraints. N In j Pareto efficiency An allocation .x A; x B ; I 1;1; : : : ; I J;1 ; I 1;2; : : : ; I J;2; y 1; : : : ; y J / is feasible if it satisfies the technology constraints and the resource constraints. A feasible allocation .x A; x B ; I 1;1; : : : ; I J;1 ; I 1;2; : : : ; I J;2; y 1; : : : ; y J / is Pareto efficient if there does not exist another feasible allocation .x 0A; x 0B ; I 1;1; : : : ; I 0J;1 ; I 01;2; : : : ; I 0J;2 ; y 01; : : : ; y 0J / such that uA.x 0A/ uA.x A/ uB .x 0B / uB .x B /; with at least one strict inequality. Note that the firm’s profits do not enter the definition, but that is OK. This model implicitly assumes that the firms are owned by the consumers, who get shares of their profits. Competitive Equilibrium: Definition  Allocation: .x A; x B ; I 1;1; : : : ; I J;1; I 1;2 ; : : : ; I J;2 ; y 1; : : : ; y J /; where i i x i D .x1; x2/; j j for i D A; B , j D 1; : : : ; J and ` D 1; 2.  Prices for the consumption goods: p D .p1; p2/  Prices for the inputs: w D .w1; : : : ; wn/ such that... j;` j;` y j D .y1 ; y2 / and I j;` D .I1 ; : : : ; In /: Competitive Equilibrium: Definition .... such that:  consumers take prices p and w as given, take firms’ profits as given and maximize their utility subject to their budget constraint;  firms take prices p and w as given and maximize their profit subject to their technological constraint;  the markets for input goods and consumption goods clear. Competitive Equilibrium: Definition Consumer maximization: A A x A D .x1 ; x2 / solves: max uA.x1; x2/ subject to p1x1 C p2x2 A A p 1 !1 C p 2 !2 C N X nD1 NA w n In C J X  j;A j j D1 and likewise for B . A A NA Here,  j designates the profit of firm j in equilibrium; !1 , !2 and In are A’s endowments; and the  j;A’s are his shares of profits. Competitive Equilibrium: Definition Firm maximization: j j j;1 j;1 j;2 j;2 y j D .y1 ; y2 /, I j;1 D .I1 ; : : : ; IN / and I j;2 D .I1 ; : : : ; IN / solves  j D max p 1 y1 C p 2 y2 N X 2 1 wn.In C In / nD1 subject to 1 1 y1 D f1.I1 ; : : : ; IN / 2 2 y2 D f2.I1 ; : : : ; IN /: Competitive Equilibrium: Definition Market clearing conditions: A x1 C B x1 A !1 C B !1 C X y1 X y2 j j A x2 C B x2 A !2 C B !2 C j j X j j;1 j;2 In C In NA NB In C In ; for n D 1; : : : ; N: First Welfare Theorem with Production Theorem. The allocation of every competitive equilibrium is Pareto efficient. First Welfare Theorem with Production Proof using Revealed Preference. J Let ..x i /i DA;B ; .y j ; I j;1; I j;2/j D1; p; w/ be a competitive equilibrium, where xi i i D .x1; x2/ yj D .y1 ; : : : ; yN / j j;1 j j;1 I j;1 D .I1 ; : : : ; IN / j;2 2;1 I j;2 D .I1 ; : : : ; IN / w D .w1; : : : ; wN / p D .p1; p2/ FWT: Proof (cont’d) Suppose the allocation of this CE is not Pareto efficient. Then, there is some feasible allocation ..x 0i /i DA;B ; .y 0j ; I 0j;1; I 0j;2// such that either uA.x A/ < uA.x 0A/ or uB .x B / uB .x 0B / or uA.x A/ uA.x 0A/ or uB .x B / < uB .x 0B /: To simplify the proof we will cheat a bit and assume that uA.x A/ < uA.x 0A/ or uB .x B / < uB .x 0B /: FWT: Proof (cont’d) Then, by consumer maximization, we must have p1 x 0 A 1 C p2 x 0 A 2 > A p 1 !1 C A p 2 !2 C X NA w n In C n where j D profit of firm j in the competitive equilibrium; and likewise for consumer B . X j jA j ; FWT: Proof (cont’d) Also, by profit maximization, we must have  j j p1 y 0 1 C X j p2 y 0 2 wn.I 0j;1 C I 0j;2/; n n n since .y 0j ; I 0j;1; I 0j;2/ satisfies the technology constraints. Therefore, p1 x 0 A 1 C p2 x 0 A 2 > A p 1 !1 C A p 2 !2 C X NA w n In n C X j  j jA p1y 01 C j p2 y 0 2 X n wn.I 0j;1 n C  I 0j;2/ n : FWT: Proof (cont’d) Since a similar inequality must hold for B , we have: p1 x 0 A 1 C p2 x 0 A 2 > A p 1 !1 C A p 2 !2 C X NA w n In n C X  j jA p1y 01 C j p2 y 0 2 X wn.I 0j;1 n X wn.I 0j;1 n  :  : C I 0j;2/ n C I 0j;2/ n n j and p1 x 0 B 1 C p2 x 0 B 2 > B p 1 !1 C B p 2 !2 C X NB w n In n C X j jB  j p1 y 0 1 C j p2 y 0 2 n FWT: Proof (cont’d) Adding the two inequalities yields B A B A p1.x 0A C x 0B / C p2.x 0A C x 0B / > p1.!1 C !1 / C p2.!2 C !2 / 2 2 1 1 X NA NB C wn.In C In / n C X j  j j .jA C jB / p1y 01 C p2y 02 X n  wn.I 0j;1 C I 0j;2/ : n n FWT: Proof (cont’d) P j !2 C j y 0 2 N P j !1 C j y 0 1 N D!1 N D!2 N ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ A A B B p1. x 0A C x 0B / C p2. x 0A C x 0B / > p1.!1 C !1 / C p2.!2 C !2 / 1 2 1 2 X NA NB wn.In C In / C „ ƒ‚ … n C X j .jA N DIn  jB / C „ ƒ‚ … D1 j p1 y 0 1 C j p2 y 0 2 X n wn.I 0j;1 n C  I 0j;2/ n : FWT: Proof (cont’d) Thus, p1.!1 C N X j y 01 / C p2.!2 C N X j N N y 0 2 / > p 1 !1 C p 2 !2 j j C X N w n In n C p1 X j y 01 C p2 j X j y 02 j XX j wn.I 0j;1 C I 0j;2/: n n n which implies 0> X n N w n In XX n D wn.I 0j;1 C I 0j;2/ n n j X n N w n In X n wn X .I 0j;1 C I 0j;2/ n n j X n N w n In X n N wnIn D 0: FWT: Proof (cont’d) Thus, we have shown that if we assume that the allocation of a competitive equilibrium is not Pareto efficient, then 0 > 0, a contradiction. Therefore, the allocation of every equilibrium must be Pareto efficient. Q.E.D. Second-midterm material ends here. Overview of monopoly behavior Let us begin with the non-discriminating monopolist.  Downward sloping (inverse) demand: P .q/  Increasing, convex cost: C.q/ (convex cost means marginal cost increasing)  Revenue: R.q/ D P .q/q  Monopolist maximizes profit R.q/ c .q/ over all q  0. Non-discriminating Monopolist. Monopolist maximizes profit ….q/ D P .q/q C .q/ over all q  0, and so the first-order condition yields: 0 D …0.q/ D P .q/ C P 0.q/q C 0.q/   q dP D P .q/ 1 C .q/ P .q/ dq … „ ƒ‚ D C 0.q/; 1=j".P .q//j since denoting by Q./ the demand function and p D P .q/ we have p dQ ".p/ D .p/ D q dp 1 : q dP .q/ p dq Non-discriminating Monopolist Therefore, P .q/ D C 0.q/ 1 1 j".q/j D M C.q/ j".q/j j".q/j 1 with j".q/j > 1 (elastic part of demand): Why do we have j".q/j > 1 at the maximum? If j".q/j < 1 then the marginal revenue at q is R0.q/ D P .q/ j".q/j 1 < 0; j".q/j and so revenue is a decreasing function near q . Therefore, by slightly decreasing q the monopolist would both increase his revenue and decrease his cost (because C.q/ decreases when q decreases). This means the monopolist cannot maximize his profit at an output level q where demand is inelastic. Non-discriminating Monopolist  linear case: p.y/ D a MR D a 2 by . by . Marginal revenue is given by  constant elasticity: q D Ap ", with 0 < " < 1. – in this case, MR D p.1 1="/. – markup on marginal cost Inefficiency of monopoly  Pareto efficiency means no way to make some group better off without hurting some other group  Pareto inefficient means there is some way to make some group better off without hurting some other group  monopoly is Pareto inefficient since p > M C  measure of deadweight loss - value of lost output. Price discrimination  first degree - perfect price discrimination, i.e., each unit of the good is sold to the individual who values it most highly at the maximum price the individual is willing to pay – gives Pareto efficient output – producer gets all surplus  second degree - nonlinear pricing – – – – two demand curves would like to charge each full surplus but have to charge bigger one less to ensure self-selection but then want to reduce the amount offered to smaller consumer  third degree - most common – two groups – max p1.y1/y1 C p1.y2/y2 c .y1 C y2/ – first-order conditions: 0 p1 C p1.y1/y1 D c 0.y1 C y2/ 0 p2 C p2.y2/y2 D c 0.y1 C y2/; – or p1 1 p2 1 1 D MC j"1j 1 D MC j"2j – Result: more elastic users pay lower prices: j"1j < j"2j implies p1 > p2 . Two-part tariffs  what happens if everyone is the same?  entrance fee = full surplus  usage fee = marginal cost Bundling  type A: willingness to pay 120 for word processor, 100 for spreadsheet  type B: willingness to pay 100 for word processor, 120 for spreadsheet  no bundling profits = 400  bundling profits = 440  reduce dispersion of willingness to pay ...
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