Arne Henningsen

Your ePortfolio

Blog Subscribe to this blog

Derivation of shadow prices in farm household models

 

I got an e-mail with a few questions regarding my paper "Modeling Farm Households' Price Responses in the Presence of Transaction Costs and Heterogeneity in Labor Markets" (Henning & Henningsen, 2007, American Journal of Agricultural Economics 89(3), p. 665-681):

  1. Specifically, I would like to understand better how you derived equation 14. As you say in the text, this is obtained by deriving the total differential on equation 3. Unfortunately, I did not find the arguments of the functions included in eq 3 (probably I missed their specification).
  2. In addition, I noticed that – as one could expect – eq 103 also includes the indirect effect of the change on C_L induced by the change in Y (following a change in P_j); this is not the case in the main text.
  3. Also, still in equation 14: for the coefficients (dXs_L/dP*_L) and (dXh_L/dP*_L), could you please refer me to the estimations tables where these values appear? I am not sure where you took these estimated values.

As my answers might be relevant to other readers of my paper, I post them here:

  1. The partial derivative d P*_L / d P_j in equation (14) is derived by applying the implicit function theorem to the time constraint (3), i.e. the numerator on the right-hand side of equation (14) is the partial derivative of the time constraint (3) with respect to P_j and the denominator on the right-hand side of equation (14) is the partial derivative of the time constraint (3) with respect to P*_L. Unfortunately, I am not sure which "arguments" and which "functions" you mean. Please clarify this.
  2. The first fraction on the right-hand side of equation (103) in the supplementary appendix is identical to both sides of equation (14) in the main paper. In equation (103) in the supplementary appendix, I have separated the total effect of a price change on the leisure time (d C_L / d P_j) into the usual Marshallian demand effect (d C_L / d P_j | Y = const) and the effect that the price has through a change of income from agricultural production ( (d C_L / d Y ) * ( d Y / d P_j ) ).
  3. The effects of the shadow price of labor on the off-farm labor demand (d X^s_L / d P*_L) and on the hired on-farm labor (d X^h_L / d P*_L) were derived from equations (18) and (19), respectively: d X^s_L / d P*_L = 1 / ( d P*_L / d X^s_L ) = 1 / beta^s_1 and d X^h_L / d P*_L = 1 / ( d P*_L / d X^h_L ) = 1 / beta^h_1, where beta^s_1 and beta^h_1 are obtained from the estimations of equations (18) and (19).

Sorry that my paper and the supplementary appendix are not sufficiently clear!

Were the price peaks of agricultural commodities caused by financial speculation?

A recent comprehensive literature review concludes that financial speculation neither increased the price level nor the price volatility of agricultural commodities:

"This literature survey comprises 35 empirical studies, published between 2010 and 2012, which analyze the influence of financial speculation on the markets for agricultural commodities: According to this current state of research, there is little evidence for the point of view that the recent increase in financial speculation has caused (a) the price level or (b) the price volatility in agricultural markets to rise. Rather, fundamental factors are made responsible for this." (http://wcms.uzi.uni-halle.de/download.php?down=26926&elem=2624087)

I think that this finding sounds reasonable. It is often claimed that financial speculators increase the market price by having mostly or only long positions, i.e. they promise to buy the physical commodity when their futures contracts expire. However, the financial speculators do not want to receive the physical commodity, because they have no experience in storing, transporting, and trading perishable and bulky agricultural commodities. Therefore, they try hard to offset (sell) all their futures contracts before the contracts expire. This massive selling of the contracts should rather reduce the price of the futures contracts when they approach maturity. Hence, even if financial speculations increase the price of non-mature futures contracts (which to my knowledge also has not yet been proven), they should not increase the maturity price of the futures contract, which is (due to possible arbitrage) closely related to the market price of the physical commodity. Hence, financial speculation should not increase the market price of the physical commodity. Furthermore, if financial speculators would increase the price volatility of agricultural commodities by buying futures contracts at high prices and selling them at low prices, they would make huge losses and finally would voluntarily abstain from speculation with commodity futures -- or otherwise, they would go bankrupt.

Although I am convinced that financial speculation does not significantly affect the prices of agricultural commodities, I do not disagree with others who claim that speculation somewhat increased the market prices of some agricultural commodities during the price peaks (e.g. Per Pinstrup-Andersen: Seminar: Food Policy in Disarray, IFPRI, Nov 21, 2011): I am sure that speculation with physical commodities rather than speculation with futures contracts increased the market prices of the physical commodities during the price peaks, because many farmers, storage firms and traders did not sell their physical commodity during the price peaks, because they expected the prices to increase even further, i.e. they speculated with the physical commodities that they owned. The very limited supply of the physical commodities met a very inelastic demand, which resulted not only in very low quantities that were traded on the market but also in the large price peaks that we observed.

Therefore, I agree with the conclusions of the above-mentioned study:

"[...] most papers are not in favour but against (c) erecting market barriers by regulation. Against this background, the public alarm, claiming that financial speculation has detrimental effects and should be forbidden, seems to be a false alarm: People who are interested in fighting global hunger should take care of fundamental factors and take appropriate measures in order to keep supply in step with the demand, which is likely to rise in the near future."  (http://wcms.uzi.uni-halle.de/download.php?down=26926&elem=2624087)

Warning: this short note is not based on a thorough scientific analysis but presents some thoughts that I had about this topic.

Applied Production Analysis - A Dual Approach: Errata

The book "Applied Production Analysis - A Dual Approach" by Robert G. Chambers (1988, Cambridge University Press) is my favourite textbook in applied production economics. Although it is very well written, it has a few typos that could confuse the reader. The following document lists all typos in this book that I know of: https://absalon.itslearning.com/data/ku/103018/teaching/chambers_errata.pdf. If you have found further typos, please let me know so that I can add them to this document.

Decomposition of Cost Efficiency with R

In the following, I estimate a Cobb-Douglas production function and decompose cost efficiency into (cost) technical efficiency and (cost) allocative efficiency as done by Coelli at al (2005, p.273, Table 10.2) using the R package frontier:

# loading the "frontier" package
library( "frontier" )
# loading the data set
data( "riceProdPhil" )
 
# calculating total costs and cost shares (ignoring "other inputs")
riceProdPhil$cost <- with( riceProdPhil, AREA * AREAP + LABOR * LABORP + NPK * NPKP)
riceProdPhil$sArea  <- with( riceProdPhil, AREA * AREAP / cost )
riceProdPhil$sLabor <- with( riceProdPhil, LABOR * LABORP / cost )
riceProdPhil$sNpk   <- with( riceProdPhil, NPK * NPKP / cost )
 
# mean-scaling output quantity and input quantities
riceProdPhil$prod <- with( riceProdPhil, PROD / mean(PROD) ) 
riceProdPhil$area <- with( riceProdPhil, AREA / mean(AREA) ) 
riceProdPhil$labor <- with( riceProdPhil, LABOR / mean(LABOR) ) 
riceProdPhil$npk <- with( riceProdPhil, NPK / mean(NPK) ) 
 
# estimating production frontier (ignoring "other inputs")
m <- sfa( log(prod) ~ YEARDUM + log(area) + log(labor) + log(npk), 
  data = riceProdPhil )
summary(m)
 
# extracting coefficients
b0 <- coef(m)["(Intercept)"]
theta <- coef(m)["YEARDUM"]
b1 <- coef(m)["log(area)"]
b2 <- coef(m)["log(labor)"]
b3 <- coef(m)["log(npk)"]
sig2 <- coef(m)["sigmaSq"]
gamma <- coef(m)["gamma"]
print( lambda <- unname( sqrt( gamma) / sqrt( 1 - gamma ) ) )
print( sigma <- unname( sqrt( sig2 ) ) )
 
# calculating technical efficiencies manually
sig2v <- sig2 * ( 1 - gamma )
sig2u <- sig2 * gamma
mustar <- - residuals(m, asInData = TRUE ) * gamma
sigstar <- sqrt( sig2u * sig2v / sig2 )
musig <- mustar/sigstar
uhati <- mustar + sigstar * dnorm(musig) / pnorm(musig)
tei <- ( pnorm(musig-sigstar) / pnorm(musig) ) * exp( sigstar^2 / 2 - mustar )
 
# calculating technical efficiencies with the frontier package
tei2 <- efficiencies( m, asInData = TRUE )
all.equal( tei, tei2 )
 
# calculating cost efficiencies
h2 <- log( riceProdPhil$sArea / riceProdPhil$sLabor ) - log(b1/b2)
h3 <- log( riceProdPhil$sArea / riceProdPhil$sNpk ) - log(b1/b3)
r <- b1 + b2 + b3
ai <- (1/r) * (b2*h2 + b3*h3) + log(b1 + b2*exp(-h2) + b3*exp(-h3))
# cost efficiency due to technical inefficiency
ctei <- exp( -uhati / r )
# cost efficiency due to allocative inefficiency
caei = exp( log(r) - ai )
# cost efficiency
cei <- ctei * caei  
 
# collecting all efficiency measures in a data frame
eff <- data.frame( tei, ctei, caei, cei )
# showing efficiencies of first and last observations
eff[ c(1:4, 339:344), ]
# showing some summary statistics of efficiencies
t(sapply(eff, function(x)
  c( N=length(x), MEAN=mean(x), ST.DEV=sd(x), VARIANCE=var(x), 
     MINIMUM = min(x), MAXIMUM = max(x), COEF.OF.VAR=sd(x)/mean(x))))
 

References

Coelli, TJ; D.S. Prasada Rao; C.J. O'Donnel and G.E. Battese (2005): An Introduction to Efficiency and Productivity Analysis, 2nd edition, Springer.

 

Non-separable Household Model

I got following e-mail from an Italian PhD student:

Currently I am working on my thesis, whose principal aim is to provide an empirical framework for the design of agricultural development strategies aiming to reduce rural poverty in Tanzania.

I would like to use a non-separable farm household model to assess the impact, in a partial equilibrium framework analysis, of specific agricultural policies on farm households' income in presence of imperfect markets. With regards to the estimation of the non-separable model, it not very clear to me how the introduction household labour in the production system and of leisure in the demand system (setting both prices at the shadow wage rate previously estimated), as you carried out in Henning and Henningsen (2007) can "overcome" the non-separability problem.

The theoretical framework is not clear to me. Indeed, an increase of shadow wage leads to a decrease in household labour (which enters as an input in the prodction system, and, thus, having a negative own-price elasticity), leading in turn to an increase in leisure on the demand-side. But the own-pice (shadow wage) elasticity of leisure is obviously negative.

Thus, I would like to know how household labour and leisure in the production and demand systems, repectively, can "overcome" the non-separability problem.

This is my answer:

Unfortunately, I am not sure if I fully understand your problem and question but I will try to answer as good as I can. Please let me know if any question remains unanwered.

For simplicity and given your email, we assume that all markets except for the labour market are perfectly functioning. For now, we assume that labour is homogeneous and the labour market is totally absent. Please note that the endogenous internal shadow price of labour cannot be directly (exogenously) changed but can be indirectly influenced by changes of exogenous market prices. In equilibrium, the internal shadow price of labour is identical to the marginal value product of farm labour and the marginal willingness to pay for leisure. If the market price of a cosumption good increases and leisure is a gross substitute to this good, the marginal willingness to pay for leisure increases. Hence, the household increases leisure and decreases farm labour so that the marginal willingness to pay for leisure (again) decreases and the marginal value product of farm labour increases. This substitution of leisure for farm labour continues until the marginal willingness to pay for leisure (again) equals the marginal value product of farm labour. As a result, leisure time and the internal shadow price of labour increase, while farm labour and agricultural production decrease. 

In contrast,  if the market price of a cosumption good increases and leisure is a gross complement to this good, the marginal willingness to pay for leisure decreases. Hence, the household decreases leisure and increases farm labour so that the marginal willingness to pay for leisure (again) increases and the marginal value product of farm labour decreases. This substitution of farm labour for leisure continues until the marginal willingness to pay for leisure (again) equals the marginal value product of farm labour. As a result, leisure time and the internal shadow price of labour decrease, while farm labour and agricultural production decrease. 

If the market price of an agricultural output increases, the marginal value product of farm labour likely also increases. Hence, the household decreases leisure and increases farm labour so that the marginal willingniss to pay for leisure increases and the marginal value product of farm labour (again) decreases. This substitution of farm labour for leisure continues until the marginal willingness to pay for leisure (again) equals the marginal value product of farm labour. As a result, the internal shadow price increases. However, there is a second effect: the increased market price of an agricultural output increases the household income, which usually results in a higher marginal willingness to pay for leisure. Hence, the household increases leisure and decreases farm labour so that the marginal willingness to pay for leisure decreases and the marginal value product of farm labour increases. This substitution of leisure for farm labour continues until the marginal willingness to pay for leisure (again) equals the marginal value product of farm labour. As the result, the internal shadow price increases. Putting both effects together, the internal shadow price of labour clearly increases but we cannot say whether the leisure time or farm labour increases.

Now we assume that the labour market is neither perfectly functioning nor totally absent but plagued by heterogeneous labour and non-proportional variable transaction costs as in the model suggested by Henning and Henningsen (2007). In equilibrium, the internal shadow price of labour is identical to the marginal willingness to pay for leisure, the marginal value product of farm labour of household members, the marginal revenue from off-farm labour and the marginal costs of hired farm labour. If the market price of a cosumption good increases and leisure is a gross substitute to this good, the marginal willingness to pay for leisure increases (as already described above). In this case, the household increases leisure so that the marginal willingness to pay for leisure (again) decreases, the household decreases farm labour of household members so that the marginal value product of farm labour of household members increases, the household decreases off-farm labour so that marginal revenue from off-farm labour increases, and the household increases hired farm labour so that the marginal costs of hired farm labour increase. This substitution of leisure and hired farm labour for off-farm labour and farm labour of household mambers continues until the marginal willingness to pay for leisure (again) equals the marginal value product of farm labour of household members, the marginal revenue from off-farm labour, and the marginal costs of hired farm llabour. As a result, leisure time, hired farm labour and the internal shadow price of labour increase, while off-farm labour, farm labour of household members, and agricultural production decrease. 

References

Henning, Christian H.C.A. and Arne Henningsen (2007): Modeling Farm Households' Price Responses in the Presence of Transaction Costs and Heterogeneity in Labor Markets. American Journal of Agricultural Economics 89(3), p. 665-681. [Authors' PDF file]

 
Powered by: itslearning