Econ Agri Production
Econ Agri Production AEB 6184
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Weldon Rau I
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This 15 page Class Notes was uploaded by Weldon Rau I on Friday September 18, 2015. The Class Notes belongs to AEB 6184 at University of Florida taught by Charles Moss in Fall. Since its upload, it has received 19 views. For similar materials see /class/206589/aeb-6184-university-of-florida in Agricultural Economics And Business at University of Florida.
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Date Created: 09/18/15
Measuring Changes in Productivity Lecture XXII 1 De ning Changes in Productivity A The most basic concept of changes in productivity is that for a given level of inputs we now get more output yl 1 p1 y2 yi yz AEB 6184 7 Production Economics Professor Charles Moss 1 Lecture XXII Fall 2005 In this gure we assume that the inputs are constant at x but the total level of outputs has increased from y1 y2 to yl39 yz39 Alternatively in input space it now takes fewer inputs to produce the same level of outputs 3 I x2 x2 x2 In both cases most would agree that a technical change has taken place Further most would agree that the technical change has increased the economic well being of society We now have more stuff for the same level of inputs 4 However there are some issues that need to be raised a First one could raise the question about embodied versus disembodied technical change i This debate regards whether there has been an increase in knowledge or whether there has been an increase in the quality of inputs ii If the increase in output has been associated with an increase in quality of an input is it technical change iii For example a large portion of the gains to research literature can be traced to Griliches discussion of hybrid corn In this case the increase in technology was associated with the improvement in an inputiseed iv More recently some of the most recently observed increases in productivity may be traced to genetically modi ed organisms GMOs v Under most concepts of productivity these increases do not represent changes in productivity in agriculture but can be traced to changes in the input bundle AEB 6184 7 Production Economics Lecture XXII Professor Charles Moss Fall 2005 b A second area of concern is whether the changes in technology are neutral with regard to the input bundle i Going back to the figure the increase in technology implies relatively more x1 it is biased toward x1 ii In the hybrid corn example additional fertilizer complemented the use of hybrid corn iii In the classic discussion Hicks developed the notion of labor or capital augmenting technological development II Measuring Technological Change A In the single inputsingle output analysis one could directly measure technical change yfxt0t13t039llj x x 1 Several factors should be considered a We know that pro tmaximizing behavior changes the point of production even in the univariate case Speci cally we know that the decision maker chooses to produce where the marginal value product equals the price of the input Thus if either the price of the input or the price of the output has changed the ratio of outputs to inputs will change Even in the single variable case we would wonder about excluded factors things beyond the decision maker s control B Extending the analysis to the multivariate world we begin by examining the productproduct relationship in the rst graph Note 90 Y39x Piyi szi Yx Piyi sz2 could be used as one measure of technical change Similarly in the input input relationship Fquot O 90 m wixi W299i Vy wlx1 wzx2 Focusing on these two formulations 1 Note rst that each of these formulations is based implicitly on Shephard duality I Y39x quotPWJ1 p1ypwt1pzy pwt1 Yx 7rpwt 0 p1y1pwt 0 ply2 pwt 0 90 V39y cwyt 1 w1x139wyt l wzx wyt l Vy cwyt 0 w1x1wyt 0 wzx2 wyt 0 2 More formally cwytminwxxe Vyt ltgt Vyt xwx2 cwytwgt0 AEB 6184 7 Production Economics Lecture XXII Professor Charles Moss Fall 2005 3 Thus by gross simpli cation we could envision a cost function cwyt a0 a39w w39Aw 6quoter y39By w39Fy 9wyt xwyt a Aw Fy VW9wyt with 9w yt being a measure of technical change 4 This formulation allows us to discuss several key features of technology measurement a First in the grossest sense technological change tends to be a measurement of factors that we don t understand From the preceding equation what is the difference between technology and a residual i One approach is to proxy technical change with a simple time trend t ii Alternatively several studies have used other proxy variables such as spending on agricultural research b This formulation allows the researcher to examine the neutrality of technical change 00w y t 0w 00wyt xwyt xwy 014 Finally it is possible to envision adjusting this measure for differences in input quality i For example if the quality of one variable increases over time then we could adjust the price of that variable upward to account for the increase in quality IIIT0tal Factor Productivity and Index Number Theory A The index number approach can be looked at as an extension of the single inputsingle product scenario above 0 xt Jr 0fxt O yfx t dt ax dt at B In a multivariate context yfxt Zafx t m dt 0xx dt at Replacing differentiation with log differences AEB 6184 7 Production Economics Lecture XXII Professor Charles Moss Fall 2005 dlnlyl ablyldhllxxl1x t dt 0nxx dt Zq dlix h m wx dlnxx T t Zpy dt x Tx tdlny72 dlnxx dt Py dt This formulation is sometimes approximated as Txtlnyt7lnyt717Kt V17t71lnxnilnxl The approximation is typically called the TomqVistTheil measure Working backward from the discussion above dlny dlnx Txt 2 dlny7 dlnxTxt d1n rxt 0 dlnQy Txt Q where Qy and Qx are index numbers representing the total quantity of inputs and outputs 1 In the TomqVistTheil index the indices were implicitly DiVisia output and input indices 92 1quot 322 wx I ijy wfx J 2 The linkage in this case is the de nition of total factor productivity yxt TFPXTFPX 0 dt x J39c 20fyxt 6xI dt 1 AEB 6184 7 Production Economics Lecture XXII Professor Charles Moss Fall 2005 IV Distance Functions A Back to the rst of the lecture remember the inputinput discussion x1 Xi x2 x2 90 m wle wzx V00 mnmn B These concepts actually de ne a distance function measure of productivity growth 1 De ne a measure 9 for a new technology based on the old technology as D5xyngn6Fx 92y AEB 6184 7 Production Economics Lecture XXII Professor Charles Moss Fall 2005 Subadditivity of Cost Functions Lecture XX 1 Concepts of Subadditivity Evans D S and J J Heckman A Test for Subadditivity of the Cost Function with an Application to the Bell System American Economic Review 741984 61523 A The issue addressed in this article involves the emergence of natural monopolies Speci cally is it possible that a single rm is the most cost ef cient way to generate the product In the specific application the researchers are interested in the Bell System the phone company before it was split up Basic concept 1 The cost function C q is subadditive at some output level if and only if US 0 Z l 1 11 which states that the cost function is subadditive if a single rm could produce the same output for less cost As a mathematical nicety the point must have at least two nonzero rms Otherwise the cost function is by de nition the same Developing a formal test Evans and Heckman assume a cost function based on two input N U ZCa1915beZgtCQIQZ iLquot n 211111 1 leb1 la120 b120 Thus each of 139 rms produce a percent of output ql and b1 percent of the output qz l A primary focus of the article is the region over which subadditivity is tested a If the sum of the disaggregated rm s cost functions are greater than the cost of the aggregated rm ZCa nb zgtc h z The cost function is subadditive and the technology implies a natural monopoly If the sum of the disaggregated rm s cost functions are less than the cost of the aggregated rm Z Car 1bi zltcqial the cost function is superadditive and the rm could save money by breaking itself up into two or more divisions Fquot AEB 6814 7 Production Economics Lecture XX Professor Charles Moss Fall 2005 0 Finally if the sum of the disaggregated cost functions is equal to the cost of the aggregated rm ZCa 1bx zc la z the cost function is additive 2 The notion of additivity combines two concepts from the cost function Economies of Scope and Economies of Scale a Under Economies of Scope it is cheaper to produce two goods together The example I typically give for this is the grazing cattle on winter wheat However we also recognize following the concepts of Coase Williamson and Grossman and Hart that there may diseconomies of scope c The second concept is the economies of scale argument that we have discussed before 3 As stated previously a primary focus of this article is the region of subadditivity a In our discussion of cost functions I have mentioned the concepts of Global versus local To make the discussion more concrete let us return to our discussion of concavity i From the properties of the cost function we know that the cost function is concave in input price space Thus using the Translog form Fquot CWayeXPl1nCl lnCa0 a39mw 1nw39Amw mnymy39smy1nw39 r1ny ii The gradient vector for the Translog cost function is then 51 0L1 Alw my VWCwyexplnC 3 explnC sH an Anw 11y iii The Hessian matrix is then 1 A1w riy a1 A1w riy 39 A11 quot 39 Am VfWCwyexplnC 3 3 explnC I ozx Alw1 y ozx Alw1 y A1 Am1 CA wy 39wy where wy is a vector of estimated shares based on input prices and output levels iv Given that the cost is always positive the positive versus negative nature of the matrix is determined by AEB 6814 7 Production Economics Lecture XX Professor Charles Moss Fall 2005 A wy 39wy V Comparing this results with the result for the quadratic function we see that VfWC w y A vi Thus the Hessian of the Translog varies over input prices and output levels while the Hessian matrix for the Quadratic does not vii In this sense the restrictions on concavity for the Quadratic cost function are globalithey do not change with respect to output and input prices However the concavity restrictions on the Translog are locali xed at a specific point because they depend on prices and output levels viiiNote that this is important for the Translog Speci cally if we want the cost function to be concave in input prices x39A wy 39wyx 0 W 3 x39Ax x39 Wy 39Wayx g 0 3 x39Ax wyx39 39Wayx g 0 But 39wyx39 39wyx 2 0 AEB 6814 7 Production Economics Lecture XX Professor Charles Moss Fall 2005 b Thus any discussion of subadditivity especially if a Translog cost function is used or any cost function other than a quadratic needs to consider the region over which the cost function is to be tested i Thus much of the discussion in Evans and Heckman involve the choice of the region for the test Speci cally the test region is restricted to a region of observed point ii De ning qu as the minimum amount of ql produced by any rm and ng as the minimum amount of qz produced we an de ne alternative production bundles as A i i gt Q1r qlM wqlt gm q 1 qL qlM17wq qm 0 g 1 g 1 0 g 1339 g 1 Thus the production for any rm can be divided into two components within the observed range of output Thus subadditivity can be de ned as AEB 6814 7 Production Economics Lecture XX Professor Charles Moss Fall 2005 iii If Sub w is less than zero the cost function is subadditive if it is equal to zero the cost function is additive and if it is greater than zero the cost function is superadditive iV Consistent with their concept of the region of the test Evans and Heckman calculate the maximum and minimum Sub w for the region 11 Composite Cost Functions and Subadditivity A Building on the concept of subadditiVity and the global nature of the exible function form it is apparent that the estimation of subadditiVity is dependent on functional form Pulley and Braunstein allow for a more general form of the cost function by allowing the BoxCox transformation to be different for the inputs and outputs 03 Pulley L B and Y M Braunstein A Composite Cost Function for Multiproduct Firms with an Application to Economies of Scope in Banking Review of Economics and Statistics 741992 22130 1 In general they rely on the BoxCox tranformation 4 7 l y 13 0 lny 3 0 This implies a cost function a T CW 7 CXPD 0 Dl39q q Aql ql l r exp6 0 39r r39Br 7 le fW qmr AEB 6814 7 Production Economics Lecture XX Professor Charles Moss Fall 2005 2 The speci cation nests a number of standard forms a If 0 7239 0 and L39 lthe form yields a standard Translog with normal share equations b If 0 and L39 l the form yields a generalized Translog lnC a0 a39ql ql yAql q LPlnr 39lnr lnr39 Blnr s qu Blnr c If 7r01390 and k110 the speci cation becomes a separable quadratic speci cation W 010 a39q q39Aq 4 C my 39mltrgtmltrgt393mltrgt s Bnr d The demand equations for the composite function is 71 s a0 a39q q39Aq q Plnr Pq Blnr Y39q C Given the estimates we can then measure Economies of Scope in two ways The rst measures is a traditional measure Cq100rC0q200rC00qnr7Cqvqpmqwr Cqpqzqmr Another measure suggested by the article is quasi economies of scope Clim7lsq1qzsqnsrCqlslimilsq2q3sqnsr7Cq1q2qnr Cq1qzqmr SCOPE QSCOPE D The Economies of Scale are then de ned as C W 0C qr Z g y 0 SCALE Differential Models of Production The Single Product Firm Lecture XXV Overview of Differential Approach A Until this point we have mostly been concerned with envelopes or variations of deviations from envelopes in the case of stochastic frontier models 1 The production function was de ned as an envelope of the maximum output level that could be obtained from a given quantity of inputs 2 The cost function was the minimum cost of generating a xed bundle of outputs based on a vector of input costs B The differential approach departs from this basic formulation by examining changes in optimizing behavior 1 Starting from consumption theory we have maXUx 3 6Ux 1px SJ p39x S Y 6x a We assume that consumers choose the levels of consumption so that these rstorder conditions are satis ed b The question is then what can we learn by observing changes in these rstorder conditions or changes in the optimizing behavior 2 First note that there are n rstorder conditions 6U x 6X A 6Ux 1 6x2 72 1p 6U x 6x
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