Econ Agri Production
Econ Agri Production AEB 6184
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Basic Notions of Production Functions Lecture I I Overview of the Production Function A The production function is a technical relationship depicting the technical transformation of inputs into outputs l The production function in and of itself is devoid of economic content 2 In the development of production functions we are interested in certain characteristics that make it possible to construct economic models based on optimizing behavior B One way to write the production function is as a function map f R a R which states that the production function f is a function that maps n inputs into m outputs By convention we are only interested in positive input bundles that yield positive output bundles The rst lecture will focus on the production function as a continuous function as students have probably seen it in previous courses The next lecture will develop the concept of the production function more rigorously 11 One Product OneVariable Factor Relationships A A commonly used form of the production function is the closed form representation where the total physical product is depicted as a function of a vector of inputs 0 yfx where y is the scalar single output and x is a vector multiple inputs 03 Focusing for a moment on the single output case we could simplify the above representation to y fx1x2 or we are interested in examining the relationship between x1 and y given that all the other factors of production are held constant Using this relationship we want to identify three primary relationships Total physical productiwhich is the original production function 2 Average physical productidefmed as the average output per unit of input Mathematically APP Z M x x Marginal physical productidefmed as the rate of change in total physical product at a specific input level Mathematically MPP dTPP dfxf39x dx dx dx LA AEB 6184 7 Production Economics Professor Charles B Moss 180 160 140 00 Corn hu acre Lecture I Fall 2005 C Given these notions of a production function we can introduce the classical shape of the production function High Yield Function Average Yield Function Low Yield Function t t l 80 100 120 160 Nitrogen lbsacre 180 This set of production functions are taken from Moss and Schmitz Investing in Precision Agriculture 1 This shape is referred to as a sigmoid shape 2 The exact functional form in this figure can be attributed to Zellner Arnold 1951 An Interesting General form for a Production Function Econometrica 19 18889 The exact mathematical form of the function is ltV19V2 3 a v1 ExpbV 1 1 V2 The average function sets v2 10 a00005433 and b001794 The total physical product graph given by Mathematica 40 AEB 6184 7 Production Economics Professor Charles B Moss U Lecture I Fall 2005 12 0 100 80 60 40 20 2 5 5 O 7 5 l O O l 2 5 l 5 O The marginal physical product and average physical product graphs for this production function become A Marginal Physical Product Average Physical Product 2 5 5 0 7 5 l 0 0 l 2 5 l 5 0 Given these relationships we can de ne the stages of production 1 While the production function itself is devoid of economic content we use the physical relationships to define the economically valid production region 2 Stage of Production a Stage I This stage of the production function is defined as that region where the average physical product is increasing In this region the marginal physical product is greater than the average physical product In this region each additional unit of input yields relatively more output on average Stage I This stage of the production process corresponds with the economically feasible region of production Marginal U AEB 6184 7 Production Economics Lecture I Professor Charles B Moss Fall 2005 physical product is positive and each additional unit of input produces less output on average c Stage III This stage of production implies negative marginal return on inputs 3 Mathematically these stages of production imply certain restrictions on the shape of the production function a The production function is a positive valued initially increasing function Further around the point of optimality the production function is concave in variable inputs 4 Elasticity of Production a Elasticities are often used in economics to produce a unitfree indicator of the shape of a function Most are familiar with the elasticity of consumer demand b In defining the production function we are interested in the factor elasticity The factor elasticity is defined as dy EfoAy y Tum dxy APP x For the Zellner production function the factor elasticity can be depicted as 2 5 5 0 7 5 l 0 0 l 2 5 l 5 0 c There is a specific relationship between the average physical product and marginal physical product when the average physical product is maximized Mathematically dTPP 7 dxAPP dAPP dx 7 dx dx Mpp APP x Thus when the APP is maximized 0 3 MPP APP x AEB 6184 7 Production Economics Lecture I Professor Charles B Moss Fall 2005 Following through on this relationship we have dilgt0 MPPgtAPPEEgtl dil03MPPAPP3El lt03MPPltAPPDElt1 In addition we know that E 0 lt3 MPP 0TPPis maximum E lt 0 ltgt MPP lt 0 Thus if E gt1then the production function is in stage I If lgt E gt 0 then the production function is in stage II If E lt 0 then the production function is in stage III III One Product TwoVariable Factor Relationships A Expanding the production we start by considering the case of two inputs producing one output In the general functional mapping notation f R Rf B This class of production functions can also be depicted as y f Xi x 1 In general the univariate production functions are simply slices out of the multivariazte production functions 11 l 150 a This graph depicts the more general version of the Zellner production function where both x1 and x2 are allowed to change Wade expbiil x AEB 6184 7 Production Economics Lecture I Professor Charles B Moss Fall 2005 2 These functions still have average physical products and marginal physical products but they are conditioned on the level of other inputs APP1 lfxl x2 x1 x1 APPZ 1 fx1x2 x2 x2 Similarly the marginal physical products are de ned by the partial derivatives MPP1 0y afx1 xl 0amp9 0amp9 0 MPPZ 0yfxl x2 0x2 Oxl 3 It may be useful at this point to brie y visit the notion of the Taylor expansion Taking the secondorder expansion of the production function yields 0f xuxz 0f xpxz dxl f9xzf9 x a dx 62 0x2 2 02fxlxz 02fx1xz 0x Mare 02fxix2 02fxlx2 dxz maxi 0x This approximation is exact in the case of either linear or quadratic production functions However if we focus on a quadratic production function it is clear that dymfl 0fx2dxl 0fx2dxl C Some Typical Multivariate Production Functions 1 The Linear Production Function y blxl l bzxz 2 The Quadratic Production Function y alxl l azxz l 6Anx12 l 2A12x1x2 l Azzxzz gm wit 3l52l 3 The CobbDouglas Production Function y Ax x 4 The Transcendental Production Function y Ax 81 x52 81 5 The Constant Elasticity of Substitution CES Production Function yAbxfg libxgiyg gm dxz AEB 6184 7 Production Economics Lecture I Professor Charles B Moss Fall 2005 D Given the multivariate nature of the production function it is possible to define isoquants or the relationship that depicts the combinations of inputs that yield the same output 1 Starting from the basic production function yfxpxz3xz fxby 2 That is we are interested in constructing a functional mapping of xi based on the level of x1 and y 3 Next we hold the level of output constant and derive the levels of x2 for any level of 9 X2 X1 4 The output for each of these isoquants are such that yl lt yz lt ys E The isoquants are useful in de ning the rate of technical substitution or the rate at which one input must be traded for the other input dxi 12 d dx 0 3 7 y M 12 2 dxz Building on the slopes of isoquants we define the isoclines and ridgelines Each of these relationships are comprised of those points that have the same rate of technical substitution Lecture I AEB 6184 7 Production Economics Fall 2005 Professor Charles B Moss Ridgeline Isocline The ridgelines are the isoclines where the rate of technical substitution is equal to zero or in nity They represent the maximum physical output for one variable while holding the other variable constant 1 Factor independence Two factors are independent if the MPP of one factor is not a function of the MPP of the other factor 2 The simplest example of this is a quadratic production function with A12 A21 0 In this case the isoquants are circles or elipses 0 i a1 Allxl y alxl l azxz l 6Anx12 Jr qzzxi3 0y a2 Aux 0x a Case I39 02y 7 0 0y 0 7 f gt 0then and x are axlaxz axl x i 12 x1 2 technically complementary b Case 11 If f12 0 then x1 and x2 are technically independent c Case 111 If f12 lt 0 then x1 and x2 are technically competitive Applied General Equilibrium Analysis Partial versus General Equilibrium A Most of our policy analysis can be characterized as partial For example if we want to examine the effect of a price oor on an industry such as agriculture we analyze P1 1 April 12 2001 Sp1p2W19W2 Dp1p2Y Assuming that Slp1p2w1wz is the supply curve for good 1 as a function of the price of good 1 p1 given the price of good 2 p2 and the input p1ices w1 and wz and that Dp1p2Y is the demand for good 1 as a function of the price of good 1 p1 given the price of good 2 and income Y we can analyze the impact of the price oor This analysis is usually cloaked in ceterus paribus assumptions regarding other prices The only problem is that we know that other prices change as we remove the price oor AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss April 12 2001 Good 1 Good 2 The demand for good 2 shifts outward if good 1 and good 2 are substitutes for the consumer the effect of the removal of the price oor is a price increase On the supply side the decreased price of good 1 leads to an outward shift in supply for good 2 given that the production of good 1 and good 2 compete for limited resources This interaction is complicated as the rst market reacts Speci cally the increased price of good 2 leads to a bftward shift in the supply curve in the market for good 1 and an upward shift in the demand for good 1 AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss Good 1 Good 2 4 These interactions are further complicated by changes in the factor market Speci cally if the industry represented a signi cant demand for the variable factor the initial elimination of the price oor implies a decline in the value of the input Hence the overall demand for the input shi s to the left B Modeling the interaction between the two markets involves moving from a partial equilibrium analysis to a general equilibrium analysis 1 April 12 2001 Early work on general equilibrium analysis involves the concept of a Walrasian equilibrium The primary idea of the Walrasian equilibrium was the concept that some price vector could be found for any endowment that equated the supply and demand or resulted in zero excess demand 51p DPWSPW p pVK 0 61p W S 0 a 1p is the excess demand for good i it is a function of the price vector Demand is determined by the initial endowment of goods W b pi ipWi is the complementary slackness conditions This condition implies that either the price of the 1 good is zero or its excess demand is zero Under the traditional formulation Walrasian equilibria were hypothesized by tatonnement Following the preceding discussion we could iterate through the equilibrium until all the excess demand curves met the Walrasian conditions Basically economist simply assumed that the process would yield a viable solution a De ne a viable solution as one were a unique set of positive prices simultaneously solved for the Walrasian equilibrium AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss 3 Wald A On Some Systems of Equations of Mathematical Economics Econometrica 19Oct 1951 3687403 As a rule 39 have t t d quot 39 with equating the number of equations and unknowns and have assumed without further investigation that the system of equations had a meaningful solution from an economic viewpoint and that this solution was unique But the equality of the number of equations and unknowns does not prove that a solution exists much less the uniqueness of a solution Wald pp 36970 Arrow K J and G Debreu Existence of an Equilibrium for a Competitive Economy Econometrica 22July 1954 265790 offer a de nitive proof of the existence of a solution to the general equilibrium problem based on what are referred to as xed point theorems C However for applied economist knowing that an equilibrium exists is insuf cient The real question is what kinds of questions can I address with the procedure and how do I nd the equilibrium 1 2 3 In general there are two primary techniques for nding the equilibrium solution The rst is attributed to Scarf H The Approximation of Fixed Points of a Continuous Mapping Journal oprplieal Mathematics 151967 1328743 This procedure involves decomposing the space into simplexs or triangularizations then searching the triangularizations by eliminating points The second set of methods is referred to as Newton methods The simplex methodologies are sure in that they will always nd the xed point However they are also dif cult to formulate The Newton methods are far more straightforward to formulate but may fail to nd a solution if the general equilibrium is poorly structured D The remainder of this lecture then focuses on three topics 1 An explanation of the proof of general equilibrium 2 Formulations of applied or computable general equilibrium problems 3 Solution of applied or computable general equilibrium problems 11 Existence of Walrasian Equilibria A Following Varian H R M icroeconomic Analysis Second Edition New York WW Norton 1984 with the notation from Shoven J B and J Whalley Applying General Equilibrium New York Cambridge University Press 1992 we begin by examining the allocation of outputs among n competing consumers 1 2 April 12 2001 In this case we assume that consumers start with an endowment of m goods The general equilibrium then involves trading the preexisting goods among themselves until an appropriate price vector is determined In this case we eliminate the production activities from our earlier discussion to yield AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss 4 April 12 2001 p51p 5107K 0 To graphically motivate this discussion consider the allocation of two goods In this case the initial allocation of the goods produces the dimensions of the traditional Edgeworth box where X W W X 2 W21 W22 given that W3 is the amount of good i held by agent j The graphical problem then becomes X2 The existence of a unique general equilibrium solution is equivalent to the statement that there exists one set of prices that determine a line from the endowment point W to the locus of optimal tangencies As a rst step we need to normalize the prices Speci cally given that demand functions are homogeneous of degree zero in prices there is no money illusion we must determine a numeraire a price equal to one or impose the restriction that the prices sum to some constant Varian takes the second approach k S pian Zpl 1 11 Restating the goal we want to show that there exists a p de ned above such that p p 0 for all goods with p gt0 and plt0 a From a mathematical point of view not that p can be written as a function of excess demand and excess demand is a function of p AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss 2 gfgjstpaaphnd epe b Fixed point theorems show that under certain conditions p can be chosen such that p p p or that the mapping if xed c Brouwer xed point theorem a continuous function from a ball of any dimension to itself must leave at least one point xediArmstong M A Basic Topology New York SpringeriVerlag 1983 p 110 i Proof for dimension 1 Assume that a function is de ned on a unit interval I01 fI 1 Further assume that the function is continuous Then there must exist a point x I such that fxx The proof is by contradiction Both xfx 1 starting from x0 and moving to the right If x0fx 0 is the xed point Otherwise fxgtx Moving to the right if fx is continuous then either fxx for some x or fxgtx If fxgtx for all point x 01 then either fl1 or fx violates the domain XX Fx ii This xed point theorem can be extended to 2 or more dimensions but these proofs require axioms beyond this brief introduction 4 Kakutani s Theorem Let the pointtoset mapping X X of the simplex S into itself be upper semicontinuous Assume that for each X IX is nonempty closed convex set Then there exists a xed point X X a It is worthwhile to develop the concept of a simplex In general a simplex is a hyperplane spanned by k points in a k dimensional space For example in E2 space two points span a line In E3 space three points span a plane In general k1 points determine a convex set called the simplex of dimension k April 12 2001 6 AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss b In the current analysis we are interested in k1 prices A given set of these k1 prices span a kt dimension hyperplane For three prices we span a two dimensional plane in E3 space P2 4 my 1 c The functional maps then involve fRK391 RK391 is then some mathematical operation that takes as its input one point on the plane and gives another point on the plane 5 The GaleNikaido mapping can then be used as a functional mapping of the simplex back onto itself Using our previous de nition of excess demand the GaleNikaido mapping as 2 pl max0 l 1 Zmax0 l 1 If the point obeys Brouwer s xed point theorem we have 2 p maX0 lp 1 2le max0 l To demonstrate the equilibrium we must show that the denominator of the right hand side is equal to one or that N i p 0 0 0 cgt 121 maxl 5 17 p 1 0 To show this let April 12 2001 AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss 0 1 max0 l 3 cp p max0 c 1p maxlo 15 Mathematically c cannot be less than one due to the max0x on the right hand side The economic context of this expression then depends ruling out cgt0 If cgt0 since pigt0 there must be an excess demand ipgt0 which would violate the complementary slackness condition in Walras law N 2 p15 in 0 11 111 Construction of a General Equilibrium Model A It is somewhat inconceivable to discuss a general general equilibrium formulation However Following Shoven and Whalley we can discuss some very general function forms typically used in general equilibrium analysis 1 CobbDouglas Demand functions at xl p 2 Constant Elasticity of Substitution N 671 01 Z 061 x1 4 11 Demand functions a I 1 x1 N R 202 P51 1 In a simple example let s assume that we produce three commodities with two factors of production labor and capital For simplicity let s assume that the demand functions are derive from a Cobb Douglas preference structure and production is consistent with a CBS function 1 Stalting with the demand functions and assuming ocs we get DJ 3 y1 P1 5 y2 P2 2 y3 P3 2 Production is determined by April 12 2001 8 AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss 1 ll 039 all all V 039 all all 3 The procedure is then that the demand equations based on a set of prices generate a set of desired outputs These desired outputs at a given set of input prices generate input demands Computing a general equilibrium then involves nding a set of prices where the excess demands for all outputs and inputs is equal to zero a One piece of the puzzle is income We assume that each household has an endowment of inputs Income is then generated through the sale of these inputs wlx1 wzx2 b An additional assumption is that no pro ts are retained in production 72quot plyl p2y2 p3y3 1x1 w2x2 0 IV Computing the Solution of an Applied General Equilibrium Model A There are two procedures for computing the general equilibrium values 1 Scarf s algorithm this algorithm develops simplexs within feasible space a Speci cally given k current solutions it determines the solution to be dropped and a better point to be added b This procedure is guaranteed to nd an optimal but is slow and tedious 2 The second algorithm is based on Newton s method Technically this procedure is not guaranteed to nd an optimum but if the supply and demand equations are well behaved typically does In addition it is easier to program a The Newton s method algorithms are actually variants of the Gauss Siedel procedure b Assume that you have a vector valued lnction based on x Gx The rst order Taylor series expansion of this function can be written as Gx Gx0VXGx0x x0 Note that by assumption Gx is of the same order as x Thus solving this expression for Gx0 implies x w lvxaltxogtraltxogt xx0vax0 16xo 0 April 12 2001 9 AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss d Given that the supply and demand relationships are well behaved the second term on the right hand side will converge to zero as GX0 converges to zero April 12 2001 Substituting One Input for Another in Production Lecture IV I Elasticity of Scale and Law of Variable Proportions A We continue to develop the economics of production through a ray from the DJ 0 origin 1 Last lecture we developed the notion of returns to scale by looking at changes in production along a ray from the origin 1x2 x1 1x1 2 Following the de nition of this ray we de ned the elasticity of scale 8 am f1x 01111 H 01nf x 7 1 0f x17 1 emu Building on these de nitions we next de ne the ray average product as WM 1 where A is a strictly positive scalar In addition we de ne the ray marginal product 6 A quot flt x am 61 H 6X To derive this result substitute AEB 6184 7 Production Economics Lecture III Professor Charles B Moss Fall 2005 lx gt 2x By extension of this result BZfUtx n n BfUtx 2 xx Ml F1 11 Byg xj I f D In order to develop the concept behind these equations we need to take a slice from the multivariate production function x2 If we focus on the slice of the production function on the ray from the origin the production function looks like a univariate production function AEB 6184 7 Production Economics Lecture III Professor Charles B Moss Fall 2005 yflx i 39l E xlx Differentiating the ray average product yields 1 6RAP 6 x aftx1 6 fax 61 M M A M i6flxflx 1 61 1 2RMP RAP Which states that the ray average product is maximum when it is equal to the ray marginal product RMP Note that this relationship is the same as the univariate relationship dAPP 7 1 dx 7 x 1 Applying these relationships to the Zellner production function 11596pr expbi 71 x2 Note that by de nition Mlxplxz afxf afxf eXpbl xl l eXpb l 1x2 x2 Thus x2 does not affect the scale economies The RMP is then MPP 7 APP AEB 6184 7 Production Economics Professor Charles B Moss RMP exp b Lecture 111 Fall 2005 3a t2 x13 x1 1 x2 2 The ray average proc uocgj can be graphed O 8 O 9 3 The ray marginal product 600 m 4 Recalling the general gr 09 1 1 can be graphed 1 11 12 aph of the ray de ning the ray average product and ray marginal product AEB 6184 7 Production Economics Lecture III Professor Charles B Moss Fall 2005 yflx A xlx a If 81at point A the production function exhibits constant returns to scale at x since af 1x mix gjanW Nix 61 z z 61111 Q fix RAP x1 1 l b If s1is to the right of A then the production function exhibits decreasing returns to scale at x since any ray from the origin to ftx for AgtA will cut ftx from below Thus the ray average product is greater than the ray marginal product c If 81to the left of A fx exhibits increasing returns to scale at x II Measures of Inputs Substitution A In the first lecture we developed the idea of the rate of technical substitution defined as the movement along an isoquant B Now we want to expand our discussion to discuss an elasticity of substitution In general we would like to define the elasticity of substitution as the percentage change in relative rate of input use However the exact nature of this elasticity is somewhat ambiguous AEB 6184 7 Production Economics Professor Charles B Moss Lecture 111 Fall 2005 C There are three general elasticities of substitution 1 Hicks de ned the rst elasticity of substitution in 1963 The Hicksian or direct elasticity of substitution 0 4 U dlf fJVx In order to develop this notion recall the general form of the isoquant x2 4 dxl f2 g x x1 Next if we want to discuss the change in this relationship AEB 6184 7 Production Economics Lecture III Professor Charles B Moss Fall 2005 x2 x1 Using the CobbDouglas as an example a flwalxz fxAxfxf xl L f Axf xz fz xl 2 6 Changing the variables such that ZWLW 2w 2w 3 dzw 2 f2 xi dw 8 gmi a q a a 9 Writing the bordered Hessian of the production surface l 0 fl fz F fl fii fiz fz fu fzz This Hessian represents the change in x1 and x2 such that y remains unchanged Based on this transformation the direct elasticity of substitution can be written as Lecture III AEB 6184 7 Production Economics Fall 2005 Professor Charles B Moss D xlfl xzfz i F 012 xle 0 12 F 7 12 f1 fu 1212 0 ii 12 Ff1 fu 1221 fwaf 2 2 12 22 11 12 flf22 ififl l fzfn flzfzz zflfzflz 7 fsz22 2 Allen Partial Elasticity of Substitution is a generalization of the matrix expression above x1 I fi 0 xxx F 0 f1 f2 f fl fii fiz fin Ffz fiz fzz on fquot fl f2quot fin 3 M orishima Elasticity of Substitution is the nal generalization f M7 J J x 039 039y039 V ifxx Estimation of Production Functions Fixed Effects in Panel Data Lecture VIII 1 Analysis of Covariance A Looking at a representative regression model y of 39x 7392 u i1N t l T Where x and z are k1 xl and k2 x1 vectors of exogenous variables and of Y and 7 are estimated parameters and u is an independently and identically distributed iid error term with mean 0 and variance 7 1 It is well known that ordinary least squares OLS regressions of ya on x and z are best linear unbiased estimators BLUE of of Y and 7 2 However the results are corrupted if we do not observe z Speci cally if the covariance of x and z are correlated then OLS estimates of the Y are biased 3 However if repeated observations of a group of individuals are available ie panel or longitudinal data they may us to get rid of the effect of z a For example if z z or the unobserved variable is the same for each individual across time the effect of the unobserved variables can be removed by rstdifferencing the dependent and independent variables y 7 yxyH 639xxt 7 xxyH 72 7 zxyH uxt 7 until Since 2 21H z y 7yH 39xt 71un 7W1 139 LN t 2 T b Similarly if z 2t or the unobserved variables are the same for every individual at a any point in time we can derive a consistent estimator by subtracting the mean of the dependent and independent variables for each individual y 77 m 7Z7Zt 72H 5 Since 2 Z yxtii 39xxtiflulti l AEB 61847Production Economics Lecture VIII Professor Charles B Moss Fall 2005 4 OLS estimators then provide unbiased and consistent estimates of Y a Unfortunately if we have a crosssectional dataset ie T 1 or a single timeseries ie N 1 these transformations cannot be used B Next starting from the pooled estimates y of 39x u we envision two sets of restrictions or hypotheses 1 Case I Heterogeneous intercepts a 7 01 and a homogeneous slope yxt or 39xlt u 2 Case 11 Heterogeneous slopes and intercepts a 7 a 3x 7 8 y a Ex u 3 A third case which is typically not addressed can be posited one of homogeneous intercepts and heterogeneous slopes y 05 xlxn u C Empirical Procedure 1 From the general model we pose three different hypotheses a H1 Regression slope coefficients are identical and the intercepts are not b H2 Regression intercepts are the same and the slope coefficients are not c H3 Both slopes and the intercepts are the same 2 Estimation of different slopes and intercepts a Estimate the average dependent and independent variables for each individual 7 iy 2 ig b Estimate the 3 vector using individual moment matrices These are called the within group estimators The i3911 group residual sum of squares RSS WWW1 WXY139Wi39xWXYx The unrestricted residual sum of squares is N 51 Z RSS1 11 AEB 61847Production Economics Lecture VIII Professor Charles B Moss Fall 2005 Covariance Matrices X X Nitrogen P39 r39 Potash X39Y beta alpha Illinois Nitrogen 12823 07194 15488 07415 07985 37917 Phosphorous 07160 06410 10156 02204 09813 Potash 15427 10174 20326 07894 02734 Indiana Nitrogen 10346 02489 07220 06577 04386 36162 Phosphorous 02348 03717 02320 00913 08905 Potash 07268 02448 06072 04587 05894 Pooled Nitrogen 23168 09683 22708 13992 05924 39789 Phosphorous 09508 10128 12475 01291 09335 38851 Potash 22695 12622 26398 12481 04098 3 Estimation of different intercepts with the same slope W WQWXY r7 7 7372 i LN N N WXX 2Wng WXY ZWXYJ 11 11 N WW 2 WW 11 The residual sum of squares Sz WW 7 WXY39WJEWXY 4 Estimation of homogeneous slopes and intercepts 1 TJEJlTXY or 7 7 7 The overall sum of square errors is then 53 TW 7 57ngY AEB 6184Production Economics Lecture VIII Professor Charles B Moss Fall 2005 5 Testing first for pooling both the slope and intercept terms Honraa1 A N S3 51 pg MNFN1K1NTNK1 NTNK 1 6 If this hypothesis is rejected we then test for homogeneity of the slopes but heterogeneity of the constants H1 l z quot39 N Sz 51 E MNFN1KNTNK 1 NTNK 1 II Dummy Variable Formulation A The analysis of covariance can be accomplished using dummy variables yl e 0 0 xi Mi 3 2 0 8 0 x2 1 Y za1aZaN z y3 0 0 e xN uN yd x111 x211 39 39 39 xle 7 y12 7 x112 x212 xm yli xii 39 y1t x117 x217 xK1T za39eM1X7 e39l l l 139 EMle u1lu11 2 MW Eux0 Euxux39039217 Euxu390139 j 1 Given this formulation we know the OLS estimation of ylt xltult a The OLS estimation of a and Y are obtained by minimizing N N S Zufu 2y 7w ex y 7w ex 11 11 b Yielding 553717 39fl iLN 1 T y fzyn xx Zxxt 11 32V im7w zgt l imizgtltymgtl AEB 61847Pr0duction Economics Lecture VIII Professor Charles B Moss Fall 2005 2 Sweeping the data Q177wee39 aExample 1000 11111777 74 0100 11117 11 001071111 1 0001 11 4 4 4 4 ex ex Thus inX space kmmwmm walkwwmza wwzltlwgtmm wimrmlhmxa 161 xzi1 14 7ee39lx xlxzxjx4 0 7x4 1 951xzJhx4 99JQJQXA Simultaneity and Other Simple Problems Lecture VI I Simultaneity and Estimation of the Production Function A The above discussion and estimates makes the experimental plot design assumption regarding the data 1 Speci cally I essentially assumed that the data are being generated from some sort of experimental design so that the errors are truly random 2 If the data are actually the result of farm level decisions the data are endogenous B The simultaneity literature starts with Hock Irving Simultaneous Equation Bias in the Context of the CobbDouglas Production Function Econometrica 264Oct 1958 56678 1 The basic rmlevel model is that we have an empirical model under the assumption of a A CobbDouglas production function and b Competition 2 In Hoch s notation the production function becomes Q X0 K0 1 1qu 41 a where X 0 is the level of output b X g is a level of physical input c a4 is the elasticity of output with respect to an input and d K0 is a constant 3 Assuming pro t maximization an output price of PO and input prices of Pg we have the pro t maximization conditions 0X0 130 g P 0X4 4 X 4 Dividing through by Pg X P Y dabquot q q q where Y0 is the total value of output and Y4 is the total value of each P 0 input a Klein demonstrates that the best linear unbiased estimate of a4 I Y AFHEEJ 11 is AEB 6184 7 Production Economics Lecture VI Professor Charles B Moss Fall 2005 b In this approach the average rm is de ned to be the optimal rm 4 As an alternative X P a X where R4 is some constant and the investigator wishes to test whether it is equal to one a The rm sets the value of the marginal product equal to the price augmented by the effect of any restrictions that exist In this formulation R4 can of course vary among rms but Fquot for a sample of rms the investigator would be interested in testing whether the average R4 is equal to one V39 Single equation estimates are biased when the equation is a member of a system of equations in the following way the system is such that some of the independent variables as well as the dependent variable are functions of the disturbance in the given equation 6 Two models of simultaneity a Model 1 Production disturbance not transmitted to the independent variables 1 If the disturbance in the production equation affects only the output and is not transmitted to the other variables in the system then there is no simultaneous equation bias Single equation estimates are consistent 2 For example if inputs are xed or are predetermined b Model 2 Production disturbance transmitted to the independent variables 1 Simultaneous equation bias arises when disturbances in the production relations affect the observed values of all variables and as a result single equation estimates are not consisten 7 Empirical setup Q X0 KOHXfU 41 11 P X 40X0V q12Q 4 R4 134 4 Taking the natural logarithms of both sides of each equation yields Q x0 k0Zaqxqu 41 xqkqx0vq q2Q where x0 lnX0 xk lnXq and kg 1na4 P q q AEB 6184 7 Production Economics Lecture VI Professor Charles B Moss Fall 2005 C Indirect Least Square Solution Kmenta J Some Properties of Alternative Estimates of the CobbDouglas Production Function Economem39ca 3212JanApr 1964 18378 1 Simplifying the general system of equations x0 k0 a1x11 a2sz V0 x11 klx01le x21 k2 x0 V2 2 Transforming the estimation problem to x01 be b1xlx Xm b2x21 x0xex yields estimates of b1 and b2 that are consistent Note by the definitions x11 Icrx01 le xrx xm 19V11 x21 k2 x01v213x217xm k2 v21 3 Working through the mechanics x01b0blxlxib1x01bZJClxib2x01ex 1b1b2x01b0b1crxbzczxex b0 b2 1 xh xz 8 1b1b2 1blb2 1blb2 1b1b2 4 Which implies that x0 13 a 1 b1 bl II Zeros in the Cobb Douglas Functional Form rl2 Moss Charles B Estimation of the CobbDouglas with Zero Input Levels Bootstrapping and Substitution Applied Economics Letters 710Oct 2000 67779 A Zeros raise several difficulties in estimating the CobbDouglas production function 1 On the theoretical side the presence of a zerolevel input is that it violates weak necessity of inputs 2 On the empirical side how do you take the natural logarithm of zero B Two assumptions 1 The existence of zeros is the result of measurement error a Agronomically production of a crop is impossible without some level of each fertilizer b Thus production occurs based on some true level of each nutrient available to the plant x xx al in this case we observe x but production is dependent on x O In fact we can think of the soil as a sponge that contains a variety of nutrients that we can augment by applying fertilizer The actual level of fertilizer used by the crop could then be a AEB 6184 7 Production Economics Lecture VI Professor Charles B Moss Fall 2005 function of what we add the weather ie if adequate moisture is not present the crop does not use the full potential etc 2 Second a production function that does not admit zero input could represent a misspecification C In this paper I consider two techniques for adjusting the zero observations 1 First I redraw from the sample averaging the result until the pseudo sample contains no zeros 2 Second I substitute a small nonzero number for those observations that contain zeros ie 01 001 0001 D The goodness of fit for each procedure is then compared using a Strobel measure of information N 111 S 1 I Where s1 is the theoretically appropriate budget share for each input and 51 is the budget share estimated using each empirical approximation of zero III Linear Response Plateau Model Constant Elasticity of Substitution in Applied General Equilibrium The choice of input levels that minimize the cost of production for any set of input prices and a xed level of production can be expressed as min 2 9 w 31 Y f Limiting for exposition purposes the set of inputs to 3 and focusing on the constant elasticity of substitution CES production function yields min wlx1 wzx2 w3x3 1 1 7x2 7x3 Y dram a3x Forming the Lagrangian for the optimization problem L wlx1 w2x2 w3xj Y 0616107 ogldxjix The rstorder conditions for this minimization problem are then 039 7 3x1 0 1 7 Giventhat01 1alg1 0 7 7 a 0 7 7 1 1 and 0 1 0 1 0 1 0 1 1 azmngua g H0 3x w Wyaw amya f 0 2 a or a C Lw3 la3 gy alxl 0 2y x2 0 3y x3 1 0 3 039 2 2 Y 0 1x167062x a310xTil 0 The ratio of the rstorder condition for the rst input to the rstorder condition for the second input yields AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss w lot64oq q axfaxf W2 whysGagammmaxj l f 0 2 x1 w H mm 1 XZ Tl xl w2 062 x1 w2 06 Simply replacing the price and comtant for X3 yields 5 5 l I Substituting these results into the rst order condition with respect to the Lagrange multiplier yields a 07 1 a 07 71 Y a1xfy oczx 99 M s 99 0 w2 06 W 051 1 3 Focusing on the term inside the brackets 1 7 71 ya 07 a2039 X7 WI 0 07 a3Ta 0 WI 0 07 061 x1 T x1 G y X1 061 0 W2 a1 a First note that 1 a 1 7 7 Thus the above expression can be simpli ed to 071 0 1 x 6 062 n 6 as n 6 0 1 x1 arkj x1 arkj WS 99 l wz Next we solve a1 05 a 1 l z zl 0 substituting this solution into the rst term yields Oil 071 a 71 06 w 1 06 W i a 9904O 3 1 9964 3 1 990 061 X 062 X wz Finally we multiply the rst term by a form of one April 12 2001 2 AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss this yields 01 71 71 0 1 W1 7 0 2 W1 0 7 0 3 W1 0 Oyw XI Olfx wzj XI afy W3 XI 061 a 1 Factoring out the terms involving awnU w1 and X1 yields 057wf391996j iam The firstorder condition from the Lagrange multiplier then becomes 170 07 3 3971 YOq Awf39lxl ZOMMJ 0 3 am Y 06JIW1 99ZOM39 11 Yal xi 3 7 wlaZalw1 7 11 Now if we assume that land labor and capital are applied to produce agricultural and manufacturing goods we can specify a production system Clef c commodity and Land Labor Capital f factor Xc 1 Otcz 0t c3 6c YaiAgriculture 070 010 020 095 Ym f iug 005 045 050 105 Given these parameters we have two production functions Ya 6870x 05 0886x 05 1838x 526 I Ym 0577990476 4675x20476 5167x3 47621 The input demands conditional on the output level can be defined based on these parameters April 12 2001 3 AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss 70Ya W195 70w1 5 10w2 5 20wS 519 10Y x2 95 05 05 05 19 w 70w1 10w2 20w3 20Y WS 70 W05 10 mg 20w3 53919 05Y 9 x3 a w11 505w 5 45w 5 50w 5 45Y m w 5 05w1 05 45w 5 50w 521 50Y wg 05w1 05 45w 5 50w 5 It is obvious that given output levels and input prices the demand equations de ne the quantity of input demanded If we assume that each factor is given some initial endowment then the excess demand for each input can be de ned as 51WYaYm x Z w Ya xm w Ym E Taking the example a step further we could write the expression in an object form as 5 wp1 X WY 117 1 99 MC 117 1 E To complete the picture we have to formulate the demand equations for the outputs Again if we assume the CES form the utility maximization problem becomes max 1lyl 3 ny VI sty1p1 yzpz 1 Forming the Lagrangian of the maximization problem L 1y1 1 1 y1p1 y2p2 gt71 1 gt71 gt71 Vail y LBIMA h x wz n HP10 l yf lyf z yl HP1 0 Syi Bzx yi l yfzwz yj l sz 0 6L EIy171 y2p2 0 April 12 2001 AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss Again taking the ratio of the first and second first order conditions yields l y l ylw ly V1 71 3262 M MW 18 V1 pl 32 y1 p2 ampampamp 32 y1 p2 yamp y 2 pl 31 1 Substituting this solution into the first order condition with respect to the Lagrange multiplier yields f ll 1 y1p1 Eyi pl 0 p2 B1 1y1p1p539lpf y1 0 1 I p1pi lpf M 31 I 1 pl 1 p2 pl 2 yl H31 W yl M31 172 P1 32 To reconcile this result with preVious results we factor pll out of the denominator yielding y I 1 1 pf p11 MP 732 or Hi 2quot Z 107 l J It is now possible to specify the excess demand for an output priceincome combination y April 12 2001 5 AEB 6184 Applied General Equilibrium Notes Professor Charles B Moss U l 0171 ltp1gt zaxtltwp1gt A Pf Z pj39l jj Letting l 535 z m65 and 175 the demand for each good becomes y 351 pa7535p2565pn25 65 ym April 12 2001 6 Comparative Statics and Duality of the Cost Function Lecture XIV Comparative Statics A Comparative statics with respect to changes in input prices 1 The most common results of the comparative statics with respect to input prices involve intuition about derived demand functions a Fquot 0 3 1 From the primal approach we expect the demand functions for each input to be downward sloping with respect to input prices Starting from the cost function 620wy x my 6w 6w 6w By Shephard s lemma In addition we know that if 139 j then by the concavity of the cost function in input prices 620wy 6x my a Vla Vl 6W1 In addition we know that by Young s theorem the Hessian matrix for the cost function is symmetric 620wy 620wy 6w 16w S0 6w6w The Hessian matrix for the cost function is also singular I Euler s Theorem Euler s theorem is based on the definition of homogeneity ftx t fx Differentiating both sides with respect to t and applying the chain rule yields N 6 Ix 6 Ix f gt lt nfx H 6tx 696 N we x rt x W A gt 11 Letting tl then yields N 6fx x r x 11 IIICoupling this result with the observation that if a function is homogeneous of degree r then its derivative is homogeneous of degree r l We know that the input demand functions are homogeneous of degree zero in prices Thus QV A AEB 6184 Production Economics Lecture XIV Professor Charles B Moss Fall 2005 N 6xwy N 620wy 0 21 6w w E 6wl6wj w IVMultiplying this expression by x w y yields N 6xwy w N j 0 2 6w xiltwy 28 11 11 V Given that we know that S lt 0 this result imposes restrictions on the crossprice elasticities 2 Brie y let us prove the homogeneity of the marginal cost function It is a useful demonstration of the use of Shephard s lemma Starting with the marginal cost differentiate with respect to each price 6cwyZN620wyw 3 N 60wy W 6y H May 6y1 6W I aim w x w I 6y1 y N CwayZxYWyWx 11 N620wy 6 11 aw y w cw y Therefore the marginal cost function is homogeneous of degree one B Comparative statics with respect to output levels Following from the restrictions on the cross price elasticities above the comparative statics with respect to output levels imply that not all inputs can be inferior or regressive 2 An inferior input is an input whose use declines as production increases while the use of a normal input increases as production increases a Like the crossprice results above we start with the sum of the differences of individual demand functions with respect to the level of output N N 2 Z 6x1wyw 26 cwyw 1 1 H 6y 11 away 60wy T 3 In order to develop the effect of output on total cost we start with the original Lagrangian from the primal problem 6fx LWx1y fx L w 1 6x1 0 LAyfx0 AEB 6184 Production Economics Lecture XIV Professor Charles B Moss Fall 2005 Solving for the output using the firstorder condition on the Lagrange multiplier and differentiating the solution then yields N yfx3dy2mdamp 1 6X From the first set of firstorder conditions we see 6f x W 6x A 1 Therefore N W N N dx dyZ alxl3Adyzlwldxl312211111 l 21 1 21 21 dy Which proves the definition of 7 consistent with the envelope theory Therefore N dxwy 1 my wil dy Given this optimum 7 we can then sum over the initial firstorder conditions N N 0fxwy wx 1 w lt y ax Introducing the notion of an elasticity of scale 8yxltwygw wyy MW 4 N wax my CW y 21 my 51y M 0y nuw eltywgt a Chambers defines lacw ly alntcm cw y 01n y as the cost exibility the ratio between the marginal and average costs Remember the elasticity of scale in the production function nyw Fquot 57 alnf1x 7 N afi g 0111M 7 ax 0y which is the ratio between the marginal physical and average physical products AEB 6184 7 Production Economics Lecture XIV Professor Charles B Moss Fall 2005 c We understood that this measured the overall response of production to inputs levels along a ray from the origin What is developed here is not quite the same but is actually the elasticity of size I It answers the question Do I build one large plant or several small ones 11 De ning yym 4w wc 3 1 cwycwmym wy gmymis related to the homogeneity of the cost function in terms of scale Taking mm mm w 111 If nyw gt 1 then syxwy lt 1 there are no efficiencies to centralization diseconomies of scale IVIf nyw lt 1 then syxwy gt 1 there are efficiencies to centralization economies of scale 1 The cost exibility also has a geometric interpretation CWNV y a Just like the MPPAPP comparison we can envision the ratio be marginal cost and average cost It is clear that 00w y a 7 7 1 CW CW g 0y nyWl CW V 0y y y b Also it is apparent that average cost equals marginal cost at the minimum of the average cost curve AEB 6184 7 Production Economics Lecture XIV Professor Charles B Moss Fall 2005 0 cm y y 00wy1 cwy100wy cwyl0 By 0y y y y 0y y y 11 Duality between the Cost and Production Functions A In our discussion of the primal we demonstrated how the production function placed restrictions on economic behavior B The question posed in duality is whether the optimizing behavior can be used to recover or reconstruct the properties of the production function 1 Minkowski s Theorem A closed convex set is the intersection of halfspaces that support it X H 35A The half space H mk is defined as Hmkxmx k 3 Thus based on the definition of a cost function we have Nwyxwx cwy This definition actually recovers the original production set Vyxwx20wyVwgt0 If VyVy the original technology can be recovered N Factor Bias Technical Change and Valuing Research Lecture XXIV Mathematical Model of Technical Change A If we start from the quadratic production function specified as fx1x2 a0 alxl azxz 411x12 2A12x1x2 A12x22 assuming an output price of p and input prices of WI and w2 for inputs x1 and x2 respectively the derived demands for each input can be expressed as A12a2pA22611pA22w1A12w2 pAA A AlchpAnazpAuw1z4an pAA A xf pawpr x pawpr B In order to analyze the possible effect of technological change we hypothesize an input augmenting technical change similar to the general form of technological innovation introduced by Hayami and Ruttan 1 Speci cally we introduce two functions where y1LJ and y2 IV are augmentation factors and w is a technological change Hence y1LJy2 4021 for any w Thus technological change N increases the output created by each unit of input Integrating these increases into the forgoing production framework the derived demands for each input becomes 2 A12a2Y1Wquot2WpA126hY1WY2WpA22quot2WW1A12Y1Wwz pA11A22 2 vfwvzw Au xP W W W AIZQIY1WYZ WpA11a2Y1wYZ Op AU WWIA22Y1WWZ 2 1 2 pA11A12 A12Y1WY w pawpwpw 3 In order to simplify our discussion we assume that the new technology does not affect the effectiveness of x2 or yz LII gt 1 Under this assumption the derived demand for each input becomes 2 141261271WpA226h71WpAzzw1 A1271Ww2 pA11A22 14122Y12 W x p w w w 141261171WpA11a271WpA12w1 A1271 W wz 2 1 2 pAuAnAuvlw pawpwpw 4 In order to examine the effect of the technological change on each derived demand we take the derivative of each of the demand curves in equation 5 with respect to w as y1LJ gt 1 yielding 0x pawpwpw 2A12a2p 412611p 412w1A12W2Yill 6w 414 Ap A12a2pA12a1pA12W2 7i 4 414 Ap 6x p W1 wz l AilalpA11a2pA12wl A11w2 7i 4 6w 414 Ap A12a1pA11a2pA11w2 7i 4 14111422 A122p 71 WPI 71 WPI II Valuing State Level Funding for Research Results for Florida A The most basic definition of productivity involves the quantity of output that can be derived from a fixed quantity of inputs For example most would agree that a gain in productivity has occurred if corn yields increased from 70 bushels per acre to 75 bushels per acre given the same set of inputs ie pounds of fertilizer or hours of labor 1 Aggregate agricultural outputs and inputs could be computed based on Divisia quantity indices Specifically let Ytbe the aggregate output index computed as Y Zamyu where r is the revenue share of output y Similarly the aggregate input index can be computed as X 2 X2 211 312x12 where s is the cost share of input x Equating aggregate output with N aggregate input yields Y2 72X 39 a Rearranging slightly yields t X2 Y b The rate of technical change can the derived from the log change in both sides In 1n 1n X2 X271 Vtrl Total Factor Productivity 3 The actual procedure used by Ball Butault and Nehring 2002 is somewhat more complex First the input and output indices are computed using Fisher chained indices Second Ball Butault and Nehring adjust for quality changes by adjusting input prices to re ect quality changes B Ball Butault and Nehring 2002 develop a detailed productivity index 1 The measure developed from these aggregate output and input measures are referred to as index numbers The index created in this case is called the Total Factor Productivity index TFP a These results indicate that TFP for agriculture in Florida rose from 0728 in 1960 to 1590 in 1999 b This represents an average annual growth rate of 201 percent The growth in total factor productivity is primarily the result of increased output c Total farm outputs grew at an annual rate of 265 percent over the sample period with greater growth in crop outputs 306 percent as opposed to increased livestock output 200 percent d Total farm inputs remained relatively steady increasing at only 065 percent per year However this stability masks an increase in intermediate inputs of 174 percent per year This increase is partially offset by declines in land and labor uses 18 7 16 7 14 7 12 7 10 7 08 7 06 7 04 7 02 7 00 1 1 1 1 1 1 1 1 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Year Alabama I Florida Georgia South Carolina Natural Logarithm of 1996 Dollars 178 7 7 05 7 04 176 7 7 03 D4 11 174 7 C 39 7 02 0 E g 172 7 7 01 2 DD o 1 17 7 7 0 E a 7 701 Z 168 7 7 702 166 7 7 703 164 1 1 1 1 1 1 1 1 1 04 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Year R amp D Stock Total Factor Productivity C The Johansen 1988 approach involves estimating a vector errorcorrection mechanism expressed as k Ax HxH Z 1 mm QDD at 1 11 where x2 is a vector of endogenous variables A19 denotes the timedifference of that vector Ax x2 x271 D2 is a vector of exogenous variables 82 is a vector of residuals and H F1 and d3 are estimated parameters 1 N If a longrun relationship eg cointegrating vector exits the H matrix is singular H 04339 The 3 vector is the cointegrating vector or longrun equilibrium Further the statistical properties of the cointegrating vector are determined by the eigenvalues of the estimated H matrix Denoting 7L1 represent the zquot11 eigenvalue in descending order of signi cance the test for signi cance of the cointegrating vector can be written as 21nQH1rH1p Tl11nl Xl which tests the hypothesis that r cointegrating vectors are present H1r against the hypothesis that p cointegrating vectors are present H1 p Hamilton 1994 p 645 D The existence of a cointegrating vector in this framework implies that the linear combination 2 of the natural logarithm of TFP and research and the P1 natural logarithm of research and development stocks RD is stationary or a longrun equilibrium between these two series exists 1 N E 4 While this cointegrating vector is not uniquely identi ed the longrun relationship can be expressed as 1000 lnTF J z 13433 0794 1nRD Building on this expression the longrun relationship can be expressed as lnTFB 07941nRD13433 2 Manipulating this result further yields d TF8 0794 TFB dRDt RD Using the geometric mean of both TFP and research and development stocks TFP increases 00302 with a one million dollar increase in the research and development stock This number appears small but it represents 113 percent of the average annual increase in productivity observed in the state Thus we are left with the conclusion that research expenditures in IFAS have a signi cant effect on agricultural productivity in the state In order to understand the possible causes of the lack of a longrun equilibrium between agricultural profitability and productivity I express the change in profit over time as DnDFDkIQ where 11 denotes profit in period t F denotes Total Factor Productivity in time t and 1 denotes the change in relative price ratio in time t 1 In order to derive this relationship we start with agricultural profit defined as n 21 pay 1 wzsz where p12 is the price of output 139 in period t y is the level of output 139 produced and sold in period t w is the price of input j in period t and x is the input j purchased in period t 2 Differentiating both sides yields dnt 21 dpny 21 pltdy 21 dw x 21x dw Rewriting this expression using logarithmic differentiation yields DOT 21 rItDplt 211 DOIt 21 S12DW 21 312D x1 where 192 d lnz This expression can be rearranged to yield DOT 2116Dy1l 211 SJDx 211 nlDpll 211 S DWf 3 The first term on the left hand side of this expression is simply the change in Total Factor Productivity as previously discussed while the second term on the left hand side is the relative change in input and output prices Stochastic Error Functions 1 Another Composed Error Lecture X 1 Concept of the Composed Error Term A To introduce the composed error term we will begin with a cursory discussion of technical ef ciency which we develop more fully after the dual B We start with the standard production function y fxxa where yx is the level of output produced by rm 139 x1 is the level of inputs used by rm i and is a vector of parameters 1 We begin by acknowledging that rms may not produce on the ef cient frontier y fxl TE where TE denotes technical inef ciency 2 We assume that TE 1 with TE 1 denoting a technically ef cient producer TE L f xw 3 The above model presents all the error between the rm s output and the frontier as technical inef ciency 4 Augmenting this model with the possibility that random shocks may affect output that do not represent inef ciency y fxu eXPV TE Where V1 is a random shock resulting from factors such as weather that are outside the control of the producer 11 Models of technical inef ciency without random shocks A Building on the model of technical inef ciency alone we could estimate the production function using a onesided error speci cation alone 1 Mathematical Programming Goal Programming First we could solve two nonlinear programming problems a First we could minimize the sum of the residuals such that we constrain the residuals to be positive minzluI SJ LtI o Z kh1xxkihlyl k u 2 0 139 Lu which approximates the distribution function for the exponential distribution with a log likelihood function mltLgt1mltagtiaiuIul AEB 61847Production Economics Lecture X Professor Charles B Moss Fall 2005 b The second speci cation minimizes the sum of square residuals such that the residual is constrained to be positive min SJ LtI o Z khlxxkihlyl k u 2 0 i 1 1 which approximates the halfnormal distribution 7 1 2 1 2 111L7 C751n039M 7 20 Zux 2 Corrected Ordinary Least Squares a In this approach we estimate the production function using ordinary least squares then we adjust the estimated frontier upward by adding a sufficient constant to the estimated intercept to make all the error terms negative 8 o m X the estimated residuals are then M u I 121 imax tx b This procedure simply shifts the production function estimated with OLS upward no information on the inefficiency is used in the estimation of the slope coefficients 3 Modified Ordinary Least Squares a A related two step estimation procedure it to again estimate the constant and slope parameters using ordinary least squares and then to fit a secondary distribution function ie the half normal gamma or exponential to the residuals b The expected value of the residuals for this second distribution is then used to adjust the constant of the regression and the residuals 0E721 fE x c In addition to the constant shift in the production function addressed above this specification does not necessarily guarantee that all the residuals will be negative 111 Stochastic Frontier Speci cations A Adding both technical variation and stochastic effects to the production model we get h1yx 0 Z kxxkvx x k Where V1 is the standard twosided typically normal distribution used in regression analysis and MI is a onesided error term AEB 61847Production Economics Lecture X Professor Charles B Moss Fall 2005 1 The overall error term of the regression is refereed to as the composed error 1nyx 0Z c1nxxkgx k 5 Vx ux N Assuming that the components of the random error term are independent OLS provides consistent estimates of the slope coefficients but not of the constant Further OLS does not provide estimates of producerspecific technical inefficiency However OLS does provide a test for the possible presence of technical inefficiency in the data a Specifically if technical inefficiency is present then ux gt0 so that the distribution is negatively skewed Various tests for significant skewness are available Bera and Jarque but in this literature 21 quot5 m2 2 E Fquot where m2 and W13 are the second and third central moments of the distribution respectively A second statistic Ly N N01 3 Z 6 I B Returning to the Illinois and Indiana dataset the ordinary least squares results are Estimated CobbDouglas Coefficients Variable Estimate Constant 4 585 82 005607 Nitrogen 001265 001 179 Phosphorous 001677 000732 Potash 001322 000629 131 quot3 22140057 m Ly 15785307 N N01 De nition and Properties of the Cost Function Lecture XIII From Previous Lectures A 03 0 U P1 In the preceding lectures we rst developed the production function as a technological envelope demonstrating how inputs can be mapped into outputs Next we showed how these functions could be used to derive input demand cost and pro t functions based on these functions and optimizing behavior In this development we stated that economist had little to say about the characteristics of the production function We were only interested in these functions in the constraints that they imposed on optimizing behavior Thus the insight added by the dual approach is the fact that we could simply work with the resulting optimizing behavior In some cases this optimizing behavior can then be used to infer facts about the technology underlying it Gorman 1976 Duality is about the choice of the independent variables in terms of which one de nes a theory Chambers p 49 The essence of the dual approach is that technology or in the case of the consumer problem preferences constrains the optimizing behavior of individuals One should therefore be able to use an accurate representation of optimizing behavior to study the technology The Cost Function De ned A B 0 The cost function is de ned as cwy mggilwgtlt x xe Vy l Literally the cost function is the minimum cost of producing a given level of output from a speci c set of inputs 2 This de nition depends on the production set Vy In a speci c instant such as the CobbDouglas production function we can de ne this production set analytically 3 Technology constrains the behavior or economic agents For example we will impose the restriction on the technology so that at least some input is used to produce any nonzero level of output The goal is to place as few of restrictions on the behavior of economic agents as possible to allow for the derivation of a fairly general behavioral response Not to loose sight of the goal we are interested in be able to specify the cost function based on input prices and output prices cwya0a39ww Aw 39yy39BywTy Is a standard form of the quadratic cost function that we use in empirical research We are interested in developing the properties under which this function represents optimizing behavior AEB 6184 Production Economics Lecture XIII Professor Charles B Moss Fall 2005 D In addition we will demonstrate Shephard s lemma which states that 60 w q wy a AwwFHy w Or that the derivative of the cost function with respect to the input price yields the demand equation for each input 111 Properties of the Cost Function A Properties 1 cwy gt 0 for w gt 0 and y gt 0 nonnegativity If w 2 w then cw y Z cwy nondecreasing in w Concave and continuous in w Ctw y tcw yt gt 0 positively linearly homogeneous If y 2 y39 then cwy Z cwy nondecreasing in y and Cw 0 0 no fixed costs If the cost function is differentiable in w then there eXists a vector of costs minimizing demand functions for each input formed from the gradient of the cost function with respect to w B In order to develop these costs we begin with the basic notion that technology set is closed and nonempty Thus Vy implies x e Vy Thus 8995 mgtiglwgtltx wgtltx x39 0ery Graphically Vy C Discussion of Properties 1 Property 2Bl simply states that it is impossible to produce a positive output at zero cost Going back to the production function it was AEB 6184 Production Economics Lecture XIII Professor Charles B Moss Fall 2005 impossible to produce output without inputs Thus given positive prices it is impossible to produce outputs without a positive cost Property 2B2 likewise seems obvious if one of the input prices increases then the cost of production increases Graphically N B cWay wch A l 2 W1 W w l i a First if we constrain our discussion to the original input bundle x1 it is clear that wlx1 ltwzx1 if w2 gtw1 Next we have to establish that the change does not yield change in inputs such that the second price is lower than the first This conclusions follows from the previous equation mggiwxxwgtltx x S0ery In other words it is impossible for wlx2 lt wlx1 w1x2ltw1x1 Taken together this results yields the fundamental inequality of cost minimization w1 w2x1 x2S0 If we focus on one price w wx x30 3 Continuous and concave in w a This fact is depicted in the above graph 1 Note that A B and C lie on a straight line that is tangent to the cost function at B 11 Movement from B to C would assume that input bundle optimal at B is also optimal at C 111 If however are opportunities to substitute one input for another such opportunities will be used if they produce a lower cost Fquot AEB 6184 Production Economics Lecture XIII Professor Charles B Moss Fall 2005 b To develop a more rigorous proof let w0 WI and w11 be vectors of prices and x1 and x11 be associated input bundles such that w0 6wl1 t9w OSBSI Thus w1 is one vector of input prices and w11 is another vector of input prices w0 is then a linear combination of input prices We then want to show that cw y2 90wlyl t9cwny Let x0 be the cost minimizing bundles associated with w0 By cost minimization 10 11 110 1111 wx wa andw x wa Therefore cw y w x t9w1 l t9w x 19wle l t9w x Z 6cwlyl t9cwny wlx0 Z CWly wlx1 wnx0 Z Cw11y wnx11 Positive Linear Homogeneity Ctway EMWW xe Vy tm2iglwaery 4 tcwy UI No fixed costs 6 Shephard s lemma a In general Shephard s lemma holds that 60 w y Bwl xi my 4 Vwcwy sz 2 5 39 x w y 60 w y 6wquot b At the most basic level this proof is a simple application of the envelope theorem 1 First assume that we want to maximize some general function fx1x2xna AEB 6184 Production Economics Lecture XIII Fall 2005 Professor Charles B Moss were we maximize f x0c through choosing x but assume that at is fixed To do this we form the first order conditions conditional on at 99Xzaquot39xnaa03 xi xf a yxfax avquotxlaaa Ma The question is then How does the solution change with respect to a change in ac To see this we differentiate the optimum objective function value with respect to ac to obtain ay 2 am 2 am aampaaf 6a 6a 11 Byq 6a 6a given 6f O v z39 699 3 am 2 am 6a 6a 11 Similarly in the case of the constrained optimum maxfxlxzxn a SJ gx1x2xna 6L g lg1 xlzxxa 3Lflg3 3 6L 0 zltagt M Again differentiating the optimum with respect to at we get 6W 6W 6fx 6ampa0f 6a 6a 6a 11 Byq 6a but L ML i6g 69q 69q Byq To work this out we also differentiate the cost function with respect to x gxaxaxaa 6g quot 6g6qa6g 6a 11 Byq 6a 6a Putting the two halves together AEB 6184 Production Economics Lecture XIII Professor Charles B Moss Fall 2005 ayltgtafltxgta ltagt am iagcmm ago 6a 11 Byq 6a 6a 11 Byq 6a 6a 6y 1 6fx16g 6xa 6f16g 6a 11 69q 69q 6a 6a 6a 6fx16g 390Vi ayi39 l 6x 6x 6a 6a 6a c Thus following the envelope theorem min CW y wlxl wzx2 SJ fx1x2y 3Lw1x1w2xzly0 fxlxz 60 my at t 3 w Bwl Bwl x1 x1 y More explicitly 60 w nyl Maiw i Bwl Bwl Bwl However by firstorder conditions af wz 1 6x2 3 aim Wiggxrigj Bwl 6x1 Bwl 6x2 Bwl However differentiating the constraint of the minimization problem we see 0 6 6f6x Bwl 6x1 Bwl 6x2 sz yofxle 3 Thus the second term in the preceding equation is zero and we have demonstrated Shephard s lemma gt 03 De nition and Properties of the Production Function Lecture 11 Overview of the Production Function Chambers The production function and indeed all representations of technology is a purely technical relationship that is void of economic content Since economists are usually interested in studying economic phenomena the technical aspects of production are interesting to economists only insofar as they impinge upon the behavior of economic agents Chambers p 7 Because the economist has no inherent interest in the production function if it is possible to portray and to predict economic behavior accurately without direct examination of the production lnction so much the better This principle which sets the tone for much of the following discussion underlies the intense interest that recent developments in duality have aroused Chambers p 7 l The point of these two statements is that economists are not engineers and have no insights into why technologies take on any particular shape We are only interested in those properties that make the production function useful in economic analysis or those properties that make the system solvable There are several interpretations of the dual Let use brie y discuss one concept Assume that we are interested in analyzing production of some crop say cotton a One approach would be to estimate a production function say a CobbDouglas production function in two relevant inputs y xix b Given this production function we could derive a cost function by minimizing the cost of the two inputs subject to some level of production min wlx1 wzx2 N SI y xf xz Forming the Lagrangian of this optimization problem we have Lwlx1 wzx2 vyixf x2 a w 1m 0 1 0amp9 xi a E 22qu 0 0x x 0L 7 ex 0 0 y x1 2 Taking the rst two firstorder conditions together we have AEB 6184 7 Production Economics Lecture 11 Professor Charles B Moss Fall 2005 g 0i 3 n x2 3 n g wz 7 x1 XI 7 W1 xi 0x Substituting this relationship into the nal firstorder condition yields 0L w a i M w 3y sz x 03x2wlwpyyy 1 EM w1 w2 By substituting this relationship back into the previous condition with respect that solves xl as a function of xi we have V a 51 w x1w1w2yyy WI Substituting both of these optimal relationships output conditional input demand curves back into the cost function yields a j w w u j Cwlawzayw1 y W1 wz y 4 AM wfM wz wl m w2 y c Thus in the end we are left with a cost function that relates input prices and output levels to the cost of production based on the economic assumption of optimizing behavior d Following Chamber s critique recent trends in economics skip the first stage of this analysis by assuming that producers know the general shape of the production function and select inputs optimally Thus economists only need to estimate the economic behavior in the cost function e Following this approach economists only need to know things about the production function that affect the feasibility and nature of this optimizing behavior 3 In addition production economics is typically linked to Sheppard s Lemma that guarantees that we can recover the optimal input demand curves from this optimizing behavior 11 Production Function Defined A Following our previous discussion we then define a production function as a mathematical mapping function f R a R However we will now write it in implicit functional form Y Z 0 AEB 6184 7 Production Economics Lecture II Professor Charles B Moss Fall 2005 This notation is sometimes referred to as a netput notation where we do not differentiate inputs or outputs In more traditional terms we differentiate inputs and outputs yielding Y yx 0 Following the mapping notation we typically exclude the possibility of negative outputs or inputs but this is simply a convention In addition we typically exclude inputs that are not economically scarce such as sunlight Finally I like to refer to the production function as an envelope implying that the production function characterizes the maximum amount of output that can be obtained from any combination of inputs 1 As such the production function is a frontier function which is somewhat at odds with some of the implications of ordinary least squares 2 The concept of production functions as frontier functions allows for the analysis of technical inefficiency III Properties of the Production Function A Properties of fx 1 Monotonicity and Strict Monotonicity a Ifx39 2 x thenfx39 2 monotonicity b Ifx39 gt x thenfx39 gt strict monotonicity 2 QuasiConcavity and Concavity DJ 0 a Vy xfx 2 y is a convex set qausi concave b f9x 179x29fx 179fx for any0 9 l concave 3 Weakly essential and strictly essential inputs a f 0 0 where 0 is the null vector weakly essential b fx1xH0xH1xn 0 for all xx strictly essential 4 The set Vy is closed and nonempty for all y gt 0 5 f x is finite nonnegative real valued and single valued for all nonnegative and finite x 6 Continuity a f x is everywhere continuous and b f x is everywhere twicecontinuously differentiable B Properties la and lb require the production function to be nondecreasing in inputs or that the marginal products be nonnegative In essence these assumptions rule out stage III of the production process or imply some kind of assumption of freedisposal One traditional assumption in this regard is that since it is irrational to operate in stage III no producer will choose to operate there Thus if we take a dual approach as developed above stage III is irrelevant N AEB 6184 7 Production Economics Lecture 11 Professor Charles B Moss Fall 2005 C Properties 2a and 2b revolve around the notion of isoquants or as redeveloped here input requirement sets 1 The input requirement set is de ned as that set of inputs required to produce at least a given level of outputs V y Other notation used to note the same concept are the level set Strictly speaking assumption 2a implies that we observe a diminishing rate of technical substitution or that the isoquants are negatively sloping and convex with respect to the origin N Vy x2 3 Assumption 2b is both a stronger version of assumption 2a and an extension For example if we choose both points to be on the same input requirement set then the graphical depiction is simply AEB 6184 7 Production Economics Lecture 11 Professor Charles B Moss Fall 2005 f6x 1 6x 26fx 1 6fx Vy x2 a If we assume that the inputs are on two different input requirement sets then f9x 179529fx07fxfxl a i N x xfx Clearly letting 9 approach zero yields f x approaches f9x 179x29 f xl however because of the inequality the lefthand side is less than the right hand side Therefore the marginal productivity is nonincreasing and given a strict inequality is decreasing As noted by Chambers this is an example of the law of diminishing marginal productivity that is actually assumed c Chambers offers a similar proof on page 12 learn it D The notion of weakly and strictly essential inputs is apparent l The assumption of weakly essential inputs says that you cannot produce something out of nothing Maybe a better way to put this is that if you can produce something without using any scarce resources there is not an economic problem Fquot 2 The assumption of strictly essential inputs is that in order to produce a positive quantity of outputs you must use a positive quantity of all resources 3 Different production functions have different assumptions on essential inputs It is clear that the CobbDouglas form is an example of strictly essential resources E The remaining assumptions are fairly technical assumptions for analysis First we assume that the input requirement set is closed and bounded This implies that functional values for the input requirement set exist for all output AEB 6184 7 Production Economics Lecture 11 Professor Charles B Moss Fall 2005 levels this is similar to the lexicographic preference structure from demand theory F Also it is important that the production function be finite bounded and real valued no imaginary solutions The notion that the production function is a single valued map simply implies that any combination of inputs implies one and only one level of output G The continuity constraints are for mathematical nicety IV Law of Variable Proportions A The assumption of continuous function levels and first and second derivatives allows for a statement of the law of variable proportions B The law of variable proportions is essentially restatement of the law of diminishing marginal returns 1 The law of variable proportions states that if one input is successively increase at a constant rate with all other inputs held constant the resulting additional product will first increase and then decrease 2 This discussion actually follows our discussion of the factor elasticity from last lecture Ay y dy x MPP E rum dx y APP x MPP dTPP dxAPP APH x dAPP dx dx d x 3 Working the last expression backward we derive 1Mpp7App dx x Or in multivariate and Chamber s notation 0 AP 1 0f y 6x1 7 61 6x1 61 V Elasticity of Scale A The law of variable proportions was related to how output changed as you increased one input Next we want to consider how output changes as you increase all inputs B In economic jargon this is referred to as the elasticity of scale and is defined as 87 amfix 01111 zl AEB 6184 7 Production Economics Lecture 11 Professor Charles B Moss Fall 2005 1 This change implies the movement along a ray drawn from the origin xlxz x1 xlxl 2 The elasticity of scale takes on three important values a If the elasticity of scale is equal to 1 then the production surface can be characterized by constant returns to scale Doubling all inputs doubles the output b If the elasticity of scale is greater than 1 then the production surface can be characterized by increasing returns to scale Doubling all inputs more than doubles the output c Finally if the elasticity of scale is less than 1 then the production surface can be characterized by decreasing returns to scale Doubling all inputs does not double the output 3 Note the equivalence of this concept to the definition of homogeneity of degree k 17W f 1x 4 For computational purposes