lecture17_slides - Econ 121 Intermediate Microeconomics...

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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 17 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 are in zero net supply  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. Resource constraints Consumption good 1: A B 1 2 J x1 C x1 D y1 C y1 C : : : C y1 D J X j y1 j D1 Consumption good 2: J 2 1 B A x2 C x2 D y2 C y2 C : : : C y2 D J X j D1 Input n D 1; : : : ; N : J X j;2 j;1 N .In C In / D In j D1 Thus, we have 2 C N resource constraints. j y2 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 / 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. Input-output economy This simple model assumes a complete separation between consumption / output goods and input goods. This seems a bit farfetched: think of real-world complicated supply chains. Even for natural resources this separation is problematic: think of crude oil vs. refined oil. Nevertheless, we will go along with this assumption for now. (Next class, we will remove this assumption.) Production possibilities frontier Set of .y1; y2/ such that NN 1 J y1 C : : : C y1 y1 D max N subject to 1 J y2 C : : : C y2 j j;1 j;1 j j;2 j;2 f1 .I1 ; : : : ; IN / f2 .I1 ; : : : ; IN / J X j D1 j;1 j;2 In C In D y2 N j D y1 ; all j D y2 ; all j D N In; all n j Production possibilities frontier equivalent to y1 D max N J X j j;1 j;1 j j;2 j;2 f1 .I1 ; : : : ; IN / j D1 subject to J X f2 .I1 ; : : : ; IN / D y2 .w/ Lagrange mult. / N D N In; .w/ Lagrange mult. n/ j D1 J X j D1 j;2 j;1 In C In Production possibilities frontier Lagrangean: LD J X j D1 J X j j;2 j j;1 j;1 j;2 f1 .I1 ; : : : ; IN / C . f2 .I1 ; : : : ; IN / y2 / N j D1 C N X nD1 FOC: j;1 MPn C n D 0 j;1 MPn0 C n0 D 0 Hence, j;1 TRSn;n0 Likewise, for consumption good 2. D j 0;1 TRSn;n0 : J X j;1 j;2 n . In C In j D1 N In / T he boundary of t he production possibilities s et is called t he production possibilities frontier. T his should be contrasted with t he production function discussed earlier t hat depicts t he relationship between t he i nput good and t he o utput good; t he production possibilities set depicts only t he set of o utpr~tgoods t hat is feasible. ( In more advanced t reatments, b oth i nputs and o utputs can b e considered p art of t he production possibilities s et. b ut these t reatments cannot easily be handled with two-dimensional diagrams.) T he sha.pe of t he production possibilities set will deperid on t he n ature of t he underlying technologies. If t he technologies for protlucing c ocon~its and fish exhibit constant r eturns t o scale t he production possibilities set will t ake a n especially simple form. Since by asslrmption t here is only one i nput t o production--Robinsml's labor--the production functions for fish and coconuts will b e simply linear functions of labor. For example. suppose t hat Robinson car1 produce 1 0 ponnds of fish per Production possibilities frontier Once we solve for the PPF, we get some implict equation: T .y1; y2/ D 0 iff .y1; y2/ 2 PPF: Marginal rate of transformation: MRT .y1; y2/ D @T .y ; y / @y1 1 2 @T .y ; y / @y2 1 2 Pareto efficiency WIth production, Pareto efficiency requires not only that the MRS of the two consumers is equalized, but also that they equal the MRT. PARETO EFFICIENCY 605 Production and the Edgeworth box. At each point on t he production possibilities frontier, we can draw a n Edgeworth box t o illustrate t he possible consumption allocations. T he Pareto set describes t he set of P areto efficient bundles given t he amounts of goods 1 and 2 available, but in an economy with production those amounts can themselves be chosen out of t he production possibilities set. Which choices from t he production possibilities set will be P areto efficient choices? Let us think about t he logic underlying t he marginal r ate of substitution condition. We argued t hat in a P areto efficient allocation, t he MRS of consumer A had t o be equal t o t he MRS of consumer B: t he r ate a t which 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 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 But, wait, the model seems to be incomplete. Where do the firms’ profits go? Who owns the firms? Well, if this is a model of the whole economy, the firms must be owned by either A or B , or both. Otherwise, we left the owner of our firm out of the model, hence our model would be incomplete. Competitive Equilibrium: Definition Thus, we need a new primitive in the model, namely the ownership shares:  j;i D fraction of firm j that is owned by consumer i; for i D A; B; j D 1; : : : ; J: We must have:  j;A C  j;B D 1; j D 1; : : : ; J: Competitive Equilibrium: Definition Also, we need to specify the endowments: i !` D endowment of consumption good ` of consumer i Ni In D endowment of input good n of consumer i Competitive Equilibrium: Definition Consumer maximization: A A x A D .x1 ; x2 / solves: max uA.x1; x2/ subject to A p1x1 C p2x2 D p1! A C p2!2 C N X NA w n In C nD1 and likewise for B . Here,  j designates the profit of firm j in equilibrium. J X j D1  j;A j 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: A x1 C B x1 D A !1 C B !1 C X j y1 j A x2 X j C B x2 D A !2 C B !2 C X j y2 j j;1 j;2 NA NB In C In D 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 Just like we did in the case of exchange economies, we will see two proofs of this result. The first proof assumes that preferences and technology are smooth and exploits first-order conditions. The second proof, which is more general, does not assume smoothness; it uses a revealed preference argument. ...
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This note was uploaded on 03/01/2012 for the course ECON 121 taught by Professor Samuelson during the Spring '09 term at Yale.

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