Chapter 5 Notes. These notes correspond to chapter 5 of Mas-Colell, Whinston, and Green.

Transcription

1 Chater 5 Notes These notes corresond to chater 5 of Mas-Colell, Whinston, and Green. 1 Production We now turn from consumer behavior to roducer behavior. For the most art we will examine roducer behavior in isolation, leaving the study of general euilibrium for the next course. We begin with a discussion of the roducer s objectives and develo some assumtions underlying roducer behavior. We will then discuss the roducer s roblem and study the seci c case of cost and suly for a roduction technology that roduces a single outut. Generally seaking we will consider our roducers to be rms, although it should be noted that the theory develoed alies eually to all tyes of roducers, whether they are called rms, roduction units, families, etc. There may be additional assumtions/restrictions that one desires to make when discussing rms that are not controlled by agents that no not have identical references. The theory develoed here alies directly to a rm comosed of a single individual or agents with identical references. Whether these assumtions are alicable for other models of rms, such as artnershis, cororations with owners and managers, etc., is a decision each researcher needs to make on his or her own. The rm is an interesting economic agent. There are many otential uestions one can ask about the rm. Such uestions could be: 1. Who owns the rm? Does the identity of the owner change the rm s objectives? 2. Who manages the rm? If the owner and the manager are di erent agents, how does this a ect the rm s behavior? 3. How is the rm organized? Do di erent organizational forms of the rm romote or inhibit e ciency? 4. What can the rm do? These are interesting and imortant uestions that an individual could turn into a long and fruitful academic career, and by no means is this an exhaustive list. Our rimary focus will be on the 4 th item, What can the rm do? In articular, we will begin by assuming that the rm can transform inuts into oututs. The rm s goal is to make ro ts seci cally, the rm s goal is to maximize ro ts. It is ossible that the rm has other goals many rms set ro t targets or sales targets (in $) or sales targets (in uantity of items sold) or wish to maximize sales. We will focus on ro t maximization as (1) it rovides a close aroximation to the goals of many rms; (2) it is consistent with utility maximization if the rm is controlled by a single individual or agents with identical references; and (3) it allows us to use the tools develoed in the study of consumer theory to solve the rm s roblem. Now, if you were to start a business one thing you would want to know is exactly how your rm would transform inuts into oututs (and ultimately ro ts). In the theory constructed, the how is assumed to be simly a black-box of roduction. We do not know how, just that there is some technology that exists that allows the rm to transform inuts into oututs. Thus, our view of the theory is fairly accurately reresented by the Underants Gnomes in South Park. 1 The Underants Gnomes have a business lan: Phase 1: Collect underants 1 Eisode 217, titled Gnomes, originally aired on 12/16/

2 Phase 2: (Gnome shrugs shoulders, suggesting he does not know) Phase 3: Pro t Well, they ski the stage where they turn the inuts into oututs, but for the most art this is dead on. Our rms transform inuts (underants, hase 1) into oututs to make ro ts (hase 3), and the exact rocess is unknown (hase 2). We will be slightly (but not much) more formal than shrugging our shoulders when discussing the roduction rocess, but will simly secify it as a roduction function, and will list certain roerties of that function. Again, should you attemt to enter the business world, most lending agents will want a little more than a shrug of the shoulders or that you have some unseci ed roduction function. However, for general theoretic uroses the generic roduction function will su ce. While the Underants Gnomes ortrayal of the theory is a little harsh, it is imortant to note that the theory does not discuss the how of roduction. It is also imortant to note that given some minimum assumtions about rms, which in essence is what we will discuss, the economy will be able to obtain a cometitive euilibrium outcome (we will discuss this in chater 10 and you will also see this next semester in discussing general euilibrium). Thus, there is beauty in the simlicity of the theory in that it allows for euilibrium to be achieved with minimal assumtions. 2 Production Sets Consider an economy with L commodities. A roduction vector (also inut-outut vector, netut vector, or roduction lan) is a vector y = (y 1; :::; y L ) 2 R L that describes the net oututs of the L commodities from a roduction rocess. In the most general notation, it is assumed that inuts are negative numbers (a rm uses 7 units of y 1, so 7 units are consumed) and that oututs are ositive numbers (a rm makes 5 units of y 2, so 5 units are created). It could be that L = 5 so y 2 R 5 and y = (7; 4; 2; 0; 6). In this case, the rm will use 4 units of y 2 and 6 units of y 5 (net) in the roduction rocess and will create 7 units of y 1 and 2 units of y 3 (net). The entry for y 4 = 0, so that there is no net roduction of y 4. Note that the rm may use 9 units of y 2 in the roduction rocess, but that the roduction rocess returns 5 of those units. Our vector y simly rovides the net roduction of the L commodities. The rst uestion to ask is which roduction vectors are ossible. This is the analog to the consumer s consumtion set which consumtion bundles are ossible. The set of all roduction vectors that constitute feasible lans for the rm is the roduction set, denoted Y R L. Any y 2 Y is ossible, and any y =2 Y is not. Note that because we are allowing for inuts and oututs that we do not make the restriction that Y is constrained to be ositive as we did in consumer theory. Like the consumer, the rm faces constraints. These constraints may be technological constraints, legal constraints, re-commitments restricted by contracts there are any number of constraints the rm may face. We can describe the roduction set Y by the transformation function F (), where Y = y 2 R L : F (y) 0 and F (y) = 0 if and only if y is an element of the boundary of Y. The set of boundary oints of Y, y 2 R L : F (y) = 0 is known as the transformation frontier. If F () is di erentiable, and the roduction vector y satis es F (y) = 0, then for any commodities ` and k, the marginal rate of transformation of y` for y k (y) =@y` MRT`k (y) =@y k The marginal rate of transformation is simly a measure of how much the net outut of y k can increase if the rm decreases the net outut of y` by one unit. 2.1 Some roerties of roduction sets The following is a list of some standard roerties of roduction sets. meant to imly that all of the roerties hold for all roduction sets. It is not a comlete list, nor is it 1. Y is nonemty just like the consumer s roblem is uninteresting if the consumer has no wealth to send, the rm s roblem is uninteresting if it cannot do anything. This assumtion simly states that the rm needs to be able to do something in order for us to study it. 2

3 2. Y is closed we know that a set is closed if its comlement is oen. Essentially, the closedness of the roduction set means that it includes the transformation frontier. 3. No free lunch We cannot have y 2 Y and y 0. If both y 2 Y and y 0 then this would mean that the rm could roduce some outut without using any inuts. Even the Underants Gnomes needed underants. 4. Possibility of inaction We can have 0 2 Y. Thus, the rm is allowed to be inactive. This seems to be a reasonable assumtion if the rm has not yet undertaken any commitments to costly activities and is only thinking about the ossibility of becoming a rm. However, it may not be as reasonable if the rm has already incurred some costs the rm may still decide to be inactive, but it has already made some commitments. 5. Free disosal holds if the absortion of any additional amounts of inuts without any reduction in outut is always ossible. The extra amount of inuts can be disosed of or eliminated at no cost. 6. Irreversibility If y 2 Y and y 6= 0, then y =2 Y. If I make oututs using inuts, I cannot reverse the roduction rocess and turn the outut back into the same amount of original inuts. 7. Nonincreasing returns to scale If Y exhibits nonincreasing returns to scale, for any y 2 Y, then y 2 Y for 2 [0; 1]. This means that the roduction rocess can be scaled down. 8. Nondecreasing returns to scale If Y exhibits nondecreasing returns to scale for any y 2 Y, then y 2 Y for 1. This means that the roduction rocess can be scaled u. 9. Constant returns to scale If Y exhibits constant returns to scale for any y 2 Y, then y 2 Y for 0. The roduction rocess can be scaled u or down. 10. Additivity (free entry) Let y; y 0 2 Y. Then y + y 0 2 Y, or Y + Y Y. This imlies that if y 2 Y, then ky 2 Y for all k = Z > 0, where Z is the set of integers. From an economic viewoint, additivity means that a rm with one roduction lant can set u a second roduction lant and that the second lant will not interfere with the rst. Also, if one rm roduces y and the other roduces y 0, then aggregate roduction is y + y 0. So roduction sets must satisfy additivity when free entry is ossible. 11. Convexity We know what convexity is. The roduction set Y is convex. If y; y 0 2 Y and 2 [0; 1], then y + (1 ) y 0 2 Y. Convexity incororates two ideas about roduction ossibilities. Number one is nonincreasing returns. If inaction is ossible, then convexity imlies nonincreasing returns because y + (1 ) 0 2 Y. Also, balanced lans are more roductive than imbalanced lans, so that if y and y 0 roduce the same amount of outut, some convex combination of y and y 0 will roduce at least as much outut as y and y 0. Again, this is similar to the consumer tyically choosing mixed bundles rather than bundles of redominantly one tye of good. 12. Y is a convex cone This occurs when Y is convex and also satis es constant returns to scale. Y is a convex cone if for any roduction vector y; y 0 2 Y and constants 0 and 0, we have y + y 0 2 Y. Proosition 1 The roduction set Y is additive and satis es the nonincreasing returns condition if and only if it is a convex cone. Proof. If the roduction set Y is a convex cone, then it is additive. Let y; y 0 2 Y. Statement Reason 1. y + y 0 2 Y 1. De nition of convex cone 2. Y is additive 2. From 1 and def. of additive Proof. If the roduction set Y is a convex cone, then it satis es the nonincreasing returns condition. y 2 Y. Statement Reason 1. y 2 Y 1. De nition of convex cone 2. Y satis es nonincreasing returns 2. From 1 and def. of nonincreasing returns Let 3

4 Proof. If Y is additive and satis es the nonincreasing returns condition, then it is a convex cone (or, y; y 0 2 Y and and are ositive constants, then y + y 0 2 Y ) Statement Reason 1. k 2 Z such that k > max (; ) 1. We know that there is no greatest integer 2. ky 2 Y; ky 0 2 Y 2. Additivity 3. y = k ky, y = k ky 3. Arithmetic 4. k < 1, k < 1 4. From 1 5. y 2 Y, y 0 2 Y 5. Nonincreasing returns 6. y + y 0 2 Y 6. Additivity This result should be considered carefully. Nonincreasing returns states that rms can scale roduction rocesses down. Additivity states that the oeration of additional lants do not interfere with one another. If both of these hold then convexity is obtained. Note that this does not say that a convex combination of the technology is better than one of the extreme technologies, just that it is available in the roduction set. 3 Pro t Maximization (PMP) The rm s goal is to maximize ro ts. For now, we assume that rms are rice-taking, meaning that the rices for the L commodities (both inuts AND oututs) are given and that the rm s behavior does not a ect these rices. The rm s ro t is then: The rm s ro t maximization roblem is then: or y = P L `=1 `y`. max y max y y s.t. y 2 Y y s.t. F (y) 0. The ro t function, (), associates to every the amount () = max fy : y 2 Y g. We also have the suly corresondence at, y (), which is the set of ro t-maximizing vectors y () = fy 2 Y : y = ()g. If the transformation function is di erentiable then we can nd rst-order conditions to characterize the solution to the consumer s roblem. To do this we can set u the Lagrangian as we did for consumer theory: The rst-order conditions are given by: max y L (y; ) = y + ( = (y = 0 for all ` = 1; :::; L Alternatively, the rst-order condition can be rearranged so that: ` (y The ratio of the rst-order condition for commodities ` and k is: ` k (y ) (y ) =@y k = MRT`k Note that this condition is similar to the one found in consumer theory that euates the marginal rate of substitution with the rice ratio when the consumer is at an otimum (and the nonnegativity constraints are not binding). 4

5 3.1 The distinct outut case The revious discussion alies to the general theory of roduction. In most cases the researcher (as well as the real-world rm) is concerned with the case of distinct inuts and distinct oututs. We retain the world of L commodities, and assume that there are M oututs and L M inuts. We denote oututs as the vector M 2 R M + and inuts as the vector z L M 2 R+ L M. Note that in the case with distinct inuts and oututs that both the oututs AND the inuts are assumed to be nonnegative, so = ( 1 ; :::; M ) 0 and z = (z 1 ; :::; z L M ) 0. The most basic case is that of multile inuts and a single outut (M = 1). This case can be described by the roduction function, f (z). The roduction function describes the maximum outut of that can be roduced using the z inuts. In the simlest economic models, the inuts are given generic identities like caital and labor. Slightly more comlex models may allow for di erent tyes of caital (old and new technologies), or di erent tyes of labor (erhas skilled and unskilled laborers). But in the general theories of economics seci c factors of roduction are tyically not seci ed. If outut is held constant at, then the marginal rate of technical substitution (MRTS) of z` for z k at z (z) =@z` MRT S`k = (z) =@z k The MRTS is a measure of how much of z k is needed to kee outut at = f (z) when z` is marginally decreased. When Y is a single-outut technology, we can write: max z0 f (z) wz where is a scalar denoting the rice of the outut good and w is a vector of inut rices. rst-order conditions we see (z) w` 0 z` and when z` > (z) z` w` = w` (z) z` Finding the Note is simly the marginal roduct of the inut z`, or MP z` z`. This result states that the MP z` is eual to the relative rice of the inut to the outut. Alternatively, if we nd the ratio of the rst-order conditions for two inuts, we get: w` (z ) =@z` w (z. ) =@z k Note )=@z k is the marginal rate of technical substitution of the inuts, or alternatively the ratio of marginal roducts. We can rewrite this as: MP k w k = MP`. w` If a rm is otimizing and using the inuts, 2 then their ratio of marginal roducts to rice must be eual. This is similar to the consumer roblem where the ratio of marginal utilities to rice are eual at the otimum. Finally, if Y is convex then the rst-order conditions are necessary and su cient conditions for determining a solution. Proosition 2 Suose that () is the ro t function of roduction set Y and that y () is the associated suly corresondence. Assume that Y is closed and satis es free disosal. 2 If an inut is not used then this is akin to a corner solution in consumer theory and this result need not hold. 5

6 1. () is homogeneous of degree one in 2. () is convex in 3. If Y is convex, then Y = y 2 R L : y () for all >> 0 4. y () is homogeneous of degree zero in rices 5. If Y is convex, then y () is a convex set for all. Moreover, if Y is strictly convex, then y () is single-valued (if nonemty). 6. (Hotelling s Lemma) If y () consists of a single oint, then () is di erentiable at and r () = y (). 7. If y () is a function di erentiable at, then Dy () = D 2 () is a symmetric and ositive semide nite matrix with Dy () = 0. Points 1, 2, and 4 are results that are fairly similar to those in the consumer s roblem. Result 1 states that if all rices double, then ro t will double (technically, if we change all rices by the same ercentage, then ro ts will change by the same ercentage). Result 2 states that convex combinations of rices roduce ro ts at least as high as the extremes. Result 4 states that if all rices change by the same ercentage, then suly will remain unchanged. Result 5 rovides results for the suly function based uon the convexity of the roduction set. Result 3 rovides an alternative descrition of the technology, similar to the indirect utility function, v (; w), roviding a descrition of references. We can then use result 6 to directly calculate suly (similar to using Roy s Identity to nd demand functions from the indirect utility function) from the ro t function. Result 7 follows from result 6 and rovides the law of suly holding all other rices constant, suly of a commodity will change in the same direction as its rice. Note that this is unambiguous, as oosed to the case of consumer theory where the demand for a good may increase if its rice increases (Gi en goods). Since there are no budget constraints (generally) in roducer theory, there are no wealth e ects to be concerned about and this matrix of second derivatives is akin to the matrix of second derivatives of the exenditure function, which is the substitution matrix in consumer theory. Thus, the matrix of second derivatives of the ro t function, D 2 (), may be regarded as a substitution matrix. 4 Cost minimization (CMP) The dual roblem for the roducer is the cost-minimization roblem. The roducer s goal with this roblem is to choose an outut level and then determine the cost-minimizing amount of inuts based uon the rices of the inuts and the roduction technology that will roduce this level of outut. Since this roblem is just the dual of the ro t-max roblem, why study cost minimization? 1. There are a number of useful results that follow from the cost-minimization roblem. 2. If a rm is not a rice-taker in the outut market then we cannot use the ro t function for analysis. But if the rm remains a rice-taker in the inut market, we can use results from the cost minimization roblem. 3. When Y has nondecreasing returns to scale, the value function and otimizing vectors of the CMP are better behaved than () and y (). Our focus is on the single-outut case. Let z be a nonnegative vector of inuts with inut rices given by w and f (z) be the roduction function that roduces outut. The rm s roblem is then: min z0 wz s.t. f (z). The otimized value of the CMP is the cost function, c (w; ). The otimizing set of inut choices, z (w; ), is known as the conditional factor demands corresondence. Note that this otimizing set of inut choices is conditional on the amount of outut roduced,. The Lagrangian can then be constructed: min z0 L (z; ) = wz + ( f (z)) 6

7 and we can use the rst-order conditions to solve for the cost function and conditional factor demands. The rst-order conditions = (z ) = 0 for all ` = 1; Rearranging we nd w` (z And taking the ratio of the rst-order conditions for commodities ` and k we nd: 1 2 w` w k (z ) (z ) =@z k = MRT S`k. It seems like overkill to kee reeating this statement, but these euality conditions (MRT S = w` w k, MRS =, MRT = ` k ) from the consumer and roducer roblems are extremely imortant for our analysis. Again, this only holds for the inuts the rms use, so if there is some inut z i such that z i = 0 in the rm s roblem, then these euality conditions do not hold. Figure 4 shows the isouant (meaning same uantity ) for f (z 1 ; z 2 ) = 10 (the curved black line) assuming the rm s roduction function is f (z 1; z 2 ) = z 1=3 1 z 1=3 2. The inut rices are assumed to be w 1 = 1000 and w 2 = 1. The cost for the line below the isouant (urle line) is 1000 while the cost for the urle line above the isouant is The cost for the red line is The idea in this roblem is similar to the consumer s exenditure minimization roblem x a level of utility and nd the lowest level of exenditure. The rm is merely xing a uantity level and nding the lowest cost at which it can roduce that uantity. z Isouant for the Cobb-Douglas roduction function of 10 = z 1=3 1 z 1=3 2 For the mechanics of the roblem, the conditional factor demands can be found by solving the rst-order conditions for the inut levels in terms of w and (do not forget the rst-order condition for, which I have not been including because we "know" we need to nd that condition). To nd the cost function, simly take the factor demands, which are functions of w and, and substitute them into wz, so that c (w; ) would look something like w 1 z 1 (; w) + w 2 z 2 (; w). Proosition 3 Suose that c (w; ) is the cost function of a single outut technology Y with roduction function f () and that z (w; ) is the associated conditional factor demand corresondence. Y is closed and satis es free disosal. 1. c () is homogeneous of degree one in w and nondecreasing in z1 7

8 2. c () is a concave function of w 3. If the sets fz 0 : f (z) g are convex for every, then Y = f( z; ) : wz c (w; ) for all w >> 0g. 4. z () is homogeneous of degree zero in w 5. If the set fz 0 : f (z) g is convex then z (w; ) is a convex set. Moreover, if fz 0 : f (z) g is strictly convex then z (w; ) is single-valued. 6. (Sheard s Lemma) If z (w; ) consists of a single oint, then c () is di erentiable with resect to w at w and r w c (w; ) = z (w; ) 7. If z () is di erentiable at w, then D w z (w; ) = D 2 wc (w; ) is a symmetric and negative semide nite matrix with D w z (w; ) w = If f () is homogeneous of degree one, then c () and z () are homogeneous of degree one in. 9. If f () is concave, then c () is a convex function of. The rst roerty states that if inut rices double then costs will double. Also, costs either remain the same or increase as uantity of the outut increases. The second roerty states that the cost function is concave in w. The third roerty rovides some results on the roduction set Y under convexity of the roduction function. The fourth roerty states that if all inut rices double, then the cost-minimizing conditional factor demands remain the same for a given level of outut. The fth roerty states that the conditional factor demands are convex if the roduction function is convex. The sixth roerty states that the conditional factor demands are the derivatives of the cost function with resect to the inut rices. This is similar to Roy s Identity and Hotelling s Lemma, where the Walrasian demand functions and the suly function can be derived from the indirect utility function and the ro t function. The seventh roerty rovides some results on the Hessian matrix of the cost function. In articular, own inut rice e ects are negative, so that if an inut rice increases, the conditional factor demand will decrease. Also, the cross inut rice derivatives are eual (symmetry) and Euler s formula holds. The eighth roerty rovides states results about the cost function and the conditional factor demands given the homogeneity of the roduction function. If doubling inuts leads to a doubling of outut in the roduction function, then doubling of outut leads to a doubling of cost as well as a doubling of the conditional factor demands. The ninth roerty rovides results on the convexity of the cost function based on the concavity of the roduction function. Now that we have de ned the cost function, the rm s ro t maximization roblem can be restated. max 0 c (w; ) This is the traditional seci cation of the rm s ro t-maximization roblem. Note that is simly revenue and that c (w; ) is the rm s cost given inut rices and its choice of. The necessary FOC for a level of to be otimizing (w; ) Thus rice must be less than or eual to marginal cost, and if > 0 then rice euals marginal cost. PLEASE REMEMBER that this is only true when the rm s outut decision has no imact on the rice of the outut (i.e. the rm is a rice-taker in the outut market). Alternatively, we can say that the rm s ro t maximizing outut occurs at the oint where marginal revenue,, euals marginal 8

9 Note that right now we are not discussing any notion of euilibrium, thus the rm may be making ositive, negative, or zero ro t. 3 Now, consider the cost function, c (w; ) = C (), of the single outut case with xed inut rices. Also, and the marginal cost function MC () = dc() d. The consider the average cost function, AC () = C() average and marginal cost functions will be di erent functional forms deending on the assumtions of our roduction technology. First consider the case of a convex roduction set where there are strictly decreasing returns to scale. Note that inactivity is ossible. The rm s cost function may look like the left-hand side of gure 4 with AC () and MC () on the right-hand side. Note that the suly at a articular rice is given by the marginal cost function. MC() C() AC() Strictly decreasing returns to scale. Next consider the case of constant returns to scale. In this case, the rm s cost function is a linear function since when we double outut we double cost (roerty 8 of the cost function). Then AC () = MC () and both are eual to some constant arallel to the x-axis, as in gure 4. Note that the suly is zero until the rice reaches that constant because the rm s ro t when rice is below that constant would be negative for an uantity choice of the rm. Once rice has surassed that constant then the suly is in nity. C() AC() = MC() Constant returns to scale. Now consider the case of a nonconvex technology. This is the tyical case considered in the rinciles of microeconomics classes as deicted in gure 4. Marginal cost begins below average cost then crosses average cost at the minimum of average cost and then remains above average cost. Suly is given by 0 until rice euals the minimum of average cost and then by the marginal cost function. 3 I will discuss the zero-ro t condition a little more in detail in chater 10. It is my belief that this condition and the concet of rationality discussed in chater 1 are two of the most misunderstood concets of basic economics. 9

10 MC() C() AC() A nonconvex technology. It is also ossible that there are nonsunk setu costs (usually called xed costs) of oeration, deicted in gure 4. The rm does not have to ay these costs unless it enters into the roduction stage, and once the rm enters into the roduction stage it ays this cost once. Note that these costs do not deend on the amount of outut roduced. For a strictly convex variable cost roduction technology, the icture is a combination of the case with strictly decreasing returns to scale and the nonconvex roduction technology. Again, suly is zero until rice euals the minimum of average cost and then it is eual to the rm s marginal cost. C() MC() AC() A technology with xed costs. 10