Are Indian Farms Too Small. Mechanization, Agency Costs, and Farm Efficiency (Appendix)

Appendix A. Labor shadow prices by regime and stage The Lagrangian incorporating the budget constraint, time constraint, and non-negativity constraints on hired and off-farm work is L  u ( x, lz1 , lz 2 )   (ag ( f , e1*, e2 *)  x  p f af * (A1)   ( wi (loi  alhi *)  pki aqi ki *))   i (l  lzi  loi  al fi *  s f al fi *  sh alhi *) i i   ( hi alhi * oi loi ) i First order conditions with respect to lio * , lif * , lih * , c and lzi are, respectively, (A2)  wi  i  oi  0 (A3)  ag ei eil   a (1  s f )  0 (A4)  (ag ei eil  wi a)   sh a  hi a  0 (A5) uc    0 and (A6) ulzi    0 . The complementary slackness conditions for the inequality constraints are (A7) oi loi  0 (A8) hi alih *  0 Solving (A1), (A2), and (A3) yields (A9) where g ei eil  1 s f 1  hi *( sh  s f ) wi , (A10) hi *  oi / (oi  hi ) We can now define three regimes. In regime (1), loi  0 so applying (A7), oi  0 and so hi *  0 . Thus (A11) g ei eil  (1  s f ) wi In regime (2), lhi  0 applying (A8), (A12) g ei eil  1 s f 1  sh  s f ) hi  0 and so hi *  1 . Thus wi . In regime (3), lhi  loi  0 , and applying (A2)-(A6) (A13) hi *  (1  uc / ulzi wi ) / ( sh  s f ) . Appendix B The reservation rental price The reservation rental price (RRP) v is the rental price per acre of land at which a farmer with a given level of land ownership and labor endowment would be indifferent to a marginal expansion of operational holdings. Note that given the potential returns to scale this need not be a global maximum. To calculate this rate incorporate a land cost va into the budget constraint in (A1): L'  u ( x, lz )   (ag ( f , e1*, e2 *)  x  va  p f af * (B1)   ( wi (loi  alhi *)  pki aqi ki *))   i (l  lzi  loi  al fi *  s f al fi *  sh alhi *) i i   ( hi alh * oi lo ) i Differentiating with respect to a yields (B2) d L'   ( g ( f , e1*, e2 *)  v  p f f *   ( g ei eia *  wlhi *  pki qi ki *)) da i  i (l fi *  s f l fi *  shlhi *) . i    hi lh * i Setting to zero, solving for v and substituting (A2), (A3), (A4) and (A5) yields (B3) v  g ( f , e1*, e2 *)  p f f * 1 sf   ( g ei eia *  wi (lhi * l fi *)  pki qi ki *) * 1  hi ( sh  s f ) i Define per acre profits as the maximum, for given (B4)  *(a)  g ( f , e1*, e2 *)  p f f *  i Then (B5) d *   g ei eia da h *i i i hi * , of 1 s f 1  h (sh  s f ) * i wi (lhi * l fi *)  pki qi ki * . and (B6) v   *(a )  d  *(a ) da hi * as first-order conditions for (B1) also satisfy those associated with maximization of (B4) with respect to machinery, fertilizer, and labor. Appendix C Technical scale effects on per-acre profits and inputs The solution to the cost minimization problem in stage i as (C1) ci (ei* , a)  ci (a)ei* and the conditional factor demands as, for example, (C2) li (ei* , a )  lic (a )ei* Implicit differentiation yields (C3) dlic  l c k c  2 1  wR  k i i ((2  ) ia  iar ) , da (1   ) i wi where 1  (1      2 . Labor used in the production of one unit of effort declines with area in the off-farm and hired-labor regimes as a result of the suitability of high-capacity machinery on larger-scale plots ( ia  0 ). The effect is strengthened in the autarchic regime if the effective price of labor is increasing in area among farmers in the autarchic regime ( wiaR  0 ). This latter result will obtain as long as the income effect associated with the demand for the home-produced good lzi is sufficiently large relative to the substitution effect, that there is decline in family labor per acre as acreage expands within the autarchic regime. We assume subsequently that this condition holds. The effect of an increase in acreage on number of machines used per unit of work is not determined. The fact that labor costs rise with area due to hiring-in results in increased machine use but since the capacity of machines rises with acreage the number of machines may either rise or fall depending on the elasticity of substitution of capital for labor. If labor and machines are sufficiently good substitutes (  close to 1) then the number of machines will rise in the off-farm and hired-labor regimes. (C4) dkic kc  wR  i [(l l c  2    2(1   )) ia  l lic ira ] da 1   i wi This result also holds both in the autarchic regime if the effective wage is increasing in area. The rising labor costs in this regime further encourage substitution towards machinery Overall the cost per unit area is (C5) dci   cia  (c  wiRlic ) ia (2  )  wiaR l c , i da This expression is negative, except possibly in the autarchic regime. In the autarchic regime the effective wage rises with land area leading to higher labor costs that may fully or partially offset any benefits from increased machine capacity. To actually determine how much work is done and the total use of labor and machinery we now return the per-acre profit-maximizing equation (14), which can now be written (C6)  * (a)  max g ( f * , e1* , e2* )  p f f *   ci (a)  ei* i The envelope condition implies (C7) d *   cia ei* da i which is positive except possibly when the autarchic regime prevails in one or more stages. Because profitability per acre may decline with acreage during the autarchic stage due to the rise in the effective wage and because as acreage increases a farm will in general transition from offfarm to autarchy to hired labor, overall profitability per acre may be lower in larger farms (in regime 2) than smaller farms (in regime 1). Because we assume work is complementary across stages, work per unit area will increase in area (C8) de j * da    e j ci i ej * cia . ci where  e j ci is the elasticity of demand for work in stage j with respect to the price in stage i and so forth. Fertilizer per unit area increases in area (C9) df * f* cia    fci da ci i because fertilizer and work are assumed to be complementary. The number of machines k per unit area will be increasing in area, for  sufficiently close to 1 because overall work increases and the number of machines per unit work increases. In particular, dki * dkic c dei *  ki  ei * (C10) da da da Using (C10), (C8) , (C5) and (C4) it follows that whether total expenditures on machinery will rise for   1 as land size increases depends on whether the pricing of machinery is sufficiently elastic to capacity. Regardless, as is evident from (16) with ia  0 , machine capacity (and thus the probability that a machine of a given capacity is used in a given stage), is increasing in area. An analogous expression holds for labor. (C11) dli * c dei * dlic  li  ei * da da da The effect of an expansion in area on the amount of manual labor used per acre is also ambiguous. Equation (C3) shows that in regimes (1) and (2) there is a decrease in labor per unit of effort associated with machinery but this is offset by an increase in total work (C8) and (C5). Thus if the demand for work is price inelastic or labor and machines are sufficiently good substitutes an increase in area will result in less work within these two regimes. In the autarchic regime there are two additional effects: the cost of work may rise with area (C5) thus leading to less overall work and the rising effective cost of labor will lead to less labor per unit of effort (C3). Both effects lead to less labor per unit area as acreage increases. Thus if the demand for work is price inelastic or labor and machines are sufficiently good substitutes an increase in area will result in less work overall. Appendix D Testing for perfect capital markets using measured profits that ignore interest costs The profit function in terms of profits that do not include interest costs (estimated profits) is given by (D1) ˆ *  g (e*, f *)  c(a)e *  p f f * , where the inputs are determined by programming problem (19). In this case we have (D2) d ˆ * de * df *  ( b  1)(c(a )  pf )  c '(a )e * da da da where ρb > 1 and the second term in parentheses is positive. Estimated profits also increase with owned landholdings. Comparing (D2) to (20) indicates that the gradient in estimated profits, as with that of true profits, is steeper than would be the case in the absence of credit market effects (C7). In the case in which there are no technical scale economies associated with machinery or supervision costs so c '( a )  0 , (C7) would be zero but (20) would be positive if  a  0 and (D2) would be positive if  ab  0 so that work and input use per acre expands in area.1 A direct test of credit market constraints can be obtained by examining the returns to owned capital assets using true or estimated profits. The marginal return to capital in terms of true profits is given by (D3) d * dr (a, b*)   b  (1  r0 )  r (a, b*)  r0  b*  0 , do * db * while the marginal return to estimated profits is (D4) dˆ * de * df *  ( b  1)(c(a )  pf ). do * do * do * The observed marginal returns to capital assets in the presence of credit constraints evidently 1 These conditions coincide in the case in which the interest rate is proportional to borrowing per acre. differ depending on how profits are computed. However, it is easily established that when r (a, b*)  r0 , that is when borrowing costs are independent of land ownership and equal to the returns on savings, the marginal return to capital assets is zero for either measure of profits. This is because variation in owned machinery at the margin has no effects on the use of production inputs, (D5) de * df *  0 . do * do * Therefore, the finding that there is a non-zero return, in terms of estimated profits, to owned capital assets would reject the hypothesis of perfect capital markets. Appendix E Farm Size and Profit Dynamics Addressing profit dynamics in a forward-looking model is complicated and thus we illustrate the basic structure using a simplified production function with one variable input, fertilizer, and assume that the production function and the cost of borrowing are quadratic in their respective arguments. In this model, farmers adjust their end-of-season savings based on unanticipated income shocks and subsequently use this savings to finance fertilizer purchases. We assume a stationary problem with state variables representing soil nutrition n* and cash on hand h*. Fertilizer levels are chosen prior to the realization of a shock θt. We define the value function recursively as follows: (E1) v( nt *, ht *)  max Et (t  g ( ft *  nt *)  r ( ft *  ht *)  ht 1 *   v(nt 1*, h *t 1 )) , where β is the discount factor and (E2) ht 1*  ht *   ( t  Ett ) , where λ denotes the extent to which unanticipated shocks are saved. For λ = 1 unanticipated shocks are fully saved as in the permanent income hypothesis and for λ = 0 cash on hand is just a constant. Soil nutrients depend positively on both the previous period’s stock of nutrients and fertilizer use and negatively on the output shock θt, (E3) nt 1*  nt *  f t * t The idea is that more rapidly growing plants, for example, will deplete the soil of nutrients relatively quickly. For example, if θt is rainfall, more nutrients are used if rainfall and soil nutrients are complements. The production and credit functions are (E4) g ( x)  g1 x  g 2 x 2 and (E5) r ( x)  x  r2 x 2 , where x are the respective arguments in (E1) and for notational simplicity we set the fertilizer price to one. All of the parameters in (E4) and (E5) are positive; that is we assume that production is characterized by diminishing returns but the cost of credit r increases at a higher rate with the amount of credit. Estimated profits in this model (again, profits that do not account for borrowing costs are): ˆ1*  1  g ( f1 *  n1*)  f1 * (E6) Farmers optimally choose their level of savings and use of fertilizer. Given the soil dynamics and savings behavior, the effect of a previous period shock on next-period’s profits is thus g  g 1r2   vnn d1 * (1   )( g1  1)r2   2  , d 0 g 2  r2   vnn g 2  (1   )r2   vnn (E7) where νnn is the second derivative from the value function, with νnn < g2 + r2 and g1 < 1 for an interior maximum. The two key parameters in (E7) are α and λ, reflecting the influence of the dynamic nutrient and savings functions. If λ = 0 so that liquidity h does not depend on unanticipated income shocks the lagged profit shock only influences profits in the next period because of nutrient depletion. A positive shock in period 0 in that case leads to greater nutrient depletion and therefore reduces profitability in period 1. Conversely, if there is no nutrient carryover so that α = 0 there is only a liquidity effect: a positive shock in period 0 induces higher savings and thus more cash on hand in the next period so that less credit is needed for fertilizer. The lower cost of borrowing increases fertilizer use and thus increase profitability in the current period. This effect vanishes if r2 = 0, that is, if borrowing costs do not rise as the demand for credit increases.