Crop minimum support price versus cost subsidy: Farmer and consumer welfare

INTRODUCTION

Agriculture is the main source of income for many in developing countries. In India, 70% of rural households depend primarily on agriculture for their livelihood, with 82% of farmers being small and marginal (FAO, 2020). According to the Ministry of Agriculture and Farmers Welfare, Government of India (2019), the average farm size in India is about 1 ha (=2.47 acres) and marginal and small landholders constitute 86.21% of the total landholders in India (Krishnan, 2018). A similar situation exists in developing countries such as Africa, Indonesia, Pakistan, Turkey, Thailand, and others. Smallholders (or small farmers) constitute a major portion of the farmers in many developing countries. Despite the small‐sized landholdings of these farmers, 500 million smallholder farmers in the world produce up to 80% of the food consumed in Africa and Asia (African Smallholder Farmers Group (ASFG), 2013).

To sustain small farmers' operations, governments of developing countries offer various programs to support and protect their income. A common farmer‐support program is input cost subsidies, wherein a government offers financial assistance to farmers by bearing a fraction of the farmers' cultivation costs. Input subsidy programs (ISPs) are adopted by many emerging economies such as Africa and India. In Africa, smallholder farmers are issued coupons that they can redeem for input packets from both government and private vendors (Jayne et al., 2018). In India, central and individual state government subsidies for seeds, irrigation, power, and fertilizers are offered to farmers (see Hemming et al., 2018, and the references therein).

Besides cost subsidy, an alternative support program is the credit‐based minimum support price (MSP; also known as deficiency payment). Under a credit‐based MSP, a government will not take any possession of the crop; instead, it will provide a monetary credit to farmers cultivating the crop if the prevailing market price of the crop is below the prespecified MSP. For example, in India, credit‐based MSPs are currently offered for eight crops (mostly oilseeds and legumes) for which the Indian government takes no (or very low) possession of crop (Bera, 2017).

Motivated by the credit‐based MSP program launched in India recently, we are interested in developing a model to compare the effectiveness of (i) cost subsidies and (ii) credit‐based MSP in the presence of (risk‐neutral) advisory bodies that guide farmers to make crop choice decision in a risk‐neutral manner. ¹ Our intent is to examine the following research question: Given cost subsidy is common, will MSP outperform cost subsidy in terms of value creation in the presence of market and crop‐yield uncertainties?

To facilitate our analysis, we develop a Stackelberg game‐theoretic model to capture the underlying dynamics between government (that sets the cost subsidy or MSP for a focused crop) and price‐taking small farmers (who decide on their production quantities in a risk‐neutral manner). ² In our model, we assume that farmers have heterogeneous crop production costs, which is a crucial factor that influences farmers' sowing decisions. The farmers are rational and strategic so that they make their crop‐sowing decisions based on the anticipated future (equilibrium) price of the crop after considering all the other farmers' decisions. To capture demand and supply uncertainties, we assume (i) exogenous market uncertainty and (ii) crop production yield uncertainty.

By comparing the equilibrium outcomes associated with cost subsidy and MSP, we find: 1.

Both cost subsidy and MSP induce more production; however, cost subsidy entices more farmers to sow a crop than MSP. The total production quantity of a crop always increases in the cost subsidy or MSP. Additionally, for any earmarked budget, the crop production is higher under cost subsidy than that under MSP.

Our paper is organized as follows: We review relevant literature in Section 2 and provide model preliminaries in Section 3. In Section 4, we examine the base case to form a benchmark and in Sections 5 and 6 we analyze the impacts of cost subsidy and MSP in the presence of price uncertainty, respectively. We also determine the optimal MSP and cost subsidy that maximize the net value generated. Then, in Section 7, we compare the performance of cost subsidy against MSP and discuss how a policymaker should choose between the two policies for any specific budget availability. Lastly, in Section 8, we examine how a stipulated budget should be apportioned between cost‐subsidy and MSP schemes in order to maximize net surplus along with net value derived from a hybrid policy. In Section 9, we conclude the paper and provide a few prospective directions for future research. We provide a few supplementary results in Appendixes A–C and provide the proofs of all our results in the Supporting Information of this paper.

LITERATURE REVIEW

We first review some papers related to farmer cost subsidies that belong to ISPs (Jayne et al., 2018). Cost subsidies can occur in different forms; for instance, the governments in Mali, Ghana, Nigeria, and other such African countries offer subsidies for seeds and fertilizers to smallholder farmers (Brooks & Wiggins, 2010; Jayne & Rashid, 2013; Jayne et al., 2018). In India, subsidies are offered not only on direct inputs such as seeds and fertilizers (Prasad, 2016) but also on other resources such as power and irrigation by both central and state governments (Bardhan & Mookherjee, 2011; Fan et al., 2008). Xu et al. (2009) find that fertilizer subsidies may reduce or surge the commercial demand for fertilizers in the area depending on whether the private sector is active or inactive. Chibwana et al. (2012) show that input subsidies can entice farmers to allocate more land to those subsidized crops in Malawi. We refer the readers to Hemming et al. (2018) for a comprehensive review of the various types of input subsidies provided to smallholder farmers.

Next, the agricultural economics literature on MSPs is vast; see Tripathi et al. (2013) for a comprehensive review. Fox (1956) develops an economic model to evaluate the impact of MSPs and finds that MSPs can mitigate the fall in gross national product during a recession. Dantwala (1967) finds that, in spite of the increasing MSPs, crop's market prices continue to rise because procurement‐based MSPs form a lower bound to market prices (Ramaswami et al., 2018; Subbarao et al., 2011). Chand (2003) presents a qualitative assessment of the ill effects of wheat‐and‐rice‐centric procurement‐based MSPs on the Indian economy. Besides those examining MSP in the Indian context, Spitze (1978) analyzes the impact of federal policy (The Food and Agriculture Act of 1977) on agriculture in the United States. Dean (1996) describes the Brannan plan (a credit‐based MSP) and Innes (1990) examines the impact of Brannan plan on a nonstrategic monopolist farmer. This observation motivated us to examine the efficacy of the credit‐based MSP scheme for achieving its intended goals (i.e., farmer welfare and crop availability) when farmers are small and strategic and compare the performance of MSP against the more common input subsidy scheme.

There has been a recent interest in operations management that examines social responsibility and public policy issues arising in agriculture. Hu et al. (2017) show that a tiny fraction of strategic farmers can stabilize the steady‐state crop prices without considering MSP. Alizamir et al. (2018) focus on the impact of two schemes (price loss coverage and agriculture risk coverage programs different from MSP) on (i) farmers' welfare, (ii) federal expenditure, and (iii) consumer's welfare. Recently, Guda et al. (2021) and Ramaswami et al. (2018) examine the role of procurement‐based MSPs in emerging economies by considering homogeneous production costs while Chintapalli and Tang (2020) examine the impact of credit‐based MSPs in the presence of market and yield uncertainties and when farmers are risk‐averse and possess heterogeneous production costs. More recently, Akkaya et al. (2021) examine the role of governmental support policies in farmers' adoption of innovative production methods in agriculture.

Unlike Guda et al. (2021) and Ramaswami et al. (2018) who focus on procurement‐based MSP scheme, we focus on credit‐based MSP scheme in the presence of crop's market‐price uncertainty when the crop cultivation costs are heterogeneous among farmers. ³ Next, although our paper is close to Chintapalli and Tang (2020), we focus on and study different issues in this paper as follows.

First, Chintapalli and Tang (2020) analyze only the MSP scheme as a standalone program, while we study both MSP and cost subsidy as well as the hybrid policy that combines both cost subsidy and MSP. Also, we compare the performance of MSP against cost‐subsidy scheme. Second, Chintapalli and Tang (2020) examine the role and impact of MSP when farmers are “highly” risk‐averse. However, they do not address the case when farmers are risk‐neutral, which is important especially with the increasing presence of farmer cooperatives, NGOs, and other governmental and educational advisory bodies that provide informational support and guidance to farmers in the choice of crop to show in a risk‐neutral manner thereby reducing their risk‐aversion. Therefore, the role of MSPs (that is also implemented by government) in confluence with such farmer‐advisory programs remains to be assessed, in order to obtain a holistic impact of MSPs when farmers take decisions in a risk‐neutral manner. Third, in our paper, apart from evaluating and comparing the impact of credit‐based MSP and input cost subsidies on all the stakeholders—farmers, consumers, and government (or policymaker)—we also analyze the scenario when both cost subsidy and MSP are offered simultaneously, through a hybrid policy.

MODEL PRELIMINARIES

To capture the characteristics of smallholder farmers in developing countries, we assume that farmers are infinitesimally small so that a farmer cannot influence the market price individually and each farmer is a price taker (Liao et al., 2017). For ease of exposition, we scale the total “input production capacity” in the market (i.e., the total input available across all farmers) to 1 unit and all farmers have equal input production capacity (i.e., the 1 unit input production capacity in the market is equally distributed among all farmers). Furthermore, we assume that each farmer is risk‐neutral and the farmers do not collude among themselves (Chintapalli & Tang, 2020).

Heterogeneous production cost

In our model, we account for heterogeneity in the unit cost of crop cultivation among farmers; that is, the cost incurred by each farmer to cultivate a “unit input capacity.” In this paper, we use the term “unit production cost” of a farmer to refer to the cost incurred by the farmer to cultivate a unit input capacity. We model the cost heterogeneity among farmers using the linear‐city model on the unit interval

[ 0 , 1 ] $[0,1]$

(Chintapalli & Tang, 2020; Liao et al., 2019; Mendelson & Parlaktürk, 2008; Tirole & Jean, 1988). Hence, we make the following assumption: Assumption 1

All farmers are uniformly distributed over

[ 0 , 1 ] $[0,1]$

with respect to their unit production cost

c $c$

.

In view of our modeling Assumption 1, we scale the total population of infinitesimally small farmers to 1 unit and, therefore, the total production capacity that is available across all the farmers together is scaled to 1, because the capacity of each farmer is scaled to 1 (Xiao et al., 2020).

Each farmer has two options: (i) to sow the focused crop (along with the quantity to sow) or (ii) to not sow the focused crop. We scale the profit of the latter option to

0 $\hskip.001pt 0$

(which corresponds to an outside option like the case when a farmer grows a “stable” crop with stable price and stable yield). The profit of a farmer is determined by ex post market price of the focused crop, while the cost of cultivation is incurred ex ante by the farmer.

Impact of agricultural policies

We examine the impact of cost subsidy and MSP on three competing measures: (1) farmer's surplus (

F $\mathcal {F}$

), (2) consumer's surplus (

C $\mathcal {C}$

), and (3) net value (

W $\mathcal {W}$

) of the corresponding program—cost subsidy or MSP. The “net value”

W $\mathcal {W}$

created by the associated program is defined as

W = F + C − S − E $\mathcal {W} = \mathcal {F} + \mathcal {C} - \mathcal {S} - \mathcal {E}$

, where

S $\mathcal {S}$

denotes the shortage cost that is incurred due to unsatisfied demand and

E $ {\mathcal {E}}$

denotes the implementation cost of a program. We let

F 0 $\mathcal F_0$

,

C 0 $\mathcal C_0$

,

S 0 $\mathcal S_0$

, and

W 0 $\mathcal W_0$

denote farmer's surplus, consumer's surplus, shortage cost, and net value, respectively, in the absence of any support program.

Demand model

In this paper, we analyze the impact of cost subsidy and MSP when the crop price is uncertain due to two types of uncertainties: (i) market uncertainty and (ii) crop‐yield uncertainty. To capture these uncertainties and to obtain tractable results, we assume that the crop's market price satisfies

p ( q , ξ , ε ) = a − ξ · q + ε . \begin{align} p(q,\xi ,\epsilon ) = a - \xi \cdot q + \epsilon . \end{align}

1

Here, the random variable

ε $\epsilon$

denotes the market uncertainty, which is an exogenous noise in the crop's price that is independent of the quantity of the crop available (Chintapalli & Tang, 2020; Huang et al., 2013; Mills, 1959; Petruzzi & Dada, 1999), and the random variable

ξ $\xi$

denotes the crop's production yield uncertainty factor at the “market level” that is resultant of the aggregation of individual farmer yields. While

ε $\epsilon$

captures the impact of various macroeconomic factors and changes in consumers' tastes,

ξ $\xi$

captures the variability in production yield due to environmental factors and agricultural practices such as rainfall and irrigation, infestation of pests and diseases, and quality of seeds and other farm inputs.

To ensure technical tractability and to focus our analysis on the impact of cost subsidy and MSP on stakeholders, we assume that all farmers have the same individual yield

ξ $\xi$

, which is also the market‐level yield. Such an assumption of “perfectly correlated” yields among farmers is commonly adopted in the literature to ensure technical tractability. Such an assumption is reasonable when the circumstances of crop cultivation are similar among farmers, as explained in Alizamir et al. (2018) and Kazaz and Webster (2011). This assumption is also justifiable in practice, especially when farmers adopt high‐quality farm inputs and improved farming practices to generate a conducive environment for crop cultivation (NITI Aayog, 2016). ⁴

Therefore, the “total output quantity” of crop available for sale in the market, when a total input capacity of

q $q$

is involved in crop production, is

ξ · q $\xi \cdot q$

. In other words, the crop's market price is given by

a − x + ε $a - x + \epsilon$

, where

x = ξ · q $x = \xi \cdot q$

is the total quantity of the crop available for a realized yield of

ξ $\xi$

.

We assume that the random variables

ε $\epsilon$

and

ξ $\xi$

are independent and we denote their density functions by

f ε ( · ) $f_\epsilon (\cdot )$

and

f ξ ( · ) $f_\xi (\cdot )$

, respectively. Furthermore, without loss of generality, we let

E [ ε ] = 0 $\mathbb {E}[\epsilon ] = 0$

, Var

( ε ) = σ ε 2 $(\epsilon ) = \sigma _\epsilon ^2$

,

E [ ξ ] = μ $\mathbb {E}[\xi ] = \mu$

, and Var

( ξ ) = σ 2 $(\xi ) = \sigma ^2$

. Therefore, the crop's average output quantity is

μ · q $\mu \cdot q$

and its average market price is

a − μ · q $a - \mu \cdot q$

when a total input capacity of

q $q$

is invested. Finally, we make the commonly used nonrestrictive assumption that the market potential

a $a$

is sufficiently high so that the crop's market price is nonnegative (Chintapalli & Tang, 2020; Petruzzi & Dada, 1999).

Lastly, throughout our analysis we adopt the following common notation that

u ( v ) = ∂ u ∂ v $u^{(v)} = \frac{\partial u}{\partial v}$

and

u + = max { u , 0 } $u^+ = \max \lbrace u,0\rbrace$

, and we denote the per unit crop shortage cost by

s $s$

.

BENCHMARK CASE (NO SUPPORT PROGRAM)

In this section, we examine the benchmark case when no support program is available. Under rational expectation, a risk‐neutral farmer with unit production cost

c ∈ [ 0 , 1 ] $c \in [0,1]$

[ 0 , 1 ] $[0,1]$

ϕ $\phi$

2

Because we had scaled the farmer population to 1 and the total market capacity to 1 unit,

ϕ $\phi$

3

Lemma 1

A farmer with cost

c ∈ [ 0 , 1 ] $c \in [0,1]$

will cultivate the crop if and only if

c ⩽ τ 0 $c \leqslant {\tau }_0$

, where

τ 0 = a μ 1 + μ 2 + σ 2 $ {\tau }_0 =\frac{a \mu }{1 + \mu ^2 + \sigma ^2}$

. Hence, the total fraction of farmers sowing the crop is

ϕ = τ 0 $\phi = \tau _0$

, the total input quantity is also

τ 0 $ {\tau }_0$

, and the expected total output quantity is

μ · τ 0 $\mu \cdot {\tau }_0$

.

c ⩽ τ 0 $c \leqslant \tau _0$

Assumption 2

( a + s ) μ ⩽ 1 + ( μ 2 + σ 2 ) $(a+s)\mu \leqslant 1 +(\mu ^2 + \sigma ^2)$

; this is true when

a $a$

and

s $s$

are small compared to the maximum cultivation cost (i.e., 1).

The above assumption is nonrestrictive and holds true for many crops, especially in countries with substantial climatic and environmental diversity. In such countries, farmers in specific geographical regions find it substantially costly to cultivate a focused crop. We will apply Assumption 2 in Proposition 1 to ensure an interior equilibrium, which is more realistic.

Now, we establish a benchmark by computing farmer's surplus

F 0 $\mathcal F_0$

C 0 $\mathcal C_0$

S 0 $\mathcal S_0$

W 0 $\mathcal W_0$

Farmer's surplus. Using the fact that all farmers with cost

c ⩽ τ 0 $c \leqslant \tau _0$

F 0 $\mathcal F_0$

F 0 = E ξ , ε ξ τ 0 · p τ 0 , ξ , ε − ∫ 0 τ 0 c d c = τ 0 2 2 , \begin{align} \mathcal F_0 = \mathbb E_{\xi ,\epsilon }{\left[ \xi \tau _0 \cdot p{\left(\tau _0 ,\xi ,\epsilon \right)} \right]} - \int _0^{\tau _0} c \; dc= \frac{\tau _0^2}{2}, \end{align}

4

Consumer's surplus. We adopt the traditional economic definition of consumer surplus, which is defined as the area beneath the inverse demand curve and the equilibrium market price. By recalling that for any realization of

ε $\epsilon$

a − x + ε $a - x + \epsilon$

x $x$

τ 0 $\tau _0$

C 0 $\mathcal C_0$

5

Shortage cost. From (1), we can interpret

( a + ε ) $(a+\epsilon )$

ξ τ 0 $\xi \tau _0$

a + ε − ξ τ 0 $a+\epsilon - \xi \tau _0$

s $s$

S 0 $\mathcal S_0$

6

Net value. Net value

W 0 $\mathcal W_0$

W 0 = F 0 + C 0 − S 0 = a a μ 2 − 2 s σ 2 + 1 2 1 + μ 2 + σ 2 . \begin{align} \mathcal W_0 = \mathcal F_0 + \mathcal C_0 - \mathcal S_0 = \frac{a {\left(a \mu ^2-2 s {\left(\sigma ^2+1\right)}\right)}}{2 {\left(1 + \mu ^2+\sigma ^2\right)}}. \end{align}

7

WHEN COST SUBSIDY

β $\beta$

IS OFFERED

In the presence of a cost subsidy

β ∈ [ 0 , 1 ] $\beta \in [0,1]$

β $\beta$

c $c$

c ∈ [ 0 , 1 ] $c \in [0,1]$

( 1 − β ) c $(1-\beta ) c$

c $c$

E ξ ϕ · p ( ϕ , ξ , ε ) ϕ ⩾ ( 1 − β ) c , \begin{align} \frac{\mathbb {E}{\left[\xi \phi \cdot {p}(\phi , \xi ,\epsilon ) \right]} }{ \phi } \geqslant (1-\beta ) c, \end{align}

8

ϕ $\phi$

β → 1 $\beta \rightarrow 1$

β $\beta$

η ( β ) $\eta (\beta )$

β $\beta$

Lemma 2

For any cost subsidy

β ∈ [ 0 , 1 ] $\beta \in [0,1]$

, the fraction of farmers who cultivate the crop is

η ( β ) = min 1 , a μ 1 − β + μ 2 + σ 2 ( ⩾ τ 0 ) . \begin{align} \eta (\beta ) = \min \left \lbrace 1, \frac{a \mu }{1 - \beta + \mu ^2 + \sigma ^2} \right \rbrace\quad (\geqslant \tau _0). \end{align}

9

All farmers with cost

c ⩽ η ( β ) $c \leqslant \eta (\beta )$

will grow the crop so that the fraction of farmers who grow the crop in equilibrium is

η ( β ) $\eta (\beta )$

. Moreover, the threshold

η ( β ) $\eta (\beta )$

(which is also the fraction of farmers cultivating the crop) is increasing in

β $\beta$

(i.e.,

d η d β ⩾ 0 $\frac{d {\eta }}{d \beta } \geqslant 0$

).

In order to focus on more pragmatic case of cost subsidies, we restrict

β $\beta$

Assumption 3

Let

β ¯ = ( 1 + μ 2 + σ 2 ) ( 1 − τ 0 ) . \begin{align} \overline{\beta } = (1+\mu ^2+\sigma ^2) (1 - \tau _0). \end{align}

10

The admissible cost subsidies are:

β ⩽ β ¯ $\beta \leqslant \overline{\beta }$

(so that

η ( β ) ⩽ 1 $\eta (\beta ) \leqslant 1$

).

In the following result, we observe the impact of cost subsidy on the variability in crop's production and its unit revenue. Corollary 1

A higher cost subsidy

β $\beta$

Performance metrics

Using the threshold

η ( β ) $\eta (\beta )$

that is given in Lemma 2, we can compute the different performance metrics associated with cost subsidy

β ∈ [ 0 , 1 ] $\beta \in [0,1]$

as follows.

Farmer's surplus. All farmers with cultivation cost

c ⩽ η ( β ) $c \leqslant \eta (\beta )$

will sow the crop and each farmer who cultivates the crop with unit cost

c $c$

is reimbursed an amount of

β · c $\beta \cdot c$

. Therefore, using Lemma 2, the farmer's surplus

F s ( β ) $\mathcal F_{\text s}(\beta )$

is computed as

11

Consumer's surplus. We define the consumer's surplus

C s ( β ) $\mathcal C_{\text{s}}(\beta )$

as the area between the price curve given in (1) and the equilibrium market price that depends on the total crop availability (i.e.,

ξ · η ( β ) $\xi \cdot \eta (\beta )$

); thus,

12

Shortage cost. The total shortage cost

S s ( β ) $\mathcal S_{\text{s}}(\beta )$

incurred by the society is

S s ( β ) = s · E ξ , ε a + ε − ξ η ( β ) = s · ( a − μ η ( β ) ) , \begin{align} \mathcal S_{\text{s}}(\beta ) = s \cdot \mathbb E_{\xi ,\epsilon }{\left[ a + \epsilon - \xi \eta (\beta ) \right]} = s \cdot (a - \mu \eta (\beta )), \end{align}

13

which always decreases as the crop's cost subsidy

β $\beta$

 increases.

Expenditure of cost subsidy program. Because government reimburses a fraction

β $\beta$

of the cultivation cost of those farmers who cultivate the crop (i.e., with

c ⩽ η ( β ) $c \leqslant \eta (\beta )$

), the total expenditure

E s ( β ) $\mathcal E_{\text{s}}(\beta )$

is

E s ( β ) = ∫ 0 η ( β ) β · c d c = β η 2 ( β ) 2 . \begin{align} \mathcal E_{\text{s}}(\beta ) = \int _0^{\eta (\beta )} \beta \cdot c \; dc = \frac{ \beta \eta ^2(\beta )}{2 }. \end{align}

14

The following lemma summarizes the impact of cost subsidy

β $\beta$

on

F s ( β ) $\mathcal F_{\text{s}}(\beta )$

,

C s ( β ) $\mathcal C_{\text{s}}(\beta )$

,

S s ( β ) $\mathcal S_{\text{s}}(\beta )$

, and

E s ( β ) $\mathcal E_{\text{s}}(\beta )$

. Lemma 3

For any admissible input cost subsidy

β ∈ [ 0 , β ¯ ] $\beta \in [0,\overline{\beta }]$

, where

β ¯ $\overline{\beta }$

is given in (10), the farmer's surplus

F s ( β ) $\mathcal F_{\text{s}}(\beta )$

, consumer's surplus

C s ( β ) $\mathcal C_{\text{s}}(\beta )$

, shortage cost

S s ( β ) $\mathcal S_{\text{s}}(\beta )$

, and expenditure

E s ( β ) $\mathcal E_{\text{s}}(\beta )$

satisfy the following: 1.

The farmer surplus

F s ( β ) $\mathcal F_{\text{s}}(\beta )$

is increasing in

β $\beta$

(i.e.,

d F s d β ⩾ 0 $\frac{d \mathcal F_{\text{s}}}{d \beta } \geqslant 0$

) if and only if

β < 1 − ( μ 2 + σ 2 ) $\beta < 1 -(\mu ^2 + \sigma ^2)$

.

2.

The consumer surplus

C s ( β ) $\mathcal C_{\text{s}}(\beta )$

is increasing in

β $\beta$

(i.e.,

d C s d β ⩾ 0 $\frac{d \mathcal C_{\text{s}}}{d \beta } \geqslant 0$

).

3.

The expenditure

E s ( β ) $\mathcal E_{\text{s}}(\beta )$

is increasing in

β $\beta$

(i.e.,

d E s d β ⩾ 0 $\frac{d \mathcal E_{\text{s}}}{d \beta } \geqslant 0$

), while the shortage cost

S s ( β ) $\mathcal S_{\text{s}}(\beta )$

is decreasing in

β $\beta$

(i.e.,

d S s d β ⩽ 0 $\frac{d \mathcal S_{\text{s}}}{d \beta } \leqslant 0$

).

4.

The total surplus

F s ( β ) + C s ( β ) $\mathcal F_{\text{s}}(\beta ) + \mathcal C_{\text{s}}(\beta )$

is increasing in

β $\beta$

(i.e.,

d d β ( F s + C s ) ⩾ 0 $\frac{d }{d \beta } (\mathcal F_{\text{s}} + \mathcal C_{\text{s}}) \geqslant 0$

).

From Lemma 2, we can conclude that offering a high cost subsidy

β $\beta$

will entice more farmers to cultivate the crop (because

d η d β ⩾ 0 $\frac{d {\eta }}{d \beta } \geqslant 0$

). Additionally, Lemma 3 shows that enticing more farmers in the above manner through a high cost subsidy

β > 1 − ( μ 2 + σ 2 ) $\beta > 1 - (\mu ^2 + \sigma ^2)$

will actually hurt farmer's surplus due to (i) supply glut of the crop and (ii) volatility in market prices. Enticing more farmers through a high cost subsidy increases the average production quantity of the crop, which decreases the crop's average market price thereby hurting farmer's surplus. Additionally, a high cost subsidy

β $\beta$

that entices more farmers to cultivate the crop exposes them to price volatility due to yield uncertainty

σ $\sigma$

thereby hurting farmer's surplus (also see Corollary 1). Hence, when

μ $\mu$

and

σ $\sigma$

are high, then cost subsidies can hurt farmer's surplus. Corollary 2

The higher the values of average crop yield

μ $\mu$

or crop‐yield dispersion

σ $\sigma$

, the lower is the viability of input cost subsidy in improving farmer's surplus (i.e., the smaller is the range of

β $\beta$

values that can improve farmer's surplus). Furthermore, input cost subsidy does not increase farmer's surplus when either the average crop yield

μ $\mu$

or the crop‐yield dispersion

σ $\sigma$

is sufficiently high so that

μ 2 + σ 2 > 1 $\mu ^2 + \sigma ^2 > 1$

.

Even though cost subsidy may not improve farmer's surplus as discussed above, it always improves consumer's surplus through lower crop market prices (as shown in statement 2 of Lemma 3) and the increase in consumer's surplus outweighs the decrease in farmer's surplus, if any, so that the total surplus always increases (as shown in statement 4 of Lemma 3).

Net value of cost subsidy. By substituting (11)–(14) in (15), we can compute the net value

W s ( β ) $\mathcal W_{\text{s}}(\beta )$

associated with cost subsidy

β $\beta$

as

15

The following proposition provides the optimal cost subsidy

β ∗ $\beta ^*$

that maximizes the net value generated. Proposition 1

Let

ρ ∗ ≡ ( a + s ) μ 1 + μ 2 + σ 2 = τ 0 + s μ 1 + μ 2 + σ 2 ( ⩽ 1 ) , \begin{align} \rho ^* \equiv \frac{(a+s)\mu }{1 + \mu ^2 + \sigma ^2} = \tau _0 + \frac{s \mu }{1 + \mu ^2 + \sigma ^2} \quad (\leqslant 1), \end{align}

16

where

τ 0 $\tau _0$

is given in Lemma 1. The optimal admissible cost subsidy

β ∗ $\beta ^*$

that maximizes the net value

W s ( β ) $\mathcal W_{\text{s}}(\beta )$

is

β ∗ = s ( 1 + μ 2 + σ 2 ) a + s ∈ [ 0 , β ¯ ] . \begin{align} \beta ^* = \frac{s(1+\mu ^2 + \sigma ^2)}{a+s} \in [0,\overline{\beta }]. \end{align}

17

The corresponding crop production threshold is

η ( β ∗ ) = ρ ∗ $\eta (\beta ^*) = \rho ^*$

and the optimal net value is

W ∗ ≡ W s ( β ∗ ) = ( a 2 + s 2 ) μ 2 − 2 a s ( 1 + σ 2 ) 2 ( 1 + μ 2 + σ 2 ) ( > W 0 ) . \begin{align} \hspace*{-7pt}\mathcal W^* \equiv \mathcal W_s(\beta ^*) = \frac{(a^2+s^2)\mu ^2-2 a s (1+\sigma ^2)}{2(1+\mu ^2 + \sigma ^2)} \quad ({>} \mathcal W_0). \end{align}

18

Although agricultural cost subsidy is more consumer‐centric and always improves consumer's surplus while a high subsidy could hurt farmer's surplus, as shown in Lemma 3, Proposition 1 shows that there exists a unique value of the subsidy (as given in (17)) that maximizes the net value generated. From (17), it is easily evident that a higher value of

s $s$

warrants a higher cost subsidy (i.e.,

d β ∗ d s ⩾ 0 $\frac{d \beta ^*}{d s} \geqslant 0$

) and a higher crop production (i.e.,

d ρ ∗ d s ⩾ 0 $\frac{d \rho ^*}{d s} \geqslant 0$

). Moreover, it is easy to confirm that

ρ ∗ ⩽ 1 $\rho ^* \leqslant 1$

and

β ∗ $\beta ^*$

are admissible due to Assumption 2.

Moreover, higher mean crop yield

μ $\mu$

and higher crop‐yield variability

σ $\sigma$

require a higher cost subsidy (i.e.,

∂ β ∗ ∂ μ > 0 $\frac{\partial \beta ^*}{\partial \mu } > 0$

and

∂ β ∗ ∂ σ > 0 $\frac{\partial \beta ^*}{\partial \sigma } > 0$

) in order to offset the decrease in farmer's surplus due to (i) lower mean crop price (due to high

μ $\mu$

) and (ii) higher erratic crop price (due to high

σ $\sigma$

). Lastly, the optimal net value increases in the crop's average yield

μ $\mu$

and decreases in the crop's yield variance

σ $\sigma$

(i.e.,

∂ W ∗ ∂ μ > 0 $\frac{\partial \mathcal W^*}{\partial \mu } > 0$

and

∂ W ∗ ∂ σ < 0 $\frac{\partial \mathcal W^*}{\partial \sigma } < 0$

), which indicates that between two crops with comparable cost and price structures, it is beneficial to offer cost subsidy to the crop with lower coefficient of variance.

We also analyze the second type of cost subsidy, which we term as additive cost subsidy, where each farmer who cultivates a crop is offered a fixed subsidy

γ ( > 0 ) $\gamma (> 0)$

. Through our analysis, we conclude that both multiplicative cost subsidy (that is discussed in the current section) and additive cost subsidy (that is discussed in Appendix B) result in the same optimal crop production

ρ ∗ $\rho ^*$

and net value

W ∗ $\mathcal W^*$

(that are given in Proposition 1). We refer the reader to Appendix B for further details about additive cost subsidy and its performance.

While a cost subsidy, being an input‐based subsidy, results in government sharing a fraction of farmers' cultivation costs thereby alleviating their financial burden, a crop's MSP is an “output‐based subsidy” that provides all farmers who cultivate the crop a minimum guaranteed crop price thereby protecting them from slumps in market price. Thus, MSP differs fundamentally from cost subsidy in its operationalization. Additionally, the fact that cost subsidy increases variability in market price (see Corollary 1), which acts against the governmental objective of protecting farmer's welfare, urges us to examine the impact of crop MSP in a detailed manner and compare its efficacy against that of crop subsidy: Can MSP offer high farmer's and consumer's surpluses and generate high net value? We examine this issue next.

WHEN CREDIT‐BASED MSP

m $m$

IS OFFERED

We now consider the case when government offers a preannounced credit‐based MSP

m $m$

m $m$

OPEN IN VIEWER

FIGURE 1

Farmer cost and capacity model

Under a credit‐based MSP scheme, when an MSP

m $m$

m $m$

p ( q , ξ , ε ) $p(q,\xi , \epsilon )$

m $m$

[ m − p ( q ( m ) , ξ , ε ) ] $[m - p(q(m),\xi , \epsilon )]$

p ( q ( m ) , ξ , ε ) $p(q(m),\xi , \epsilon )$

m $m$

p ( q ( m ) , ξ , ε ) $p(q(m),\xi , \epsilon )$

p ̂ ( m , ξ , ε ) = max { m , p ( q ( m ) , ξ , ε ) } $\hat{p}(m,\xi , \epsilon ) = \max \lbrace m, p(q(m),\xi , \epsilon )\rbrace$

c ∈ [ 0 , 1 ] $c \in [0,1]$

c $c$

19

ϕ $\phi$

ϕ $\phi$

m $m$

Lemma 4

For any

ϕ ∈ [ 0 , 1 ] $\phi \in [0,1]$

, MSP

m $m$

reduces the variance of the revenue of a farmer who cultivates the crop.

The above result provides an inkling that MSP could potentially mitigate the variability in farmers' earnings, which we will revisit in Corollary 3 after characterizing the equilibrium in Theorem 1. Using the concept of rational expectation and (19), we obtain the fraction of farmers who sow the focused crop in equilibrium as follows: Theorem 1

For any

ϕ ∈ ( 0 , 1 ] $\phi \in (0,1]$

m $m$

20

τ ( m ) $\tau (m)$

21

c ⩽ τ ( m ) $c \leqslant \tau (m)$

c > τ ( m ) $c > \tau (m)$

τ ( m ) $ \tau (m)$

m $m$

d τ d m ⩾ 0 $\frac{d {\tau }}{d m} \geqslant 0$

It is important to note that

τ ( 0 ) = τ 0 $\tau (0) = \tau _0$

ω ( τ 0 , 0 ) = 0 $\omega (\tau _0,0) = 0$

Next, as before, in order to avoid the unrealistic case that all farmers are enticed to grow the crop due to MSP, which is unusual in practice, we restrict

m $m$

Assumption 4

m ⩽ m ¯ $m \leqslant \overline{m}$

, where

m ¯ $\overline{m}$

is the unique value that satisfies

1 + μ 2 + σ 2 − ω ( 1 , m ¯ ) = a μ $1 + \mu ^2 + \sigma ^2-\omega (1,\overline{m}) = a \mu$

(so that

τ ( m ) ⩽ 1 $\tau (m) \leqslant 1$

).

The following result revisits the variability in unit revenue of farmers in equilibrium: Corollary 3

A higher MSP increases the variance in crop's production quantity. However, increasing

m $m$

1.

The rate of increase in the crop production that is caused by

m $m$

is sufficiently low.

2.

m $m$

is sufficiently high.

3.

The yield and market potential variability (i.e.,

σ $\sigma$

and

σ ε $\sigma _\epsilon$

) are sufficiently low.

Although MSP

m $m$

m $m$

m $m$

m $m$

m $m$

m $m$

ξ $\xi$

ε $\epsilon$

m $m$

m $m$

Performance metrics

For any given admissible MSP

m $m$

, we compute the performance metrics—that is, farmer's surplus

F m ( m ) $\mathcal F_{\text{m}}(m)$

, consumer's surplus

C m ( m ) $\mathcal C_{\text{m}}(m)$

, shortage cost

S m ( m ) $\mathcal S_{\text{m}}(m)$

, and expenditure

E m ( m ) $\mathcal E_{\text{m}}(m)$

—to assess the impact of MSP.

Farmer's surplus. For any admissible MSP

m $m$

, Theorem 1 shows that only farmers with cost

c ⩽ τ ( m ) $c \leqslant \tau (m)$

will grow the crop. Therefore, farmer's surplus

F m ( m ) $\mathcal F_{\text{m}}(m)$

can be computed as

F m ( m ) = E ξ , ε ξ τ ( m ) · p ̂ ( m , ξ , ε ) − ∫ 0 τ ( m ) c d c = τ 2 ( m ) 2 , \begin{align} \mathcal F_{\text{m}}(m) = \mathbb E_{\xi ,\epsilon }{\left[ \xi \tau (m) \cdot \hat{p}(m,\xi , \epsilon ) \right]}- \int _0^{\tau (m)} c \; dc = \frac{\tau ^2(m)}{2}, \end{align}

22

where the second equality is obtained using Theorem 1 (see (5) in the Supporting Information, proof of Theorem 1).

Consumer's surplus. We can determine the consumer's surplus

C m ( m ) $\mathcal C_{\text{m}}(m)$

under MSP

m $m$

as follows:

23

Shortage cost. The shortage cost

S m ( m ) $\mathcal S_{\text{m}}(m)$

when MSP

m $m$

is offered for the crop is

S m ( m ) = s · E ξ , ε a − ξ τ ( m ) + ε = s ( a − μ τ ( m ) ) , \begin{align} \mathcal S_{\text{m}}(m) = s \cdot \mathbb E_{\xi ,\epsilon } {\left[ a - \xi \tau (m) + \epsilon \right]} = s (a - \mu \tau (m)), \end{align}

24

which is always decreasing in

m $m$

.

Expenditure of MSP program. For a credit‐based MSP that kicks in only when the realized market price is less than the preannounced MSP

m $m$

, the total expenditure

E m ( m ) $\mathcal E_{\text{m}}(m)$

of MSP

m $m$

incurred by government is

[ m − p ( τ ( m ) , ξ , ε ) ] + $[m - p(\tau (m),\xi ,\epsilon )]^+$

per unit of the crop produced. Therefore, the total expected expenditure when MSP

m $m$

is offered can be computed as

25

where

ω ( ϕ , m ) $\omega (\phi ,m)$

is given in (20).

The following result explains the impact of MSP on the performance metrics

F m ( m ) $\mathcal F_{\text{m}}(m)$

,

C m ( m ) $\mathcal C_{\text{m}}(m)$

,

S m ( m ) $\mathcal S_{\text{\em {m}}}(m)$

, and

E m ( m ) $\mathcal E_{\text{m}}(m)$

: Lemma 5

Farmer's surplus

F m ( m ) $\mathcal F_{\text{\em {m}}}(m)$

, consumer's surplus

C m ( m ) $\mathcal C_{\text{\em {m}}}(m)$

, and expenditure

E m ( m ) $\mathcal E_{\text{\em {m}}}(m)$

are increasing in MSP

m $m$

while the shortage cost

S m ( m ) $\mathcal S_{\text{\em {m}}}(m)$

is decreasing in

m $m$

.

Unlike input cost‐subsidy program that can hurt farmer's surplus when the subsidy is high, that is,

β > 1 − ( μ 2 + σ 2 ) $\beta > 1 - (\mu ^2 + \sigma ^2)$

as shown in Lemma 3, a credit‐based MSP always improves farmer's surplus along with consumer's surplus, as shown in Lemma 5, because it insurers all farmers who cultivate the crop from low crop prices caused by crop's yield uncertainty.

Net value of MSP. The net value of MSP

m $m$

is

26

Using the expressions for

F m ( m ) $\mathcal F_{\text{m}}(m)$

,

C m ( m ) $\mathcal C_{\text{m}}(m)$

,

E m ( m ) $ \mathcal E_{\text{m}}(m)$

, and

S m ( m ) $ \mathcal S_{\text{m}}(m)$

given above along with the base net value

W 0 $\mathcal W_0$

given in (6), we obtain the following result: Proposition 2

The optimal MSP

m ∗ $m^*$

that maximizes the net value

W m ( m ) $\mathcal W_{\text{m}}(m)$

is the unique solution of the following equation ¹¹ :

τ ( m ∗ ) · ω ( τ ( m ∗ ) , m ∗ ) = s μ . \begin{align} \tau (m^*) \cdot \omega (\tau (m^*),m^*) = s \mu . \end{align}

27

The corresponding crop production threshold is

τ ( m ∗ ) = ρ ∗ ( ⩽ 1 ) $\tau (m^*) = \rho ^* (\leqslant 1)$

, which is given in (16), and the optimal net value is

W m ( m ∗ ) = W ∗ , \begin{align} \mathcal W_{\text{\em {m}}}(m^*) = \mathcal W^* , \end{align}

28

where

W ∗ ( > W 0 ) $\mathcal W^* (>\mathcal W_0)$

is defined in (18).

From (28) in Proposition 2 and (18), we can conclude that MSP generates the same optimal net value as a cost subsidy, that is,

W m ( m ∗ ) = W s ( β ∗ ) = W ∗ > W 0 $\mathcal W_{\text m}(m^*) = \mathcal W_{\text s}(\beta ^*) = \mathcal W^* > W_0$

. Hence, the question remains: Is MSP more beneficial than cost subsidy? In order to answer this question, we introduce the following result that compares the farmer's surplus that MSP provides against the surplus provided by cost subsidy. Proposition 3

The optimal MSP

m ∗ $m^*$

offers higher farmer surplus than the optimal cost subsidy

β ∗ $\beta ^*$

while incurring a commensurately higher expenditure than cost subsidy (i.e.,

F m ( m ∗ ) − F s ( β ∗ ) = E m ( m ∗ ) − E s ( β ∗ ) > 0 $\mathcal F_{\text m}(m^*) - \mathcal F_{\text s}(\beta ^*) = \mathcal E_{\text m}(m^*) - \mathcal E_{\text s}(\beta ^*) > 0$

). Hence, administering optimal MSP

m ∗ $m^*$

requires a higher budget than administering optimal cost subsidy

β ∗ $\beta ^*$

.

Proposition 3 proves that, even though MSP and cost subsidy offer the same optimal net value, MSP benefits farmers more than cost subsidies in terms of farmer's surplus. It also shows that MSP is efficient in transferring more value to farmers by commensurately increasing governmental expenditure without any loss of value.

By considering the results reported in Sections 5 and 6, we find that 1.

MSP always improves both farmer's and consumer's surpluses. However, cost subsidy benefits consumers but it can hurt farmers cultivating the crop. This is because, although a cost subsidy reduces the deterministic input costs of the farmers, it exposes farmers to market price uncertainty and low market price.

2.

Both MSP and cost subsidy improve the total surplus (i.e., the sum of farmer's and consumer's surpluses), and offer the same optimal net value (i.e.,

W m ( m ∗ ) = W s ( β ∗ ) = W ∗ $\mathcal W_{\text m}(m^*) = \mathcal W_{\text s}(\beta ^*) = \mathcal W^*$

), as shown in Propositions 1 and 2.

3.

Although cost subsidy always increases the variance in unit revenues of the farmers, Corollary 3 reveals that MSPs can reduce the variability as long as they are properly set to not entice a huge number of farmers to cultivate the crop.

PROGRAM CHOICE UNDER BUDGET CONSTRAINT ON IMPLEMENTATION COST

In this section, we consider the case when policymakers are given a budget

B $B$

P s ${\bf P}_{\text{s}}$

P m ${\bf P}_{\text{m}}$

OPEN IN VIEWER

TABLE 1

Comparing cost subsidy and MSP with respect to net value for an investment

B $B$

The following result compares different performance metrics associated with both MSP and cost‐subsidy programs ¹² : Proposition 4

For any given budget availability

B $B$

¹³ : 1.

Crop production. Cost subsidy induces more farmers to cultivate the crop than MSP (i.e.,

η ( B ) ⩾ τ ( B ) $ \eta (B) \geqslant \tau (B)$

).

2.

Farmer's surplus. MSP generates a higher farmer's surplus than cost subsidy. Furthermore, overinvesting in cost subsidy can hurt farmer's surplus.

3.

Consumer's surplus. Cost subsidy generates a higher consumer's surplus than MSP.

4.

Shortage cost. Cost subsidy incurs a lower shortage cost than MSP.

5.

Total surplus (i.e., sum of farmer's and consumer's surpluses). MSP generates a higher total surplus than cost subsidy. ¹⁴

6.

Net value. Cost subsidy generates a higher net value than MSP (i.e.,

W s ∗ ( B ) ⩾ W m ∗ ( B ) $\mathcal W_{\text s}^*(B) \geqslant \mathcal W_{\text m}^*(B)$

). ¹⁵

By subsidizing the cultivation costs through cost subsidy, government will entice more farmers to cultivate the crop than MSP, at a given budget level

B $B$

. This larger crop supply caused by cost subsidy will generate a higher consumer's surplus than MSP because cost subsidy substantially decreases the crop's market price. The consumer's surplus graphs are shown in Figure 2b. (The increased crop supply generated by cost subsidy will also reduce the shortage cost, as discussed in statement 4 of the proposition.)

OPEN IN VIEWER

FIGURE 2

Comparison between cost‐subsidy and MSP schemes

Despite higher consumer's surplus and lower shortage cost, cost subsidy can hurt farmer's surplus by exposing more farmers to lower and more variable market prices (as shown in Corollary 4 in Appendix C). On the contrary, MSP insulates all farmers who sow the crop from price falls through a floor price thereby improving the farmer's surplus greatly. Hence, statement 2 of the proposition reveals that cost subsidy will generate a lower farmer's surplus than MPS, as shown in Figure 2a. Furthermore, the lower market price that is due to the increased crop production triggered by MSP (as shown in Theorem 1) will, apart from increasing the consumer's surplus (as shown in Figure 2b), also increase the likelihood of farmers claiming the MSP. Therefore, government incurs a higher expenditure by offering MSP for a larger quantity of crop more frequently.

Finally, MSP offers higher farmer's surplus at higher expenditure as discussed above, but it also yields lower consumer's surplus and higher shortage cost. Hence, as statement 6 of the proposition reveals, MSP generates a lower net value than cost subsidy as shown in Figure 2d. (We provide the complete piecewise definition of

W s ∗ ( B ) $\mathcal W_{\text s}^*(B)$

and

W m ∗ ( B ) $\mathcal W_{\text m}^*(B)$

for different values of

B $B$

, in Lemma 7 in Appendix C.)

HYBRID POLICY

( m , β ) $(m,\beta )$

: COMBINING COST SUBSIDY AND MSP

In the previous section, we compared the performance of cost subsidy and MSP as standalone programs. We now develop and analyze a hybrid policy

( m , β ) $(m,\beta )$

that allows the use of both MSP

m $m$

and cost subsidy

β $\beta$

policies simultaneously. ¹⁶ Specifically, we ask the question: How should the available budget

B $B$

be allocated between cost‐subsidy and MSP schemes in order to maximize the net value generated?

In the presence of a hybrid policy

( m , β ) $(m,\beta )$

that comprises cost subsidy

β $\beta$

and MSP

m $m$

, the effective cost of a farmer with cost

c ∈ [ 0 , 1 ] $c \in [0,1]$

is

( 1 − β ) c $ (1-\beta ) c$

and the effective per unit revenue is

max { m , a − ξ ρ + ε } $\max \lbrace m,a-\xi \rho + \epsilon \rbrace$

, where

ρ $\rho$

denotes the fraction of farmers who cultivate the crop. Hence, the farmer will sow the crop if and only if

E ξ , ε ξ ρ · max { m , a − ξ ρ + ε } ρ ⩾ ( 1 − β ) c . \begin{align} \frac{\mathbb E_{\xi ,\epsilon } {\left[ \xi \rho \cdot \max \lbrace m,a-\xi \rho + \epsilon \rbrace \right]}}{\rho } \geqslant (1-\beta ) c. \end{align}

29

In this case, using the fact that

c ∼ U [ 0 , 1 ] $c \sim U[0,1]$

, the equilibrium threshold

ρ $\rho$

is given by the following fixed point equation:

ρ = E ξ , ε ξ · max { m , a − ξ ρ + ε } ( 1 − β ) . \begin{align} \rho = \frac{\mathbb E_{\xi ,\epsilon } {\left[ \xi \cdot \max \lbrace m,a-\xi \rho + \epsilon \rbrace \right]}}{(1-\beta )} . \end{align}

30

Next, using the distributions of

ξ $\xi$

and

ε $\epsilon$

to simplify the above expression, we can implicitly determine

ρ $\rho$

as the unique value that satisfies:

ρ = min 1 , a μ 1 + μ 2 + σ 2 − β − ω ρ , m , \begin{align} \rho = \min \left \lbrace 1 , \frac{a \mu }{1 + \mu ^2 + \sigma ^2 - \beta - \omega {\left(\rho ,m\right)}} \right \rbrace , \end{align}

31

where

ω ( · , · ) $\omega (\cdot , \cdot )$

is given in (20) and

ρ ≡ ρ ( m , β ) $\rho \equiv \rho (m,\beta )$

, whose arguments

( m , β ) $(m,\beta )$

we suppress to improve readability in our analysis. Then, the net value of the hybrid policy

( m , β ) $(m,\beta )$

is ¹⁷ :

W h ( m , β ) = F h ( m , β ) + C h ( m , β ) − S h ( m , β ) − E h ( m , β ) , \begin{eqnarray} \mathcal W_{\text h}(m,\beta ) = \mathcal F_{\text h}(m,\beta ) + \mathcal C_{\text h}(m,\beta ) - \mathcal S_{\text h}(m,\beta ) - \mathcal E_{\text h}(m,\beta ),\nonumber\\ \end{eqnarray}

32

where the farmer's surplus, consumer's surplus, and shortage cost can be computed as

F h ( m , β ) = ( 1 − β ) 2 · ρ 2 $\mathcal F_{\text h}(m,\beta ) = \frac{(1-\beta )}{2}\cdot \rho ^2$

,

C h ( m , β ) = ( μ 2 + σ 2 ) 2 · ρ 2 $\mathcal C_{\text h}(m,\beta ) = \frac{(\mu ^2+\sigma ^2)}{2} \cdot \rho ^2$

, and

S h ( m , β ) = s ( a − μ ρ ) $\mathcal S_{\text h}(m,\beta ) = s (a - \mu \rho )$

, respectively. The expenditures incurred through the cost‐subsidy and MSP programs are

E s ( β , ρ ) = β ρ 2 2 $\mathcal E_{\text s}(\beta , \rho ) = \frac{\beta \rho ^2}{2}$

and

E m ( β , ρ ) = ω ( ρ , m ) ρ 2 ( m , β ) $\mathcal E_{\text m}(\beta , \rho ) = \omega (\rho ,m) \rho ^2(m,\beta )$

, respectively; therefore, the total expenditure incurred by the hybrid policy

( m , β ) $(m,\beta )$

is

E h ( m , β ) = E s ( β , ρ ) + E m ( m , ρ ) = β ρ 2 2 + ω ( ρ , m ) ρ 2 . \begin{align} \mathcal E_{\text h}(m,\beta ) = \mathcal E_{\text s}(\beta , \rho ) + \mathcal E_{\text m}(m,\rho ) = \frac{\beta \rho ^2}{2} + \omega (\rho ,m) \rho ^2. \end{align}

33

By substituting the above values into (32), and using (31), we can simplify the net value

W h ( m , β ) $\mathcal W_{\text h}(m,\beta )$

as

W h ( m , β ) = ( a + s ) μ ρ − 1 + μ 2 + σ 2 2 ρ 2 − a s . \begin{align} \mathcal W_{\text h}(m,\beta ) = (a+s) \mu \rho - {\left(\frac{1 + \mu ^2 + \sigma ^2}{2}\right)} \rho ^2 - a s. \end{align}

34

The following result characterizes an optimal hybrid policy and provides the “menu” of cost subsidy

β $\beta$

and MSP

m $m$

values that comprise it. Proposition 5

Let

( m h ∗ , β h ∗ ) $(m^*_{\text \em {h}},\beta ^*_{\text \em {h}})$

denote an optimal hybrid policy. The frontier of optimal hybrid policies, which we denote by

H ∗ $\mathcal H^*$

, satisfies the following equation:

ρ m h ∗ , β h ∗ = ρ ∗ ( ⩽ 1 ) , for all m h ∗ , β h ∗ ∈ H ∗ , \begin{align} \rho {\left(m^*_{\text \em {h}},\beta ^*_{\text \em {h}}\right)} = \rho ^* \; (\leqslant 1),\quad \text{ for all } {\left(m^*_{\text \em {h}},\beta ^*_{\text \em {h}}\right)}\in \mathcal H^*, \end{align}

35

where

ρ ( m h ∗ , β h ∗ ) $\rho (m^*_{\text \em {h}},\beta ^*_{\text \em {h}})$

is implicitly defined by (31) and

ρ ∗ $\rho ^*$

is given in (16).

The corresponding optimal net value is

W h ( H ∗ ) = W ∗ , \begin{align} \mathcal W_{\text{\em {h}}}(\mathcal H^*) = \mathcal W^*, \end{align}

36

where

W ∗ $\mathcal W^*$

is defined in (18).

Even though the policymaker has two levers

β $\beta$

and

m $m$

to operate with under a hybrid policy, the second statement (36) of the above proposition reveals that the combination of two programs does not create additional net value. In the absence of a budget constraint, a hybrid policy is unwarranted and a standalone cost‐subsidy program will suffice to maximize the net value generated. (Moreover, in the presence of a budget constraint, Proposition 7 in Appendix C proves a stronger result that a hybrid policy generates the same optimal net value as a standalone cost subsidy does, for any budget

B $B$

. Therefore, using Propositions 4 and 7 (stated in Appendix C) we can conclude that, for any budget

B $B$

, standalone cost subsidy alone suffices to maximize the net value that is generated.)

The above conclusion prompts us to raise the following important question: Do hybrid and MSP policies offer any additional benefit over cost subsidy? To explore this question, we first graphically illustrate the frontier of optimal hybrid policies, which we denote by

H ∗ $\mathcal H^*$

, in Figure 3. As shown in the figure, the optimal hybrid policies vary from optimal standalone cost subsidy (i.e., point

P $P$

) to optimal standalone MSP (i.e., point

Q $Q$

). We can easily show that farmer's surplus

F h ( m h ∗ , β h ∗ ) $\mathcal F_{\text h}(m_{\text h}^*,\beta _{\text h}^*)$

and expenditure

E h ( m h ∗ , β h ∗ ) $\mathcal E_{\text h}(m_{\text h}^*,\beta _{\text h}^*)$

increase at the same rate as we move from point

P $P$

to

Q $Q$

along the frontier (i.e., by increasing the MSP and decreasing the cost subsidy while retaining the resultant policy optimal), while consumer's surplus

C h ( m h ∗ , β h ∗ ) $\mathcal C_{\text h}(m_{\text h}^*,\beta _{\text h}^*)$

and shortage cost

S h ( m h ∗ , β h ∗ ) $\mathcal S_{\text h}(m_{\text h}^*,\beta _{\text h}^*)$

remain unchanged because the crop production is retained at

ρ ∗ $\rho ^*$

along the frontier, as shown in (34). This implies that the total surplus (i.e.,

F h ( m h ∗ , β h ∗ ) + C h ( m h ∗ , β h ∗ ) $ \mathcal F_{\text h}(m_{\text h}^*,\beta _{\text h}^*) + \mathcal C_{\text h}(m_{\text h}^*,\beta _{\text h}^*)$

) and net surplus (i.e.,

F h ( m h ∗ , β h ∗ ) + C h ( m h ∗ , β h ∗ ) − S h ( m h ∗ , β h ∗ ) $ \mathcal F_{\text h}(m_{\text h}^*,\beta _{\text h}^*) + \mathcal C_{\text h}(m_{\text h}^*,\beta _{\text h}^*) - \mathcal S_{\text h}(m_{\text h}^*,\beta _{\text h}^*)$

) also increase from

P $P$

to

Q $Q$

along the frontier. (We establish the technical details about the profile of

H ∗ $\mathcal H^*$

and other statements in Lemma 6 in Appendix C.) This observation enables us to “refine” the set of optimal policies for any budget

B $B$

using the net surplus

F h ( m h ∗ , β h ∗ ) + C h ( m h ∗ , β h ∗ ) − S h ( m h ∗ , β h ∗ ) $ \mathcal F_{\text h}(m_{\text h}^*,\beta _{\text h}^*) + \mathcal C_{\text h}(m_{\text h}^*,\beta _{\text h}^*) - \mathcal S_{\text h}(m_{\text h}^*,\beta _{\text h}^*)$

to break the ties. Specifically, we define an efficient policy for a given budget

B $B$

, which we denote by

( m ̂ ( B ) , β ̂ ( B ) ) $(\hat{m}(B), \hat{\beta }(B))$

, as follows:

37

That is, among all the hybrid policies that yield the maximum net value, the efficient policy

( m ̂ ( B ) , β ̂ ( B ) ) $(\hat{m}(B),\hat{\beta }(B))$

is the one that offers the highest net surplus. Therefore, using the fact that the net surplus is increasing along the frontier

H ∗ $\mathcal H^*$

from point

P $P$

to point

Q $Q$

in Figure 3, we can conclude that an efficient policy is unique and its expenditure is equal to

B $B$

. Now, we introduce the following result that explains how the unique efficient policy can be implemented for any budget

B $B$

. Proposition 6

Let

B ∼ = s ( a + s ) μ 2 ( 1 + μ 2 + σ 2 ) $\tilde{B} = \frac{s (a+s) \mu ^2}{ (1 + \mu ^2+\sigma ^2)}$

. For any budget

B $B$

, we obtain: 1.

If

B $B$

is low, that is,

B < B ∼ 2 $B < \frac{\tilde{B}}{2}$

, then the efficient policy is to invest

B $B$

in standalone cost subsidy (so that

m ̂ ( B ) = 0 $\hat{m}(B) = 0$

).

2.

If

B $B$

is moderate, that is,

B ∼ 2 ⩽ B ⩽ B ∼ $\frac{\tilde{B}}{2} \leqslant B \leqslant \tilde{B}$

, then the efficient policy is to invest

B ∼ 2 $\frac{\tilde{B}}{2}$

in cost subsidy and

B − B ∼ 2 $B - \frac{\tilde{B}}{2}$

in MSP (so that

β ̂ ( B ) > 0 $\hat{\beta }(B) > 0$

and

m ̂ ( B ) > 0 $\hat{m}(B) > 0$

).

3.

If

B $B$

is high, that is,

B > B ∼ $B > \tilde{B}$

, then the efficient policy is to invest

B ∼ $\tilde{B}$

in standalone MSP (so that

β ̂ ( B ) = 0 $\hat{\beta }(B) = 0$

) and retain

B − B ∼ $B - \tilde{B}$

unused.

OPEN IN VIEWER

FIGURE 3

Optimal hybrid policies

H ∗ $\mathcal H^*$

Proposition 6 reveals that the actual choice of the efficient policy is solely driven by the amount of budget

B $B$

that is available. Specifically, when

B $B$

is moderate or high, the hybrid and standalone MSP policies offer the additional advantage of increasing the net surplus, without deteriorating the maximum net value. ¹⁸

CONCLUSIONS

Our paper represents an initial attempt to examine the impact of a credit‐based MSP on farmers and consumers and compare its performance against a more common cost‐subsidy scheme.

We developed a stylized game‐theoretic model to evaluate the impact of cost subsidy and credit‐based MSP on (i) farmer's surplus, (ii) consumer's surplus, and (iii) the net value that the farmer‐support programs deliver. By assuming that farmers are infinitesimally small, rational, and strategic with heterogeneous crop production costs, we analyzed the impact of cost subsidy and MSP in the presence of market and crop production yield uncertainties and compared the performance of both the support schemes for any budgetary investment.

We found that, although optimal standalone MSP, hybrid (i.e., mixture of MSP and cost‐subsidy policies), and standalone cost‐subsidy policies all generate the same maximum net value, they are ranked in the decreasing order of their efficiency in achieving higher net surplus. However, the choice of the appropriate policy is determined by the availability of budget.

In addition to the factors that we discussed in the current paper, there are other aspects of MSPs to explore as future research. First, a natural extension of our work is to examine the performance of MSPs against cost subsidies when they are offered to multiple crops. It will be interesting to capture and understand the interaction between competing crops when farmers make the sowing decisions. Second, it is known that farmers tend to grow crops that are highly subsidized even though such crops are unsuitable to be grown in specific environmental conditions. This leads to adoption of unsustainable farming practices thereby resulting in depletion of natural resources. Examining such secondary impacts of agricultural subsidies, and how cost subsidies perform against MSPs to address the above environmental concerns, are both relevant and crucial. Third, given the complexity of the effects and interactions of various subsidy policies, policymakers often find it challenging to arrive at the efficient design of policies that should be offered and the optimal portfolio of crops for which the policies should be offered. Hence, addressing the large scale problem of multicrop MSPs, along with cost subsidies, is an interesting direction. The above are some problems pertaining to MSPs that are of high relevance and importance, especially in the context of emerging economies.

Footnotes

AGGREGATE MARKET‐LEVEL YIELD MODEL FOR DISCRETE NUMBER OF FARMERS

WHEN ADDITIVE COST SUBSIDY γ $\gamma$ IS OFFERED

AUXILIARY RESULTS

1

Besides MSPs, governments, NGOs, and a few firms in developing countries are also providing technical and scientific informational support to farmers when making cropping decisions (Chen & Tang, ). For instance, the Indian government had set up Krishi Vigyan Kendras (or Agriculture Knowledge Centers) that are run by the Indian Council of Agricultural Research, which provide necessary information and advice to farmers on farming. There are also many NGOs (such as Centre for Collective Development, Centre for Sustainable Agriculture, Krishiyodha, and others), mobile apps (such as AgriApp, Kisan Yojana, IFFCO Kisan, RML, and others), and companies (such as Ninjacart, KrishiHub, and others) that give inputs and assistance to farmers.

2

Many farmers in India often make their decisions based on the advice provided by the (risk‐neutral) advisory bodies.

3

In general, the cost of cultivating a crop can vary among farmers depending on the soil and natural resources locally available, the climatic conditions, and the farming practices they employ, which makes it imperative to consider cost heterogeneity when formulating a model.

4

We will defer the case of “partially correlated” yields to future research.

5

We adopt the standard Hotelling linear city for capturing the heterogeneous cultivation costs of the farmers (Tirole & Jean, ).

6

For tractability, we assume that every farmer “anticipates” the yields of all farmers (including his own) are “identically distributed” so that the ex ante average revenue earned by a farmer is the ratio of the total expected revenue earned by all the farmers who cultivate the crop to the number of farmers who cultivate the crop, as stated in ().

7

To explicate our analysis associated with the rational expectation concept, we analyze the case of

N $N$

discrete farmers in Appendix A.

8

It is also obvious that if farmers are offered cost subsidy or MSP when

τ 0 = 1 $\tau _0 = 1$

, then still all farmers will continue to grow the crop because they gain higher profit, thereby making the problem trivial.

9

It can be easily observed that offering more cost subsidies when all farmers are cultivating the crop always improves farmer's surplus because it is akin to offering free money; we avoid this trivial case that is of little practical relevance and focus on the more realistic case where not all farmers are enticed to cultivate the crop through cost subsidies, in order to capture the interactions between cropping decisions and farmer's surplus.

10

It can be shown that () is also equivalent to 38

11

It is easy to confirm that

m ∗ $m^*$

satisfies Assumption 4 due to Assumption .

12

Here, although we discuss choosing between MSP and cost subsidy when exactly one of them has to be chosen, we will introduce hybrid policy in the next section where both MSP and cost subsidy can be implemented simultaneously.

13

Note that it suffices to consider the investment in the implementation cost because there exists a one‐to‐one correspondence between the implementation cost (i.e.,

E $\mathcal {E}$

) and the total cost (i.e.,

E + S $\mathcal {E+S}$

) via the equilibrium threshold (i.e.,

η $\eta$

or

τ $\tau$

as the case may be). Hence, we can always check if the overall total budget constraint is satisfied for any budget invested in implementing the policy.

14

We define total surplus as the sum of farmer's and consumer's surpluses and net surplus as total surplus less the shortage cost; moreover, to recall, net value is net surplus less the expenditure.

15

We provide the mathematical expressions for net values generated by cost subsidy and MSP, for any given budget

B $B$

, in Lemma  in Appendix C.

16

We denote a hybrid policy that simultaneously offers cost subsidy

β $\beta$

and MSP

m $m$

by an ordered pair

( m , β ) $(m,\beta )$

.

17

We use the subscript “h” to denote hybrid policy.

18

We also provide a few supplementary results in Appendix C on how crop characteristics influence the efficient policy. We refrain from including them here because they are not the main focus of our discussion in the paper.

19

It is important to note that because the subsidy

γ $\gamma$

is offered per unit input capacity that is invested by a farmer, who is infinitesimally small (i.e., his production alone cannot influence the market price), there is no scope for moral hazard. In other words, a farmer has no incentive to invest less than the input capacity that he holds and obtain the commensurately scaled input subsidy, so long as () holds true.

20

We use subscript

a $a$

to denote additive cost subsidy.

ORCID

Prashant Chintapalli

Christopher S. Tang

References

1.

African Smallholder Farmers Group (ASFG) (2013). Supporting smallholder farmers in Africa . http://www.fao.org/family‐farming/detail/en/c/1109849/

Go to Reference

2.

Akkaya D. Bimpikis K. Lee H. (2021). Government interventions to promote agricultural innovation. Manufacturing & Service Operations Management, 23(2), 437–452.

Go to Reference

3.

Alizamir S. Iravani F. Mamani H. (2018). An analysis of price vs. revenue protection: Government subsidies in the agriculture industry. Management Science, 65(1), 32–49.

+ Show References

4.

Bardhan P. Mookherjee D. (2011). Subsidized farm input programs and agricultural performance: A farm‐level analysis of West Bengal's green revolution, 1982–1995. American Economic Journal: Applied Economics, 3(4), 186–214.

Go to Reference

5.

Bera S. (2017). Madhya Pradesh launches new farm scheme to hedge price risks in farming . https://www.livemint.com/Politics/uDdclMv4VKUhGvpSEqtmqL/Madhya‐Pradesh‐launches‐new‐farm‐scheme‐to‐hedge‐price‐risks.html

Go to Reference

6.

Brooks J. Wiggins S. (2010). The use of input subsidies in developing countries. In Global forum on agriculture (pp. 29–30), November 29–30. OECD Headquarters, Paris.

Go to Reference

7.

Chand R. (2003). Minimum support price in agriculture: Changing requirements. Economic and Political Weekly, 38, 3027–3028.

Go to Reference

8.

Chen Y.‐J. Tang C. S. (2015). The economic value of market information for farmers in developing economies. Production and Operations Management, 24(9), 1441–1452.

Go to Reference

9.

Chibwana C. Fisher M. Shively G. (2012). Cropland allocation effects of agricultural input subsidies in malawi. World Development, 40(1), 124–133.

Go to Reference

10.

Chintapalli P. Tang C. S. (2020). The value and cost of crop minimum support price: Farmer and consumer welfare and implementation cost. Management Science, 67. https://doi.org/10.1287/mnsc.2020.3831

+ Show References

11.

Dantwala M. (1967). Incentives and disincentives in Indian agriculture. Indian Journal of Agricultural Economics, 22(2), 1.

Go to Reference

12.

Dean V. W. (1996). Why not the Brannan plan? Agricultural History, 70(2), 268–282.

Go to Reference

13.

Fan S. Gulati A. Thorat S. (2008). Investment, subsidies, and pro‐poor growth in rural India. Agricultural Economics, 39(2), 163–170.

Go to Reference

14.

FAO . (2020). India at a glance . http://www.fao.org/india/fao‐in‐india/india‐at‐a‐glance

Go to Reference

15.

Fox K. A. (1956). The contribution of farm price support programs to general economic stability. In Policies to combat depression (pp. 295–356). NBER.

Go to Reference

16.

Guda H. Dawande M. Janakiraman G. Rajapakshe T. (2021). An economic analysis of agricultural support prices in developing economies. Production and Operations Management . https://ssrn.com/abstract=3103334

+ Show References

17.

Hemming D. J. Chirwa E. W. Dorward A. Ruffhead H. J. Hill R. Osborn J. Langer L. Harman L. Asaoka H. Coffey C. Phillips D. (2018). Agricultural input subsidies for improving productivity, farm income, consumer welfare and wider growth in low‐and lower‐middle‐income countries: A systematic review. Campbell Systematic Reviews, 14(1), 1–153.

+ Show References

18.

Hu M. Liu Y. Wang W. (2017). Socially beneficial rationality: The value of strategic farmers, social entrepreneurs and for‐profit firms in crop planting decisions. Management Science, 65. https://doi.org/10.1287/mnsc.2018.3133

Go to Reference

19.

Huang J. Leng M. Parlar M. (2013). Demand functions in decision modeling: A comprehensive survey and research directions. Decision Sciences, 44(3), 557–609.

Go to Reference

20.

Innes R. (1990). Government target price intervention in economies with incomplete markets. Quarterly Journal of Economics, 105(4), 1035–1052.

Go to Reference

21.

Jayne T. S. Rashid S. (2013). Input subsidy programs in sub‐Saharan Africa: A synthesis of recent evidence. Agricultural Economics, 44(6), 547–562.

Go to Reference

22.

Jayne T. S. Sitko N. J. Mason N. M. Skole D. (2018). Input subsidy programs and climate smart agriculture: Current realities and future potential. In Climate Smart Agriculture (pp. 251–273). Springer, Cham.

+ Show References

23.

Kazaz B. Webster S. (2011). The impact of yield‐dependent trading costs on pricing and production planning under supply uncertainty. Manufacturing & Service Operations Management, 13(3), 404–417.

Go to Reference

24.

Krishnan V. B. (2018). What the agriculture census shows about land holdings in India . https://www.thehindu.com/sci‐tech/agriculture/indian‐farms‐getting‐smaller/article25113177.ece

Go to Reference

25.

Liao C.‐N. Chen Y.‐J. Tang C. S. (2017). Information provision policies for improving farmer welfare in developing countries: Heterogeneous farmers and market selection. Manufacturing & Service Operations Management, 21. https://doi.org/10.1287/msom.2016.0599

Go to Reference

26.

Liao C.‐N. Chen Y.‐J. Tang C. S. (2019). Information provision policies for improving farmer welfare in developing countries: Heterogeneous farmers and market selection. Manufacturing & Service Operations Management, 21(2), 254–270.

Go to Reference

27.

Mendelson H. Parlaktürk A. K. (2008). Competitive customization. Manufacturing & Service Operations Management, 10(3), 377–390.

Go to Reference

28.

Mills E. S. (1959). Uncertainty and price theory. Quarterly Journal of Economics, 73(1), 116–130.

Go to Reference

29.

Ministry of Agriculture and Farmers Welfare, Government of India (2019). Categorisation of farmers . https://pib.gov.in

Go to Reference

30.

NITI Aayog (2016). Evaluation study on efficacy of minimum support prices (MSP) on farmers . http://niti.gov.in/writereaddata/files/writereaddata/files/document_publication/MSP‐report.pdf

Go to Reference

31.

Petruzzi N. C. Dada M. (1999). Pricing and the newsvendor problem: A review with extensions. Operations Research, 47(2), 183–194.

+ Show References

32.

Prasad A. M. (2016). Subsidy reform: Seeding change through direct benefit transfer . https://indianexpress.com/article/india/india‐news‐india/subsidy‐reform‐seeding‐change‐through‐direct‐benefit‐transfer/

Go to Reference

33.

Ramaswami B. Seshadri S. Subramanian K. V. (2018). The welfare economics of storage‐based price supports. Working Paper . https://doi.org/10.2139/ssrn.3262407

+ Show References

34.

Spitze R. (1978). The food and agriculture act of 1977: Issues and decisions. American Journal of Agricultural Economics, 60(2), 225–235.

Go to Reference

35.

Subbarao D. (2011). The challenge of food inflation (Working Papers id:4594). eSocialSciences.

Go to Reference

36.

Tirole J. Jean T. (1988). The theory of industrial organization. MIT Press.

+ Show References

37.

Tripathi A. K. (2013). Agricultural price policy, output, and farm profitability—Examining linkages during post‐reform period in India. Asian Journal of Agriculture and Development, 10(1), 91–111.

Go to Reference

38.

Xiao S. Chen Y.‐J. Tang C. S. (2020). Knowledge sharing and learning among smallholders in developing economies: Implications, incentives, and reward mechanisms. Operations Research, 68(2), 435–452.

Go to Reference

39.

Xu Z. Burke W. J. Jayne T. S. Govereh J. (2009). Do input subsidy programs “crowd in” or “crowd out” commercial market development? Modeling fertilizer demand in a two‐channel marketing system. Agricultural Economics, 40(1), 79–94.

Go to Reference