Centre for Efficiency and Productivity Analysis

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1 Centre for Efficiency and Productivity Analyi Working Paper Serie No. WP04/00 Etimating State-allocable Production Technologie When there are two State of Nature and State Allocation of Input are Unoberved C.J. O Donnell & S.Shankar Date: November 00 School of Economic Univerity of Queenland St. Lucia, Qld. 407 Autralia ISSN No

2 ESTIMATING STATE-ALLOCABLE PRODUCTION TECHNOLOGIES WHEN THERE ARE TWO STATES OF NATURE AND STATE ALLOCATIONS OF INPUTS ARE UNOBSERVED by C.J. O Donnell and S.Shankar The Univerity of Queenland Centre for Efficiency and Productivity Analyi Bribane 407, Autralia Abtract: Chamber and Quiggin (000) have ued tate-contingent production theory to etablih important reult concerning economic behaviour in the preence of uncertainty, including problem of conumer choice, the theory of the firm, and principal-agent relationhip. Empirical application of the tate contingent approach ha proved difficult, not leat becaue mot of the data needed for applying tandard econometric method are lot in unrealized tate of the world. O'Donnell and Griffith (006) how how a retrictive type of tate-contingent technology can be etimated in a finite mixture framework. Thi paper how how Bayeian methodology can be ued to etimate more flexible type of tate-contingent technologie. Paper preented at the 009 Annual Conference of the Autralian Agricultural and Reource Economic Society (AARES), Cairn. The project wa completed while O Donnell wa viiting the Univeritat Autonoma Barcelona. Financial upport wa provided by the Generalitat de Catalonia and the Autralian Reearch Council.

3 . INTRODUCTION State-contingent production theory allow economit to apply the analytical tool of modern microeconomic in a tochatic production etting, provided ex ante preference and production technologie are properly defined. Chamber and Quiggin (000) have ued the theory to etablih important reult concerning problem of choice under uncertainty, including the problem of moral hazard, incentive regulation and portfolio choice. Unfortunately, empirical implementation of the theory in a production context ha proven difficult, not leat becaue the ex ante production choice of firm are only partially oberved. O'Donnell and Griffith (006) have hown how to empirically etimate output-cubical tate-contingent technologie in a finite mixture framework. Unfortunately, output-cubical technologie are inconitent with important tylized fact concerning behaviour in the preence of rik. The purpoe of thi paper i to how how the oberved input and output of firm can be ued to econometrically etimate more flexible tate-allocable tate-contingent technologie. The tructure of the paper i a follow. Section and 3 decribe ome of the important characteritic of tochatic technologie and the producer optimization problem in the preence of rik. Section 4 develop an econometric model for recovering the parameter of a two-tate tochatic technology when allocation of input to diffierent tate of Nature are unoberved. Section 5 ue noiele imulated data to demontrate that the methodology can be ued to recover unknown parameter and other economic quantitie of interet without error. Section 6 uccefully applie the methodology to a real-world data et and recover etimate of the (rikneutral) probabilitie farmer aign to different tate. The paper i concluded in Section 7.. STOCHASTIC TECHNOLOGIES We begin by conidering a firm that ue a ingle non-tochatic input to produce a ingle tochatic output. We aume production activitie take place over two time period: in period 0 the producer chooe the input in the face of uncertainty; in period, Nature reolve uncertainty by chooing from a et of tate Ω = {,..., S}. If Nature chooe Ω then the ex pot realization of tochatic output i given by the tate- production function (.) z = f( x, β ) where β i a vector of parameter permitted to vary acro tate, and x 0 i the amount of input allocated to production in tate. We aume the function f i everywhere continuou and atifie tandard regularity propertie, including monotonicity and quai-concavity. Chamber and Quiggin (000) call uch a technology tate-allocable. To illutrate the concept of tate-allocability, Chamber and Quiggin (000) provide a implified cropping example in which a producer make a pre-eaon allocation of a fixed amount of effort to the development of irrigation infratructure and/or flood-control facilitie. If the producer allocate more pre-eaon effort to irrigation than to flood control then output will be relatively high if realized rainfall i low, and relatively low if rainfall i high. Thu, different allocation of pre-eaon effort imply a trade-off between output realized in a low-rainfall tate and output realized in a high-rainfall tate. Indeed, we can think of the producer a allocating the input to production in different tate, and of reallocating the input between tate in order to effect a ubtitution between tate-contingent output. Figure depict a two-tate technology where the total amount of the input ued in the production proce ha been fixed at x. Rightward movement along the horizontal axi in panel (a) in Figure correpond to a reallocation of thi fixed amount of input from production in tate to production in tate. The downwardloping line in thi panel how how output in tate decreae a the amount of the input allocated to that tate decreae; the upward-loping function how how output in tate increae a the amount of the input allo-

4 cated to that tate increae. Panel (b) imply depict the aociated production poibilitie frontier in tatecontingent output pace. Oberve that by allocating x A A A unit of the input to tate and x = x x unit to tate the firm can eliminate rik (z = z at point A). However, any other allocation of x involve rik. For example, if the input i equally-allocated between tate the firm will obtain a higher output in tate than in tate (z > z at point B). The biector in panel (b) give the locu of all rikle tate-contingent output pair. The line paing through point C i a fair-odd line that will be dicued later in the paper. Aociated with (.) i the tate-pecific input requirement function x = f ( z, β ). It follow that production of the tate-contingent output vector z = ( z,..., z S ) require an input commitment of (.) x f ( z, β ). Ω The input ditance function i 3 x (.3) DI ( x, z, β ) f ( z, β ) = Ω where β contain the ditinct element of β,..., β S. Thi functional repreentation of the technology i the invere of a Farrell (957) meaure of technical efficiency. Other tandard repreentation of the production technology are alo available, including cot and output ditance function 4. In each of thee alternative repreentation, the vector of tate-contingent output i treated in the ame way a we treat vector of multiple output when production i non-tochatic. 3. FIRM BEHAVIOUR Given a normalized input price of w > 0, the net return in tate of Nature i y z wx. We aume the firm eek to maximie a general welfare function that i non-decreaing in tate-contingent net return. Then it optimization problem can be written (3.) max { W( y) : D ( x, z, β ) } z I where y = ( y,..., y S ) and W i a welfare function with the property W( y) W( y)/ y 0. The firt-order condition for efficient firm behaviour are 5 Strictly peaking, the firm doe not produce z. Rather, it commit the input in uch a way that z i produced if Nature chooe from. 3 The input ditance function i defined a DI ( xz,, β ) = max{ ρ : x / ρ can produce z }. Let ρ be the maximum factor by which a firm can contract it input vector and till produce the ame output vector. That i gz (, β) x/ ρ = 0. It follow that DI ( xz,, β ) = x/ gz (, β ). 4 Given a normalized input price of w > 0, the cot function i cwz (,, β ) = wgz (, β ). To derive the output ditance function, let δ be the larget factor by which a firm can expand it output vector while holding it input vector fixed. Then g( δz, β) x = 0. If f i / b / b homogeneou of degree then g i homogeneou of degree b, o that δ= g(, z β ) x. The output ditance function give the b invere of the larget factor by which a firm can expand it output vector while holding it input vector fixed. Thu, / b / b DO ( x, z, β ) = / δ= g( z, β ) x. 5 See Chiang (984, pp. 0-0). The partial total derivative of W( y) with repect to z 0 i ym ( zm wg( z, β)) gz (, β) Wm( y) = Wm( y) = W( y) w Wm( y). z z z m Ω m Ω m Ω The inequality in (3.) i due to the non-negativity retriction z 0. 3

5 (3.) π wm( z, β ) 0 for all Ω where (3.3) mz β f z β z and (, ) (, )/ > 0 π W ( y) (0,). W ( y) Ω Becaue the π term lie in the unit interval and um to one, they can be interpreted a rik-neutral probabilitie the ubjective probabilitie a rik-neutral firm would need to have if it were to elect the ame production plan a a rational firm with preference W( y ). Equation (3.) implie that any efficient choice for a rational firm with an objective function defined over net-return can be viewed a though it were generated by a rik-neutral firm with ubjective probabilitie given by ( π,..., π S ). Thu, without lo of generality, we can retrict our attention to the rik-neutral cae. Before olving the firt-order condition (3.) for a pecific tochatic technology, it i ueful to conider an efficient rik-neutral firm eeking to olve the optimization problem (3.) ubject to the additional contraint that the input level i fixed at x. The contrained optimization problem can be written (3.4) max πz : z = f( x, β ) for all ; x = x x,..., xs Ω Ω and ha an interior olution that atifie 6 z π m (3.5) =, z π m for all m Ω,. Thu, x i optimally allocated (i.e., expected output i maximized) when negative odd ratio are equated to marginal rate of ubtitution between tate-contingent output. Panel (b) of Figure depict the 6 The Lagrangean i L = π f( x, β) ψ x x Ω Ω The firt-order condition are () () L = x x = Ω 0 and ψ L f( x, β) =π +ψ= 0 x x From (): (3) f( x, β ) x π = f( x, β ) x π m m m m and from (): x x x =. m m Thu, x / x = and equation (3) collape to equation (3.5). m 4

6 cae where an optimal allocation of x place the efficient firm at point C on the efficient frontier. The traight line paing through point C i the locu of all point with the ame expected output. It ha lope π/ π and i known a the fair-odd line. Pictorially, optimization involve chooing that fair-odd line that i furthet from the origin and hare a point in common with the production poibilitie et. Finally, Figure allow u to illutrate the importance of properly defining the tochatic technology. Suppoe the amount of input allocated to tate {, } i fixed at 0.5 x. Then the efficient firm i operating at point B in panel (b) of Figure. Free dipoability of tate contingent output, together with the fact that the firm ha no capacity to reallocate the input between tate, mean the production poibilitie frontier i the rectangle with vertice at the origin and point B. For thi technology, the fair-odd line that olve the firm' optimization problem will alway pa through point B, implying the firm will not (cannot) alter the mix of tatecontingent output in repone to change in the rik-neutral probabilitie. Even when the firm believe that tate i a near-certainty, it will not (cannot) re-allocate input to the production of output in that tate. Thi i implauible. Retrictive technologie of thi type (i.e., one that do not allow ubtitution between tatecontingent output) are aid to be output-cubical. Thi term derive from the fact that when S = 3 the production poibilitie et can be repreented a a cube in tate-contingent output pace. 4. ESTIMATION IN THE TWO STATE CASE In ome empirical application, input allocation to tate of Nature and realized tate of Nature are both readily oberved. For example, O'Donnell, Chamber and Quiggin (006) decribe a ugar-cane production ytem in which producer plant different varietie of ugar cane (either high-yielding but dieae-uceptible, or loweryielding and dieae-reitant) in the face of uncertainty about the incidence of ugar-cane mut dieae. Acreage planted to different varietie of cane (input allocation) and level of dieae infetation (realized tate) can both be oberved ex pot. In thee cae, conventional etimation method, including data envelopment analyi (DEA) and tochatic frontier analyi (SFA), can be ued to recover the parameter of the production technology. In ome other empirical context, only the input allocation are oberved. For example, medical reearcher can uually oberve the different type of influenza vaccin adminitered by medical practitioner (input allocation), but cannot oberve the number of patient expoed to different train of influenza viru (realized tate). In thee cae, if the technology i output-cubical, the parameter of the production technology can be etimated within the finite mixture framework developed by O'Donnell and Griffith (006). Thi paper develop methodology for etimating the parameter of the production technology in a third empirical context, namely when there are two obervable tate of Nature but input allocation to thee tate are unoberved. Underpinning our etimation methodology i the aumption that firm are rational and technically efficient in production. The efficiency aumption, which can be eaily relaxed, mean that the relationhip between total input uage and tate contingent output i of the form (4.) x f ( z, β ) = 0. Ω The rationality aumption mean that an interior olution to the firm optimiation problem i given by (4.) π = wm( z, β ) for all Ω. Equation (4.) i epecially important for two reaon. Firt, if the invere of mz (, β ) exit then we can expre tate-contingent output a a function of normalied input price and rik-neutral probabilitie: (4.3) z m ( w, ) = π β for all Ω. 5

7 Second, in the two-tate cae, equation (4.) allow u to expre rik-neutral probabilitie a function of normalied input price, realized tate of Nature, and oberved output: (4.4) = e[ wm( q, ) ] + e[ wm( q, ) ] (4.5) π = e[ wmq (, β )] + e[ wmq (, β )] π β β and where e = if = (and 0 otherwie). Equation (4.4) and (4.5) can be ubtituted into equation (4.3), and the reult can then be ubtituted into equation (4.). Thi yield a poibly nonlinear relationhip between total input, normalized input price, realized tate of Nature, oberved output, a well a the unknown parameter of the production technology. Etimation involve embedding thi relationhip in a tochatic framework and applying an appropriate econometric etimator, uch a nonlinear leat quare (NLS). Importantly, equation (4.) cannot be ued on it own to recover the parameter of the technology. To ee thi, imply note that for any ( z, β ) pair there exit a π that will atify (4.) exactly. Thi mean that the parameter and rik-neutral probabilitie cannot be eparately identified unle additional information i introduced into the etimation proce. In thi paper, thi additional information come in the form of equation (4.). 5. EXAMPLE SIMULATED DATA O'Donnell, et al. (006) demontrate that conventional approache to efficiency meaurement may be ytematically and eriouly biaed in the preence of uncertainty. For illutrative purpoe, they conider a tateallocable tate-contingent production function of the Cobb-Dougla type: (5.) z = x a / b / b where b and a 0 for Ω = {, }. In term of the quantitie introduced in Section to 4: (5.) (5.3) (5.4) f ( x, β ) = x a / b / b f ( z, β ) = a z b mz (, β ) = f ( z, β )/ z = baz b (5.5) m ( w π, β ) π b = bwa b (5.6) ( b ) ( = e wbaq + e wbaq ) π and b (5.7) ( b π ) ( = e wbaq + e wbaq ) where β = ( a, b ). For thi technology, the relationhip between total input, normalized input price, realized tate of Nature and oberved output i of the form:. (5.8) b b b b b b b b ( ) + ( ) ( ) + ( ) e wba q e wba q e wba q e wba q x a a = 0 bwa bwa An aociated econometric etimating equation i: 6

8 (5.9) b b b nt nt nt nt nt nt nt nt nt nt nt nt b b b b b ( ) + ( ) ( ) + ( ) e w baq e w ba q e w ba q e w baq xnt = a + a + υ nt bwnta bwnta where the ubcript n and t repreent firm and time period repectively ( n=,..., N; t =..., T), and υ nt i a random variable repreenting tatitical noie. We have ued NLS to etimate thi conditional input demand function uing the imulated data reported in Table 4 of O'Donnell, et al. (006). The value of the unknown parameter ued to generate that table were b =, a =.5 and a = 0.5. Our NLS etimate of thee parameter were bˆ =, aˆ =.5 and a ˆ = 0.5 with tandard error of zero. The aociated rik-neutral probabilitie and unoberved tate-contingent output were alo recovered without error. Implementing an NLS algorithm involve chooing tarting value for the parameter of the technology that are compatible with rik-neutral probabilitie lying in the unit interval. Indeed, thi requirement alo need to be met on each iteration of the algorithm. Our experience with the imulated data wa that the NLS algorithm failed if we choe tarting value that were too far from the true value. Thi i likely to a problem in real-world ituation where the true value are, of coure, unknown. In the following ection we overcome the problem by etimating the model in a Bayeian framework. 6. EXAMPLE RICE DATA O'Donnell and Griffith (006) ue rice data to etimate an output-cubical tate-contingent production frontier. The data conit of more than 300 obervation on the input and rice output of farmer in the Tarlac region of the Philippine. The decriptive tatitic reported in Table reveal a large amount of variation in the data et. The ample farmer have no acce to irrigation, o at leat ome of the variation in the output variable can be attributed to variation in rainfall. Oberve from Table that data on rice input ha been aggregated into a ingle input index. Thi i convenient becaue it allow u to work with the following trivial generalization of the technology given by (5.): (6.) z = c+ x a / b / b where c 0, b and a 0 for Ω = {, }. The aociated econometric etimating equation i identical to (5.9) except that qnt i replaced by qnt c. The dummy variable e nt in (5.9) wa et to one if rainfall wa oberved to fall below the firt ample quartile (865 mm). We begin by writing the full et of NT equation repreented by (5.9) in the more compact form: (6.) x = g( q, w, β ) + υ where x = ( x, x,..., x NT ), β = (, ca, a, b) and the remaining definition are obviou. The error are aumed to be independent and identically ditributed a N(0, h ). Thu, the likelihood function i h NT NT / (6.3) px ( β, h) = f ( x gqw (,, β), h I ) h exp [ x gqw (,, β) ] [ x gqw (,, β) ] N where I NT i an identity matrix of order NT and fn ( a b, C ) denote the probability denity function (pdf) of a multivariate normal random vector with mean b and covariance matrix C. We ue the following improper prior: (6.4) p h h I R I h ( β, ) ( β ) ( 0) 7

9 where I (.) i an indicator function that take the value if the argument i true and 0 otherwie, and R i the region of the parameter pace where the retriction dicued in Section 5 are atified. That i, R i the region where c 0, a 0, a 0, b > and all three parameter are uch that the rik-neutral probabilitie (defined by equation 5.6 and 5.7, but with q replaced by q c) lie in the unit interval. Thu, the poterior pdf i (6.5) p h x h f x g q w h I I R I h ( β, ) = N( (,, β), NT) ( β ) ( 0) Conditional poterior pdf that can be ued within a Gibb Sampler are: h p( β x, h) exp x g( q, w, β) x g( q, w, β) I( β R) (6.6) [ ] [ ] (6.7) p( h x, β ) = f ( h h, NT) G and where (6.8) h = NT [ x g( q, w, β )] [ x g( q, w, β )] and fg ( a b, c ) denote the pdf of a gamma random variable with mean b and degree of freedom c. Simulating from the gamma denity (6.7) i traightforward uing random number generator available in mot tatitical oftware package. However, imulating from (6.6) i lightly more complicated becaue it i a truncated pdf. To imulate from (6.6) we ued a random-walk Metropoli-Hating algorithm with a multivariate normal propoal denity. For detail concerning thi algorithm, ee Koop (003). During the tranition, or burn-in, phae of the algorithm, the covariance matrix of the propoal denity wa et to a calar multiplied by an identity matrix. The calar wa et by trial and error to yield an acceptance rate in the range After the burn-in, to improve the efficiency of the algorithm, we ued the covariance matrix of the burn-in obervation a the covariance matrix in the propoal denity. In thi paper, we imulated 0,000 obervation from the conditional poterior (6.6) and (6.7) and dicarded the firt 0,000 draw a a burn-in. Figure preent convergence plot for each of the element of β and h. We did not ue tatitical tet to confirm convergence of the MCMC chain becaue the convergence plot are quite concluive inofar a they how abolutely no ign of non-tationarity. Etimate of the unknown parameter are preented in Table. The point etimate are the mean of the MCMC ample and are optimal Bayeian point etimate under quadratic lo. The inequality retriction in the prior (6.4) enure that all the etimate in Table are theoretically plauible. The tandard error are the tandard error of the MCMC ample and ugget that only b ha been etimated with any reliability. However, etimated tandard error can be mileading. A more complete picture of the level of uncertainty urrounding the unknown parameter i preented in Figure 3. Thi figure preent etimated marginal poterior pdf for each of the parameter. A feature of thee pdf i that the etimated pdf for a, a and b are aymmetric. Thi i a direct reult of the inequality information contained in the prior. A econd remarkable feature i that the etimated pdf for b ha no upport beyond.5, indicating a high degree of ubtitutability between tatecontingent output. Third, the etimated pdf for c i rectangular. Thi parameter ha only been contrained to be non-negative, o it i omewhat urpriing that the etimated denity function ha been upper-truncated at Thi upper truncation i quite poibly a conequence of contraining the rik-neutral probabilitie to the unit interval. Finally, the etimated parameter can be ued to recover etimate of the latent variable in the model, including unrealized tate-contingent output, input allocation to different tate of Nature, and the rik-neutral probabilitie aigned to different tate of Nature by individual firm. For example, Table 3 preent etimate of π for Firm to 0 in Year, 3, 6 and 8. The etimate preented in thi table reveal that all rice farmer plauibly tend to aign imilar (rik-neutral) probabilitie to tate of Nature in any given year (e.g., in year, we etimate that the firt 0 farmer all aeed π in the range 0.66 to 0.8). Furthermore, farmer may attach very different probabilitie to future tate of Nature from one year to the next (e.g., in year 8, we etimate that 8

10 Firm to 0 aeed π in the range 0.09 to 0.9). Importantly, rik-neutral probabilitie are utility-deflated probabilitie, o variation in thee probabilitie reflect variation in the probabilitie attached to different tate of Nature a well a variation in attitude toward rik. 7. CONCLUSION Empirical etimation of flexible tate-contingent production technologie i complicated by the fact that data on tate-contingent output and allocation of input to different tate of Nature are often unoberved. Thi paper how how to overcome the problem of lack of data in the two-tate cae. In theory, the econometric model developed in the paper can be etimated uing either ampling theory or Bayeian methodology. In an application to Philippine rice data, the ampling theory approach broke down due to an inability to dynamically control a nonlinear leat quare optimiation algorithm. Etimating the model in a Bayeian framework proved much more traightforward and yielded plauible etimate of economic quantitie of interet. REFERENCES Chamber, R.G. and Quiggin, J. (000) Uncertainty, Production, Choice and Agency: The State-Contingent Approach, Cambridge, UK: Cambridge Univerity Pre. Chiang, A.C. (984) Fundamental Method of Mathematical Economic, 3rd edn., Blacklick, OH: McGraw-Hill. Farrell, M.J. (957) 'The Meaurement of Productive Efficiency', Journal of the Royal Statitical Society, Serie A (General), 0 (3): Koop, G. (003) Bayeian Econometric, Chicheter: John Wiley and Son. O'Donnell, C.J., Chamber, R.G. and Quiggin, J. (006) 'Efficiency Analyi in the Preence of Uncertainty', Rik and Uncertainty Program Working Paper, Univerity of Queenland. O'Donnell, C.J. and Griffith, W.E. (006) 'Etimating State-contingent Production Frontier', American Journal of Agricultural Economic, 88 ():

11 Table. DESCRIPTIVE STATISTICS Mean SD Min Max Q X.99e e e+004 W E Table. ESTIMATED PARAMETERS 5th 95th Coef Mean St.Dev Pctile Pctile c a a b h.3e-009.0e-00.6e e-009 0

12 Table 3. ESTIMATED (RISK-NEUTRAL) PROBABILITIES ASSIGNED TO STATE 5th 95th Ob Year Firm P(=) St.Dev Pctile Pctile : : : : : : : : : : : : : : : : : : : : :

13 q B q A q B q A q C B A biector fair odd line 0 A x A x 0 45 q B q A q 0.5x 0.5x (a) (b) Figure. A State-Allocable State-Contingent Technology

14 Figure. Convergence Plot 3

15 Figure 3. Etimated Poterior Pdf 4