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Measurement of profit efficiency using behaviouraland stochastic frontier approachesFarmal Ali a , Ashok Parikh a & Mir Shah aa University of East Anglia, Norwich, UKVersion of record first published: 28 Jul 2006.

To cite this article: Farmal Ali , Ashok Parikh & Mir Shah (1994): Measurement of profit efficiency using behavioural andstochastic frontier approaches, Applied Economics, 26:2, 181-188

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Applied Economics, 1994, 26, 18 1-1 88

Measurement of profit eficiency using behavioural and stochastic frontier approaches

F A R M A N A L I , A S H O K P A R I K H and M I R K A L A N S H A H

Uniurrsily of Eust Anglia, Norwich, UK

The objectives of this study are to estimate profit inefficiency of farms in the North- West Frontier Province of Pakistan using stochastic frontier and the behavioural profit functions. The derived measure of inefficiency, based on a half-normal distribu- tion of the stochastic error term, is related t o socio-economic variables; and of these, the size of holding, fragmentation of land, subsistence needs and the higher age of farmers contribute positively to inefficiency. The behavioural approach satisfies most of the assumptions of the dual profit function and the likelihood ratio test rejects the market efficiency hypothesis. It is also shown that use of manures, labour and fertilizers is below the optimum. Small farms seem to have higher productive efficiency than large farms and, therefore, small farm ownership needs to be encouraged.

I. I N T R O D U C T I O N

Farm efficiency is one of the most important subjects in developing countries' agriculture. The question of how to measure farm efficiency, in the presence of differing factors leading to inefficiency, has drawn enormous attention. Three distinct approaches focus on cost, profits and produc- tion. Farrell (1957) developed the concepts of technical efficiency, based on input and output relationships. Tech- nical inefficiency arises when actual or observed output from a given input mix is less that the maximum possible, while allocative inefficiency arises when the input mix used is not consistent with cost minimization.

This paper uses a profit function approach and combines the concepts of technical and allocative efficiency in the profit relationship. Any errors in the production decision would have to translate into lower profits or revenue for the producer. The use of a translog profit function permits returns to scale t o be variable and, indeed, we find non- constant returns.

Consider the following Cobb-Dougl;~s production func- tion which allows for technical inefficiency:

log Yj=a+X/jklogXk+Vj-Uj

where Y and X, itre output and inputs respectively, while U refers to technical inefficiency and V the stochastic factor. Cost minimization for a given output, will yield cost as a function of input prices and the value of output. This is defined as:

log C i=K+( l / r ) log Yj+Zyk log Wk-(l/r) (Vj-U,)

r = X /i, and y, =P,/r

Thus, economies of scale ( r > 1) tend to dampen the effect of technical efficiency (Schmidt, 1977). For this simple case, the effect of economies of scale can be removed by rescaling the estimated disturbance, but the correction is less obvious in more involved specifications such as the translog'.

Two distinct frameworks are used in this this study: (a) the stochastic frontier approach where a translog profit

'This means that the Farrell measure of inefficiency may be underestimated (biased) when there are positive scale economies. Farrell's method is based on estimating a convex hull of the observed input coefficients in the input coefficients space when assuming production functions homogeneous ofdegree one, and expanding the space to include output when assuming increasing returns to scale i.e. an efficient surface is obtained for each value ofoutput. A disadvantage ofthis method is that a frontier production function is not estimated in a form yielding explicit representation of the production function.

0003-6846 0 1994 Chapman & f loll 181

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function is specified and direct estimation conducted with- out using share equations, and (b) the behavioural ap- proach, where a combination of technical and allocative efficiency are related to the socio-economic, demographic and farm size variables. One objective is to compare the measures of profit inefficiency obtained from the two ap- proaches. Another is to analyse profit inefficiency and relate i t to social, economic, demographic and land size variables to see the consistency between the two approaches. The behavioural approach uses the concepts of shadow price and of market imperfections that are assumed to be a func- tion of socio-economic, demographic and land size vari- ables. The strict interpretation of technical inefficiency in the sense of Farrell may be problematic here, but it seems counterproductive to let this be a straitjacket. The argument that there is a frontier profit function that would apply to any farmer should have no less validity. Deviation of actual profits from the frontier profit function could then be inter- preted as the aggregate of technical and allocative ineffic- iency. The stochastic profit function approach uses this definition of inefficiency, called profit inefficiency.

The two approaches mentioned above are distinct. In the first approach, an assumption of two error terms is made and one of them is assumed to have a half normal distribu- tion when we estimate the inefficiency index. In the second approach, the farmers are assumed to equate the marginal value products of inputs to their shadow prices rather than to actual input prices as assumed in the first case.

In Section 11, data of the 436 farms in North-West Fron- tier Province (NWFP) for the year 199691 are discussed, a brief introduction to the stochastic translog prof11 frontier given and the results presented. In Section 111, a model based on the behavioural approach is presented. In Section IV, profit inefficiency a t a farm level is related to social. economic and demographic factors. In Section V, con- clusions and policy suggestions are discussed.

11. D A T A A N D F R O N T I E R M O D E L

The Institute of Development Studies in Peshawar (Pakis- tan) conducted a sample survey in NWFP. The sample consists of all districts of Peshawar Division i.e. Peshawar, Charsada and Mardan. This division constitutes the back- bone of the regional economy and has as its major crops, wheat, maize, sugarcane and vegetables.

Survey data was collected for the year 1990-91. Informa- tion regarding the household composition, farm size, inputs, outputs, prices, cost, expenditure on food and nonfood items and subsistence needs was obtained including a break- down of inputs and output by crop. Crops are aggregated to derive the overall value of output. Inputs include seed. manure, fertilizer, irrigation, human labour, animal labour and tractor use. In each case, both quantities and prices were recorded.

The Stochastic Trunslog Profit F~inction

A translog production frontier approach was popularized by Aigner et a/ . (1977) and Meeusen and van den Broeck (1977). Translog profit frontiers were recently estimated by Ali and Flinn (1989) and Kumbhakar and Bhattacharya (1992). In the profit function Equation I , there are two errors, one to represent the efficiency component and the other the random component.

log n =n+2,. log L + z a, l o g p i + E 1 Y , j logpi logpj

where FI is normalized profit, E~ is a disturbance term, P, are normalized input and output prices and are denoted as 0 to 4, and L is the (fixed) land input. The error term is assumed to behave in a manner consistent with the frontier concept. i.e.

The compound disturbance (E) is the sum of symmetric- ally distributed variables and a one-sided one. V, is norm- ally distributed to reflect random Factors such as the weather, while U,, a one-sided disturbance term, is used to represent the inefficiency component. For U, we assume U = 1 UI and U - N (o, ah) while V. N (o, a t ) and U and Yare assumed to be independent of each other.

Assume that the symmetric error V is distributed as V- N ((I, a t ) , and the non-negative error U as the absolute value of a normal distribution. i.e. half normal. The population aver- age inefficiency is

where F is the standard normal distribution functions. Measurement of farm level inefficiency, e-", requires

prior estimation o f the non-negative error U , i.e. decomposi- tion o f & into its two individual components. Jondrow et a/ . (1982) suggested a technique for this decomposition using the conditional distribution of U given e. Given the normal distribution of V and the half-normal distribution of U, the conditional mean of U given 6 is shown to be:

'Ju where ).=-and 0 2 = a ~ + u : , and /and F a r e the standard 0,.

normal density function (PDF) and the standard normal distribution (CDF) respectively. For 6, i. and a , the esti- mated values are used to evaluate the density and distribu- tion function.

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Measurement of profit efJiciency

Table I. Maximum likelihood estimales of translog profit rronrier

Name of the variables Parameters Coefficients I-ratio

Constant Land Fertilizer Human Labour Animal Labour Manure Output Land x Land Land x Fertilizer Land x Human Land x Animal Land x Manures Land x Output Fertilizer x Fertilizer Ferilizer x Human Fertilizer x Animal Fertilizer x Manure Fertilizer x Output Human x Human Human x Animal Human x Manures Human x Output Animal x Animal Animal x Manure Animal x Output Manure x Manure Manure x Output Output x Output

Log Likelihood = 109.09

*=significant at 5% level. The subscripts L, F, H, A, M and O stand for land, price of fertilizer, human labour, animal labour, manure and output re- spectively.

Maximum-likelihood (ML) estimates ofthe profit frontier (2.1) are given in Table 1. The equation was estimated by LIMDEP, an econometric package developed by Greene (1992). The ratio of the standard errors of U and V (A) is 2.5837. This implies that the one sided error term U domin- ates the symmetric error V. The discrepency between ob- served profit and the frontier profit is due to both technical and allocative inefficiencies. The average inefficiency estim- ated from Equation 3 is 24.4%, which suggests that on an average, 24.4% of the profit is lost due to inefficiency. In Table 2 we show the frequency distribution of the estimates of the inefficiencies for each farmer. The result shows a wide variation in the inefficiency across farms. The maximum and

'The profit function is nondecreasing in output prices and nonincreas prices, and convex in output and input prices.

minimum inefficiency levels were 90.0 and 3.8% respective- ly. These estimates are important in the sense that they provide detailed information for policy makers.

111. B E H A V I O U R A L A P P R O A C H T O T H E M E A S U R E M E N T O F T E C H N I C A L A N D A L L O C A T I V E I N E F F I C I E N C I E S

Farm households maximize profits in competitive markets by equating the value of the marginal product of each factor to its price. Price efficiency is said to occur when this condition is satisfied for each input. If, however, market prices for inputs differ from their shadow prices and the households equate the value of a factor's marginal product to its shadow price, we have price inefficiency in the input markets. Lau and Yotopoulos (1971) approximated the shadow price for input j , Pf, by kjPj where the factor of proportionality kj is input specitic. The term 'kjPj' is referred t o in the literature as an effective price. kj is a measure of inefficiency for input j and is assumed to be non-negative. For example, if kj is equal to one for all j , then households are price efficient in input use. If kj is less than unity the household uses more of input j than a profit maximizing household would employ. According to Lau and Yotopoulos (1971), the divergence of ki from unity may be due to: (i) constant under o r over valuation of the opportun- ity cost of various resources; (ii) satisficing behaviour; (iii) divergence of expected and actual normalized prices; and (iv) divergence of the subjective probability distribution of normalized prices from its objective distribution.

The Model

The approach adopted here is to specify a translog behavi- oural profit functionz defined in terms of shadow prices rather than market prices and normalized using the market price of one of the inputs, namely tractor usage. The behavi- oural profit function (nb), therefore, takes the following form:

logIlb=constant +a . log (k ,P , )+~ pi log (kipi)

ing in input prices, homogeneous oldegree zero in output and input

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Table 2. Frequency disfribufion o f f i r m specific privil ineflriencirs in srochasti~ rransloy pro j t frontiers

Inefficiency Number of index farmers I % )

Inefficiency index Yo

Number of farmers

%

ALL MEAN STD MIN MAX

F . Ali. A . Purikh and M . K . Shah

where kipi is the shadow (normalized) price of the ith input, prices (p). These are given by: k,P, the shadow (normalized) output price. There are five dn, (kP) variable inputs: manure, fertilizer, human labour, animal x.- I - -- . i=1.2 ... 1 1 - 1

dk,P, (6) labour and tractor usc. Their prices are used as variables in the profit function. Wc assume that the shadow3 prices are m, (LP) Q=- input, but not farm, specific. ak, p, 3 (7)

We seek to test the hypothesis that, in determining em- n- 1

ployment levels for the various inputs, farmers act as if they Xn=kopnQ- 1 n h , are faced with a vector of shadow prices rather than a vector i = L

(8)

of market prices. Applying Hotelling's lemma to the profit ~ ~ t ~ ~ l profi t (nu), is given by: function, defined in terms of shadow prices. enables input

0 - 1 demand functions and the associated output supply func- n , = P , Q ( x ) - 1 P i x , - X . . lions to be derived as functions of the vector of shadow i = I

(91

'The assumption of constant k i is relaxed in Stefanou and Saxena (1988). Later in the section we allow the k,'s to be determined by farm size, age, credit and subsistence needs.

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Measurement of profit efficiency

Substituting Equations 6, 7 and 8 into Equation 9, we obtain

Substituting the behavioural translog profit function (Equa- tion 5) and its derivatives with respect to log k ,P , and log k.P, into Equation 10, we obtain (after some manipula- tions) the following translog actual profits function:

+ ~ a O j l o g ( k j P j j

i , j = l , 2 .... n-I (1 1)

Equation 1 1 contains a large number of parameters and hence our chosen alternative is to use profit share and output equations and to estimate thewhole system, i.e. the translog profit equation, the profit share equations for four inputs and the output equation. Since the tractor price is used as the numeraire, the parameters of the tractor equa- tion can be derived from the parametric constraints of

Table 3. Parameters esrimrrrrs uhrainedfium rhr resrrictrd ,,~odcd

( k i = I )

Parameter Coefficient r-ratio Parameter

additivity, across thc profit share equations. These input share equations are:

P - X - M,=>- - -[kI]-' [ I +4[y][{$I n.

The output share equation can be written in ;in analogous manner. (A useful reference on a model with allocative efficiency is Higgins (1986)).

Results

First, we discuss the results of the procedures adopted for estimating two modcls: (a) one with the restriction that there is allocative eficicncy in both the product market and in all the input markcts i.e.. k , = I, k,= I, i = 1 ,.... 4 and (b) a second model in which no restrictions are imposed on the parameters other than the usual cross-equation and additi- vity restrictions.

The LSQ procedure was used to obtain estimates of the parameters of the translog profit function, and of the output and input share equations. In the first model convergence was achieved after 16 iterations with the log of likelihood function equal to 4468.69 (Table 3).

I n the second model, the profit inefficiency parameters (the k's) werr assumed to be linearly related to age, land size,

Coefficient 1-ratio

20 3.895 68.45' /lo" 0.159 47.78* h -0.174 - 12.25' /I* 0.030 2.55' ~ F F -0.048 - 7.62' PAA 0.004 0.70 BHF -0.002 -0.41 BAM 0.002 0.62 BAF 0.007 1.67 BAO 0.024 10.68* BMF -0.008 -2.75' a -0.320 - 29.49* POF 0.097 37.78' /IMM -0.102 - 20.57* OH -0.497 -37.79' / b ~ 0.134 3 154* BHH -0.124 -25.33' !& 2.300 104.15' AH 0.0009 0.24 Boo 0.447 59.23* BMH -0.016 -4.95' Log of likelihood=4468.69 Number of observations=436

*=significant at 5% level. The subscripts F, H, A, M and 0 stand for fertilizer, human labour, animal labour, manure and output prices respectively.

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credit and subsistence needs, and were modelled as follows.

This model was estimated using a two step iterative procedure. Using the parameter estimates from the re- stricted model, the conditional likelihood function was maximized in order to obtain estimates of the market efiici- ency parameters. Using the values thus obtained, as it was, the system of equations was re-estimated, and using the new parameter values we again estimated the market etliciency parameters. This procedure was repeated more than 500 times until the absolute differences between the parameters obtained from two consecutive estimations were less than 0.0001. These results are presented in Table 4. This model also yielded convergence, and the estimates were different from those in the first model.

Homogeneity and symmetry were tested using an LM test in the model where k i # I . The computed x 2 value (0.02056) was below the critical value a t the 5% level with 25 degrees

Table 4. Parameters esrimates of unrestricted model ( k , # I )

of freedom and so these restrictions were not rejected. The absence of heteroskedasticity was tested using White's (1980) procedure. No evidence of heteroskedasticity was found on the basis of the LM test. Convexity of the profit function in prices was also tested for by calculating the characteristic roots of the matrix of second order derivat- ives. We found that all the latent roots of this matrix were positive a t the mean values of prices thereby demonstrating convexity of the profits function at these mean values. Monotonicity of the function was also checked by examin- ing the predicted values of the output and input shares. We found that the fitted values were positive for output and negative for input shares for all observations so that the monotonicity property was satisfied4.

There is strong evidence to suggest that shadow prices were significantly smaller than observed prices for inputs and output. The estimated parameters, 2's related with age, land size, credit and subsistence needs are highly statistically significantly different from zero. A larger set ofvariables was used and insignificant ones were dropped thereafter. The best estimated model is presented in Table 5. In all the

Parameter Coefficient 1-ratio Parameter Coefficient t-ratio

. ~~ . ..... ~ H H -0.089 -27.77. Bo 8 AH -0.0003 -0.17 Boo A,. -0.016 -7.27* . ,.... AF 1.222 17.20* &A

>.GF 0.0018 1.61 &A

>.LF 0.0047 215* AM &F -0.00003 -018 ~ G M

~ N F 0.0002 2.46* ~ L M

2" 1.2407 7.30* ~ C M

~ G H 0.00004 0.01 ~ N M

~ L H -0.0024 -0.47 2.0 ACH 0.00023 0.44 ).GO

ANH -0.00005 0.15 ).LO

LA 0.79089 14.16* )co

& A -0.0016 - 1.80 L ~ o LA 0.0009 0.54 Log of likelihood =4566.61 ~'(25)=0.02956 (test for homogeneity and symmetry)

I - ' -significant at 5% level. The subscripts F, H, A. M and 0 stand for fertilizer, human labour, animal labour. manure and output prices respectively while G. L. C and N stand for age, land size. credit and subsistence needs of the farmers respectively.

'It is now well known that (empirically) most flexible functional forms do not satisfy the properties ofmonotonicity and convexity globally. For details see Lopez (1985).

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Table 5. Relationship ofprofit inflciency with farm characrerisrics

Name of the variables Coefficients f-ratio

Constant 0.1 7927 4.33' Family size (FSZ) (Numbers) -0.00502 -2.96' Age (AGE) (Years) 0.00102 2.45* Education iEDU) (Years of School) -0.00334 -2.11. OK Farm Work (OFF) (Hours permonth) 0.00029 1.1 1 Farm Asset (FAST) (Rs. Per Acre) -0.8437 x - 0.64 Non Farm Assets (NFAST) (Rs, per acre) -0.0127 -0.94 Working Animals (ANI) (Numbers) -0.5665 x -0.66 Credit (CRT) (Rs. per acre) -0.2075 x 10.' - 1.29 Fragmentation (FRG) (No. per acre) 0.01341 1.47 Extension Visits (EXT) (Number of contacts) -0.005896 -2.31. Land Size (HLD) (Acres) 0.001 54 2.00' Subsistence (SUBS) (Rs. ~ e r acre) 0.1721 x lo-' 2.14.

L& i. 296.25 Mean of TIi 0.244 White HET 90.64

(90).05 113.00

equations, &s, the constant term associated with k,.s was highly significant and excepting the animal labour and manures, its magnitudes were near unity. The means of the estimated ki.s were significantly different from unity for output (1.39), manures (1.22), hired labour (1.032) and fertilizers (1.13). If it is greater than unity for inputs, the household uses less of that particular input than profit- maximizing farmers. On the other hand, if it is less than unity, the household uses more ofthat particular input than the profit-maximizing farmer. The latter seems to be the case with animal labour (0.72) input. Both the likelihood ratio and Lagrangian multiplier tests rejected the efficiency hypothesis.

IV. P R O F I T I N E F F I C I E N C Y A N D F A R M C H A R A C T E R I S T I C S

In this section, the measure of profit inefficiency obtained in Section 11 is related to various explanatory variables using data from 436 farms. One of the important features of developing country agriculture is subsistence needs. These force farmers to produce crops which do not take them to the frontier. The inefficiency can arise from socio-economic, demographic or environmental factors and governments in these countries could possibly adopt policies to remove it. We estimated the following equation.

P l ;=ao+al FSZi+a2AGEi+a, EDUi+a, OFF,

The estimated parameters, t-ratios and other statistics are presented in Table 4. The results indicate that greater family size, education of the household head and greater wealth, assets or credit tend to reduce inefficiency, while age of the household head, land fragmentation and holding size exacerbate it. Extension visits expose farmers to better techniques and so contribute to greater efficiency. Iffarmers' subsistence needs are satisfied by their own production of food crops, these tend to contribute positively to inefficiency.

V. S U M M A R Y A N D C O N C L U S I O N

The objectives of this study were to test the hypothesis of technical and allocative efficiency using the farm level sur- vey data from the Peshawar division of the North-West Frontier province of Pakistan. We estimated profit ineffic- iency using a translog profit frontier. The results show that farmers are both technically and allocatively inefficient. The average inefficiency is 24.4% with a wide variation (max- imum of 90.0% and minimum of 3.8%). This shows that a substantial amount of profits is lost due to inefficiency.

The behavioural approach of profit maximization is also in line with our translog stochastic profit frontier approach and suggests that farmers are inefficient in the use of their resources and that there is an obvious unexploited potential for increasing the farm income without making any substan- tial additional investment in the farm enterprise. The study suggests that better rural education, extension services for expansion and propagation of modern techniques of pro- duction, as well as availability of agricultural credit, can

+ a l z EXTi+&; (14) bridge the gap between the efficient and inefficient farms.

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Moreover , larger holding size increases inefficiency accord- ing t o o u r results.

A C K N O W L E D G E M E N T S

T h e a u t h o r s would like t o thank Dr. Robin Cubi t t a n d the referee of t h e journa l for improv ing the overall presentation.

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