Lecture11-2011 - Outline Normal-Half Normal Model Maximum...

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Unformatted text preview: Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Stochastic Production Functions II - Maximum Likelihood: Lecture XI Charles B. Moss1 1 University of Florida September 29, 2011 Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values 1 Normal-Half Normal Model 2 Maximum Likelihood Results Comparison 3 Two Applications of the Primal Basic Imputed Value Formulation Estimates Conclusions 4 Euler Equation and Land Values Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Normal-Half Normal Model Assumptions about errors ￿ ￿ 2 ν￿ ∼ N 0, σν i ￿ 2 u∼ N ∗ 0, σu νi and ui independent (1) The normal distribution νi follows the standard normal (mean zero) formulation ￿ ￿ 1 ν2 f (ν ) = √ exp − 2 (2) 2σ ν 2πσν The half-normal distribution is represented by ￿ ￿ 2 u2 g (u ) = √ exp − 2 2σu 2πσu Charles B. Moss (3) Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Assuming independence ￿ ￿ 2 ν2 u2 f (u , ν ) = f (ν ) g (u ) = exp − 2 − 2 2πσν σu 2σ ν 2σu (4) Since ￿ = ν − u , or by definition of the composed error term ￿ ￿ 2 u2 (￿ + u )2 f ( u , ￿) = exp − 2 − (5) 2 2πσu σν 2σu 2σ ν Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Integrating out u , we obtain the marginal distribution function for ￿. From Weinstein ( X − µX ) σX ( Y − µY ) y= σY x= x is distributed normal, while y is distributed half-normal ￿ 2￿ 1 x f (x ) = √ exp − 2 2π ￿ ￿y f σ ￿ ￿ a ￿￿ , y ≥ −a σ 1−F − σ g (y ) = a 0 y < −a Charles B. Moss (6) (7) Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Continued Where σY σ= ￿ uσX F (u ) = f (x ) dx (8) −∞ Note that if a = 0 f ￿y ￿ σ σ ￿ − F (0)] [1 ￿ y f σ = σ [1 − 0.50] ￿y ￿ ￿ ￿ 2f y2 σ =√ 2 = exp − 2 σ 2πσx σ (σσX ) g (y ) = Charles B. Moss (9) Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Substituting in σ= σY σX y2 exp − ￿ ￿2 σY σY 2πσX σX σX σX ￿ ￿ 2 y2 =√ exp − 2 σY 2πσY g (y ) = √ 2 Charles B. Moss (10) Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values By integration Q (t ) = 1 ￿ t dz −∞ ￿ t ￿ ∞ −∞ g (z − x ) f (x ) dx ￿ z +a ￿ z −x σ ￿ ￿ ￿ a ￿￿ dz f f (x ) dx −∞ −∞ σ 1−F − σ ￿ ￿ t +a ￿ ￿ ￿ a ￿￿ 1 t −x ￿ a ￿￿ =￿ F −F − f (x ) dx σ σ −∞ 1−F − σ (11) = Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Again note that if a = 0 ￿ ￿ ￿t ￿ ￿ 1 t −x Q (t ) = F − F (0) f (x ) dx [1 −￿F (0)] ￿ ￿ −∞ ￿σ ￿ t t −x =2 F − 0.50 f (x ) dx σ ￿ ￿ ￿ −∞ ￿ ￿ t 1 t −x =2 2F − 1 f (x ) dx σ −∞ 2 (12) ￿∞ f (￿) = f (u , ￿) du 0 ￿ ￿ ￿￿ ￿ ￿ 2 ￿λ ￿2 (13) =√ 1−Φ exp − 2 σ 2σ 2πσ ￿ ￿ ￿ ￿ λ ￿ 2 =φ Φ− σ σ σ Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Thus ￿ 2 2 σu + σν σu λ= σν σ= (14) 2 2 Note that as λ → 0, either σν → ∞ or σu → 0. The symmetric error dominates the one-sided component. 2 2 Note that as λ → ∞, either σu → ∞ or σν → 0. The symmetric error dominates the one-sided component. Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Results Comparison Maximum Likelihood The parameters of the model can be estimated by maximizing ￿￿ ￿￿ N ￿i λ 1 ￿2 ln (L) ∝ N ln (σ ) + ln Φ − −2 ￿i σ 2σ i =1 i =1 N ￿ Charles B. Moss (15) Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Results Comparison Results Parameter A0 A1 A2 A3 σ λ Charles B. Moss Estimate 4.99564∗∗∗ (0.03574) 0.00903 (0.00706) 0.00504 (0.00500) 0.00452 (0.00424) 0.45639∗∗∗ (0.02126) 5.08765∗∗∗ (0.77545) Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Results Comparison More mess σu ⇒ σu = λσν σν 2 2 ∴ σu = λ2 σν 2 22 2 σ = λ ￿ ν + σν￿ σ 2 σ 2 = σ ν λ2 + 1 σ2 2 ￿ ∴ σν = ￿ 2 λ +1 2 2 ∴ σu = σ 2 − σν λ= (16) In this problem σν = 4.98 σu = 1.04 Charles B. Moss (17) Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Results Comparison Comparison Stochastic Frontier Parameter Estimate A0 4.99564 (0.03574) A1 0.00903 (0.00706) A2 0.00504 (0.00500) A3 0.00452 (0.00424) Charles B. Moss Ordinary Least Squares Parameter Estimate A0 4.58582 (0.05570) A1 0.01265 (0.01171) A2 0.01677 (0.00728) A3 0.01322 (0.00625) Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Basic Imputed Value Formulation Estimates Conclusions Two Appliations of the Primal The issue of farmland valuation for agricultural purposes is a perennial topic of interest for agricultural policymakers and farmers. Between 1960 and 1999, farmland in the United States accounted for 70 percent of the agricultural assets. Efforts to model land values as functions of the returns to farmland; interest rates and other factors have typically found that significant unexplained variation remains, particularly in the short-run. Schmitz (1995) indicates that while the present value formulation holds in the long run, significant correlation in the residuals points to the existence of short-run disequilibria. This study proposes a different approach to farmland valuation based on Ricardian rents (adjusted for other input fixities). This study demonstrates how the presence of multiple quasi-fixed factors such as machinery, labor and management, affects the measurement of residual rents. Functions II - Maximum Likelihood: Lec Charles B. Moss Stochastic Production Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Basic Imputed Value Formulation Estimates Conclusions Basic Imputed Value Formulation The literature on asset valuation is typically developed along two lines: the use of cash rents and imputed returns. Researchers are attempting to determine “what is the value of land in production?” The use of cash rents is based on the assumption that the value of land in a perfectly operating market can be observed as the price reached by a buyer and seller. This approach does not address how the renter determines the value of farmland. The concept of imputed cash returns follows from the Ricardian notion of cash rents as that amount left over after all other factors of production have been paid. Following this basic notion, the appropriate return to farmland is the revenue less all variable costs minus an appropriate return for other factors such as labor, management, and capital. Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Basic Imputed Value Formulation Estimates Conclusions “What is the appropriate price for labor and capital?” max x 1 ,x 2 , x 3 , x 4 pf (x1 , x2 , x3 , x4 ) − w1 x1 − w2 x2 0 0 s . t . x3 = x3 , x4 = x4 (18) From the Lagrangian L = pf (￿ 1 , x2 , x3￿ x4 ) − ￿ 1 x1 − w2 x2 + x , w ￿ 0 0 λ3 x3 − x3 + λ4￿ x4 − x4 ￿ ￿ ￿ ∂f ∂f dL = p − w1 dx1 + p − w2 dx2 + ￿ ∂ x1 ￿ ￿ ∂ x2 ￿ ∂f ∂f p − λ3 dx3 + p − λ4 dx4 ∂ x3 ∂ x4 Charles B. Moss (19) Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Basic Imputed Value Formulation Estimates Conclusions Therefore ￿ ￿ ￿ ￿ dL ∂f dx1 ∂f ∂ x2 =p − w1 +p − w2 dx3 ∂ x1 ∂￿ 2 x dx3 ￿ dx3 ∂f ∂f dx4 +p − λ3 + p − λ4 =0 ∂ x3 ∂ x4 dx3 (20) Imposing profit-maximizing first-order conditions on the variable inputs ￿ ￿ ∂f ∂f dx4 λ3 = p +p − λ4 (21) ∂ x3 ∂ x4 dx3 Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Basic Imputed Value Formulation Estimates Conclusions Under this representation, the shadow value of x3 is correctly imputed if either the return to x4 is set equal to its true value, or the production function is separable in the inputs. ￿ ￿ ∂f ∂f dx4 λ3 = p +p − λ4 + w4 − w4 ∂ x3 ￿ ∂ x4 ￿ dx3 ∂f ∂f dx4 dx4 λ3 = p +p − w4 + ( w4 − λ4 ) ∂ x3 ∂ x4 dx3 dx3 ∂f p − w4 = 0 ∂ x4 (22) Comparing this expression with the assumption that x4 is a variable input yields λ3 = p ∂f dx4 + ( λ4 − w4 ) ∂ x3 dx3 Charles B. Moss (23) Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Basic Imputed Value Formulation Estimates Conclusions Using these results, we see that the marginal value of input 3 is overstated if λ4 > w4 and the two inputs are compliments. Alternatively, the shadow value of input 3 is understated if λ4 < w4 and the two inputs are compliments. Following the standard methodology, we choose a flexible function form based on some second-order expansion of the profit function: 1 π (p , w , z ) = α0 + αg (p ) + g (p )￿ Ag (p ) + β g (w ) 2 1 g (w )￿ Bg (w ) + g (p )￿ Γg (w ) + g (z ) φ+ 2 (24) g ( z ) ￿ Φg ( z ) + g ( p ) ￿ Ψ 1 g ( z ) + g ( w ) ￿ Ψ 2 g ( p ) Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Basic Imputed Value Formulation Estimates Conclusions Empirically λ i = φ i + Φ ·i z + Ψ 1 , ·i p + Ψ 2 ,·i w p ￿ y − w ￿ x = κ0 + κ1 Land + κ2 Labor+ κ3 Intermediate Charles B. Moss (25) Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Basic Imputed Value Formulation Estimates Conclusions Estimates Parameter Estimates Year 1999 Intercept -21026.00 1998 -22997.00 1997 -20277.00 1996 -25576.01 Labor 10.82∗∗ 1.98 10.81∗∗ 1.65 12.57∗∗ 4.61 16.08∗∗∗ 5.73∗ Charles B. Moss Land 89.81∗∗∗ 89.59∗∗∗ 103.24∗∗∗ 101.67∗∗∗ 127.52∗∗∗ 129.83∗∗∗ 154.08∗∗∗ 149.18∗∗∗ Int. Capital 0.9797∗∗∗ 0.9812∗∗∗ 0.4195∗∗∗ 0.4191∗∗∗ 0.7000∗∗∗ 0.7036∗∗∗ 0.6061∗∗∗ 0.6233∗∗∗ Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Basic Imputed Value Formulation Estimates Conclusions Conclusions The empirical results of this study have several implications for the farmland market in the United States. The results directly imply that the shadow value of farmland differs from its observed cash rental value. Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Euler Equation and Land Values The Euler Theorem was initially developed as a part of the debate regarding the distribution of returns across factors of production. Clark (1923) and Wicksteed (1933) used the Euler Theorem result to infer that that distribution of factor returns generated by the market was optimal. Eulers Theorem is a relationship between the partial derivatives of a function and its homogeneity N ￿ ∂ f (tx ) xi ∂ xi i =1 N ￿ ∂ f (tx ) ￿: tf (x ) = xi ∂ xi t k −1 f ( x ) = (26) i =1 Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Taking the limit of Equation 26 as t → 1 and multiplying each side by the output price yields pf (x ) = N ￿ ∂ f (x ) p xi ∂ xi (27) i =1 Finally using the profit maximization condition that the value of the marginal product pf (x ) = N ￿ r i xi (28) i =1 If we let inputs 1 through N − 1 be variable inputs and input N represent the land input, the appropriate factor payment for farmland can be derived by subtracting the payments to other factors from gross returns ∂ f (x ) Charles B. Moss N −1 ￿ Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Next, we assume that two inputs are quasi-fixed (land and labor ). r N xN = p N −2 ￿ ∂ f (x ) ∂ f (x ) xN = py − r i xi − p xN − 1 xN ∂ xN − 1 N −2 i =1 ￿ xi ∂ f (x ) y ∂ f ( x ) xN − 1 p =p − ri −p ∂ xN xN xN ∂ xN − 1 xN = py − ˜ N −2 ￿ i =1 (30) i =1 ri xi∗ − p ˜ Charles B. Moss ∂ f (x ) xN − 1 ˜ ∂ xN − 1 Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Separating the last term p ￿ ￿ N −2 ￿ ∂ f (x ) ∂ f (x ) = py − ˜ r i x i − r N − 1 xN − 1 − p ˜ ˜ − r N − 1 xN − 1 ˜ ∂ xN ∂ xN − 1 i =1 ￿ ￿ N −1 ￿ ∂ f (x ) py − ˜ r i xi − p ˜ − r N − 1 xN − 1 ˜ ∂ xN − 1 i =1 (31) An alternative to formulating the imputed value approach to farmland would be to derive the imputed value of labor r N − 1 xN − 1 = p N −2 ∂ f (x ) ￿ ∂ f (x ) − r i xi − p xN ∂ xN − 1 ∂ xN Charles B. Moss (32) i =1 Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values p N −2 ￿ ∂ f (x ) y ∂ f ( x ) xN =p − r i xi − p ∂ xN − 1 xN − 1 ∂ xN xN − 1 N −1 ￿ i =1 ∂ f (x ) xN ˇ ∂ xN i =1 ￿ ￿ N −2 ￿ ∂ f (x ) ∂ f (x ) p = py − ˇ r i xi − r N xN − p ˇ − r N xN ˇ ∂ xN − 1 ∂ xN = py − ˇ r i xi − p ˇ (33) i =1 Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values 50 45 40 Residual Return to Assets 35 Imputed Return to Labor and Management $Billion 30 25 20 15 10 5 0 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 Year Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec ...
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