1 Introduction

Understanding monetary policy design in emerging markets and developing economies (EMDEs) is a growing area of research. One missing aspect in this literature is how distortions in the agriculture sector translate into output and inflation dynamics, and their implications for monetary policy setting. In particular, central banks in EMDEs often grapple with understanding the inflationary impact of a shock emanating from the agriculture sector because the precise relationship between aggregate inflation and the terms of trade may be unknown. To address these questions, we develop a three-sector (grain, vegetable, and manufacturing) closed economy NK-DSGE model for the Indian economy to understand how one major distortion—the procurement of grain by the government—affects overall inflationary pressures in the economy via changes in the inter-sectoral terms of trade. Our main contribution is to identify the mechanism through which changes in the terms of trade due to procurement leads to aggregate inflation, changes in sectoral output gaps, sectoral resource allocation, and the economy wide output gap. We then calibrate the model to India to discuss the role of monetary policy in such a set-up.

Many developing countries, including India, have a large agriculture sector which is inherently volatile. In India, the combined agriculture sector (agriculture, forestry and fishing) comprises 17 % of GDP in 2013–2014 (Reserve Bank of India 2014).Footnote 1 The employment share of the agriculture sector in India is also large: 47 % in 2013–2014 (Government of India 2013–2014). The Indian government periodically intervenes in the agricultural sector, especially in the food grain market, by directly procuring grain from farmers to create a buffer grain stock to smooth price volatility and for redistribution to the poor.Footnote 2 Non-procured grain becomes available in the market for consumption. By acting like a demand shock in the grain sector, higher procurement increases the market price for grain, because it creates a shortage for open market grain. Procurement also alters the terms of trade between grain and other agricultural goods as well as between agriculture and manufacturing. Changes in the terms of trade have both demand side and supply side effects in the other sectors of the economy thereby affecting economy wide output and inflation dynamics.Footnote 3

The question that arises—for a central bank like the Reserve Bank of India—is how monetary policy should respond to changes in the inter-sectoral terms of trade that stem from a procurement shock. In this paper, we analyze how a procurement shock transmits through changes in the terms of trade, and affects sectoral wages, marginal costs, sectoral inflation rates, generalized inflation, sectoral output gaps, resource (labor) re-allocation, and ultimately generalized inflation and the economy wide output gap.

We address these issues with a three sector model that has both standard and non-standard features. There are four entities in the economy: a representative household, firms, a government, and a central bank. Households consume open market grain, vegetable, and the manufacturing good. They supply labor to all three sectors. Labor is assumed to be perfectly mobile across sectors. The labor market is assumed to be frictionless. The manufacturing sector (M) is characterized by staggered price setting and monopolistic competition. The agricultural sector (A), which is also monopolistically competitive, is disaggregated into a grain (G) and a vegetable (V) sector, both of which are characterized by flexible prices. The reason for this disaggregation in the agriculture sector is to incorporate additional imperfections in the agricultural market that are specific to the Indian economy.

We assume that the grain sector has a procurement distortion, which creates a wedge in the price-setting equation of the firms in the grain market. Procuring grain is distortionary because this leads to a shortage of grain in the open market leading to overall inflationary pressures. In India, as part of the procurement policy, the government announces minimum support prices (\(\textit{MSP}\)) before every cropping season for a variety of agricultural commodities. Minimum support prices are the prices at which a farmer can sell the agricultural commodity to the government, and this is typically set above the market price. The procured grain is then stored in Food Corporation of India (FCI) warehouses, from where a part of it is distributed to poor households. The rest of the procured amount remains in warehouses unconsumed and serves as a buffer stock to offset future supply shocks.

To model the institutional environment in which procurement takes place in India, we follow Basu (2011) and Anand et al. (2016).Footnote 4 We assume that consumers purchase grain at the price prevailing in the open market for grain. This price is determined by the supply and the combined demand for grain by consumers and the government for procurement. In Fig. 1, this is represented by the total demand for grain schedule, PP. The demand for grain by consumers is given by the schedule, OO. A positive procurement shock leads to an increase in the total demand for grain, which shifts the demand schedule outward from OO to PP. The increase in demand leads to a change in the market equilibrium from point X to Z. The open market price rises from \(P^{*}\) to \(P_{\textit{OG}},\) where the new market clearing price, \(P_{\textit{OG}}\), is equal to the MSP. At \(P_{\textit{OG}}\), the supply of grain increases from \(\textit{OE}\) to \(\textit{OA}\). However, the open market grain left for the consumer reduces from \(\textit{OE}\) to \(\textit{OB}\), with the rest of the grain, \(\textit{AB}\), procured. A farmer sells the quantity, \(\textit{AB}\), to the government at the \(\textit{MSP}\) (or at \(P_{\textit{OG}}\) in our model as explained above). Thus, a procurement shock acts like a demand shock in the grain sector, which leads to a higher open market grain price and a lower open market grain quantity. However, the government stops purchasing grain once it meets its targeted amount. We later show that a shock to the public procurement of grain because of an increase in the demand for grain is equivalent to a time varying mark-up shock in the grain sector, i.e., higher procurement raises the mark-up charged by grain sector firms. Procurement therefore acts like a tax on grain consumers.

Fig. 1
figure 1

Effect of procurement policy on open market grain price and output

Full size image

To close the model, the central bank implements monetary policy via a simple Taylor-style interest rate rule.

1.1 Main Results

The theoretical contribution of our paper is to provide a rigorous understanding of the general equilibrium effects of procurement shocks using a closed economy NK-DSGE model. In particular, we seek to uncover the transmission mechanism of a positive procurement shock and a negative productivity shock on output and inflation dynamics, and compare their implications for monetary policy design for the Reserve bank of India and other emerging market central banks. We consider these two cases because they typify the kind of shocks experienced by the Indian agriculture sector such as an upward increase in procurement (positive procurement shock) or a bad monsoon (negative productivity shock).

1.1.1 Procurement Shock

On impact, a one period positive procurement shock increases the price of open market grain. This increases the terms of trade (i) between grain and vegetable (intra-sectoral terms of trade), and (ii) between the agriculture sector and the manufacturing sector (inter-sectoral terms of trade), making other sectoral goods (vegetable and manufacturing) relatively cheaper. Also, a procurement shock immediately raises the demand for labor in the grain sector leading to higher nominal wages in the labor market since the grain sector pulls labor away from other sectors. Because labor is mobile across sectors, nominal wages increase and equalize in all the sectors. The vegetable and manufacturing sector firms raise the prices of their goods in response to higher nominal wages, leading to generalized inflation.

Moreover, the manufacturing sector is a sticky price sector and thus only a fraction of firms revise their prices and this creates a positive output gap on impact. As a response to the rise in inflation and positive output gap the central bank raises the nominal interest rate. The real interest rate, which is the nominal interest rate adjusted for one period ahead expected inflation, also rises. A rise in the real interest rate induces a fall in aggregate consumption because of the inter-temporal substitution effect. From the aggregate goods market clearing condition, this would imply that the output produced for consumption (non-procured grain, vegetable, and manufactured goods) will fall. However, because the rise in procured output exceeds the reduction in output produced for consumption, aggregate output increases.

On impact, from the demand side, the reduction in consumption is consistent with a reduction in the sectoral demand for goods. The income effect reduces proportionately the demand for each sectoral good because aggregate consumption falls and sectoral demands are proportionate to aggregate consumption. On the other hand the substitution effect induces an increase in the demand for the manufacturing and the vegetable sector goods as both are now relatively cheaper compared to grain. In the net, the income effect dominates the substitution effect. Moreover, due to sectoral goods market clearing, the lower sectoral demand for manufacturing, open market grain, and vegetable, leads to less labor employed in these sectors. However, because aggregate output increases, lower employment in the open market grain \((\textit{OG})\) sector, the manufacturing (M) sector, and the vegetable (V) sector, is more than offset by an increase in labor demand for producing procured grain \((\textit{PG})\). Therefore total employment rises. Over time, the real interest rate falls back to its long run value, and consumption rises back to its steady state value. Hence, output approaches its steady state and the output gap goes to zero. As the effect of the procurement shock dampens, the real wage falls over time back to its steady state value, and the sectoral consumption shares, sectoral employment shares, and the intra-sectoral and inter-sectoral terms of trade fully adjust to their original pre-procurement shock levels.

In sum, a one period positive procurement shock leads to aggregate inflation, a positive output gap and labor reallocation away from the manufacturing and the vegetable sectors.

1.1.2 Productivity Shock

On impact, a one period negative productivity shock decreases grain output and increases grain prices. This increases the terms of trade (i) between grain and vegetable (intra-sectoral terms of trade), and (ii) between the agriculture sector and the manufacturing sector (inter-sectoral terms of trade), making other sectoral goods (vegetable and manufacturing) relatively cheaper. The demand for vegetable and manufacturing sector goods increases. The vegetable and manufacturing sector goods firms respond to this by increasing their output, which increases their demand for labor. A higher demand for labor in these two sectors leads to higher nominal wages across the economy. The vegetable and manufacturing sector firms raise the prices of their goods in response to higher nominal wages, leading to generalized inflation.

Moreover, the manufacturing sector is a sticky price sector and thus only a fraction of firms revise their prices and this creates a negative output gap on impact. At the same time the economy wide output gap also falls slightly. Monetary policy responds to this increase in inflation and slightly negative output gap by an increase in the nominal interest rate. The real interest rate rises. A rise in the real interest rate induces a fall in aggregate consumption because of the inter-temporal substitution effect.

On impact, from the demand side, the reduction in consumption is consistent with a increase in the sectoral demand for goods (vegetable and manufacturing) because the substitution effect due to the increase in the intra-sectoral and inter-sectoral terms of trade offsets the income effect due to a downward reduction in consumption. The income effect reduces the demand for each sectoral good. On the other hand the substitution effect increases the demand for the manufacturing and the vegetable sector goods as both are relatively cheaper. Because of sectoral goods market clearing, the higher sectoral demand for manufacturing and vegetable leads to more employment in these sectors. As the effect of the productivity shock dampens, the nominal wage falls over time back to its steady state value, and the sectoral consumption shares, sectoral employment shares, and the intra-sectoral and inter-sectoral terms of trade fully adjust to their original pre-shock levels. In sum, a one period negative productivity shock leads to aggregate inflation, a slightly negative output gap and labor reallocation towards the manufacturing and the vegetable sectors.

1.1.3 Comparison Between Both Shocks

While both the shocks lead to aggregate inflation, a one period procurement shock leads to a positive economy-wide output gap while a one period negative productivity shock leads to a slightly negative economy-wide output gap. The transmission of both the shocks from the grain sector to the other sectors also differs. A positive procurement shock is basically a demand shock in the grain sector which raises the wages in the other sectors. In contrast, a negative productivity shock in the grain sector is a negative supply shock which increases the demand for the other two sector goods and also raises the wages in the other sectors. However, while the procurement shock reallocates the labor away from the vegetable and the manufacturing sector, a negative productivity shock reallocates the labor towards the vegetable and the manufacturing sector.

When we calibrate the model to the Indian data we show that, higher is the share of the household’s expenditure on the agricultural sector good, higher is the impact on inflation from both shocks.

1.1.4 NKPC and DIS Equations

We show that the presence of procurement (under an economically intuitive sufficient condition) changes the aggregate NKPC and DIS curves which affects monetary policy design. A positive steady state procurement level makes the aggregate NKPC steeper which means a given output gap is associated with higher inflation compared to the case when there is no procurement. At the same time a positive steady state procurement level affects the economy wide DIS equation and makes the DIS curve steeper. This implies that the response of the real economy to changes in the real interest rate becomes less strong, thus requiring a stronger monetary response to curb inflation, for a given output gap. This happens because procurement creates a wedge between the output produced and the output consumed. The changes in the real rate of interest affects only output consumed which is a constant proportion of total output. Hence, procurement weakens monetary policy transmission since monetary policy only affects consumed output. Moreover, a positive steady state procurement level distorts the steady state level of all the endogenous variables which makes aggregate inflation higher and the economy-wide output gap higher. Since monetary policy follows a simple Taylor rule in our model, monetary policy is directly affected by the government’s procurement policy.

1.2 Literature Review

Our model is most closely related to the seminal work by Galí and Monacelli (2005) and Aoki (2001). The main difference between our model and these papers is that Galí and Monacelli have an open economy set-up while our model assumes a closed economy. In terms of Aoki (2001), while he does not model procurement, in his two sector model, the flexible price sector (the food sector) is distortion free, while in our model the flexible price sectors are not distortion-free. However, similar to Aoki (2001) we explain the transmission of inflation from a shock in the flexible sector to the other sectors because of a change in the terms of trade.Footnote 5 Our paper also discusses reasons behind the labor allocation induced in the economy due to these shocks which is not a focus in Aoki (2001). In our framework, a grain sector shock not only shifts the aggregate NKPC (as in Aoki 2001), but it also changes the slope of the NKPC. In particular, we show that procurement leads to a steepening of the NKPC and DIS curve under a sufficient condition. The procurement distortion therefore affects the responsiveness of the economy to changes in the interest rate which affects the monetary policy response.

A multi-sector model with different sectors has the advantage of allowing one to understand the transmission of sectoral shocks across the economy. A multi-sector setting affects the design of monetary policy depending on the presence of sectoral nominal rigidities and frictions (see Aoki 2001; Benigno 2004; Huang and Liu 2005 and Erceg and Levin 2006). Importantly, shocks in a multi-sector setting affect relative prices or the terms of trade which have real affects on the economy. Our paper is different from the above papers as much of the literature on terms of trade shocks in multi-sector settings assume a small open economy set-up (see Hove et al. 2015; Ortega and Rebei 2006 and De Resende et al. 2010). Although terms of trade shocks in an open economy set-up are important, inter-sectoral terms of trade shocks are also a key concern of monetary policy setting in emerging and developing economies.

2 The Model

There are four entities in the economy: a representative household, firms, the government, and a central bank. Households consume open market grain, vegetable, and the manufacturing good. They supply labor to all three sectors. Labor is assumed to be perfectly mobile across sectors. The labor market is assumed to be frictionless. There is a manufacturing sector (M)—which is characterized by staggered price setting and monopolistic competition—and an agricultural sector (A). The agricultural sector, which is also monopolistically competitive, is further disaggregated into a grain (G) and a vegetable (V) sector, which are both characterized by flexible prices. The government sector procures grain. The central bank sets the short term interest rate using a Taylor (1993) style rule. We discuss each sector in detail.Footnote 6

2.1 Households

An infinitely lived household gets utility from a consumption stream, \( C_{t}, \) and disutility from labor supply, \(N_{t}\). At time 0,  the household maximizes its expected lifetime utility,

$$\begin{aligned} E_{0}\mathop {\displaystyle \sum }\limits _{t=0}^{\infty }\beta ^{t}\left[ U(\Gamma _{t}C_{t})-V(N_{t})\right] , \end{aligned}$$
(1)

where \(\beta \in (0,1)\) is the discount factor, and \(\Gamma _{t}\) is the preference induced demand shock which is assumed to be the same across households and follows an AR(1) process. The utility function is standard and specified as:

$$\begin{aligned} U(\Gamma _{t}C_{t})\equiv & {} \frac{\left( \Gamma _{t}C_{t}\right) ^{1-\sigma }}{1-\sigma } \end{aligned}$$
(2)
$$\begin{aligned} V(N_{t})\equiv & {} \frac{(N_{t})^{1+\psi }}{1+\psi } \end{aligned}$$
(3)

where \(\sigma ,\) is the inverse of the inter-temporal elasticity of substitution and, \(\psi ,\) is the inverse of the Frisch elasticity of labor supply. Aggregate consumption, \(C_{t},\) is a composite Cobb–Douglas index of consumption of manufacturing, \(C_{M,t},\) and agriculture sector goods, \( C_{A,t},\) and is defined as:

$$\begin{aligned} C_{t}\equiv \frac{\left( C_{A,t}\right) ^{\delta }\left( C_{M,t}\right) ^{1-\delta }}{\delta ^{\delta }\left( 1-\delta \right) ^{\left( 1-\delta \right) }},\quad 0<\delta <1, \end{aligned}$$
(4)

where \(\delta \) is the share of total consumption expenditure allocated to agriculture sector goods. Agricultural goods, \(C_{A,t},\) is again a composite Cobb–Douglas index of consumption of grain bought by the consumers in the open market, \(C_{\textit{OG},t},\) and vegetable, \(C_{V,t}\), and is defined as:

$$\begin{aligned} C_{A,t}\equiv \frac{\left( C_{V,t}\right) ^{\mu }\left( C_{\textit{OG},t}\right) ^{1-\mu }}{\mu ^{\mu }\left( 1-\mu \right) ^{\left( 1-\mu \right) }}, \quad 0<\mu <1, \end{aligned}$$
(5)

with \(\mu \) being the share of total food expenditure allocated to vegetable sector goods. Consumption in each of the three sectors, \(C_{M,t}\), \(C_{\textit{OG},t}\) and \(C_{V,t}\) is a CES aggregate of a continuum of differentiated goods in the respective sector indexed by \(j\in \left[ 0,1\right] \): \(C_{M,t}\equiv \left( \int _{0}^{1}C_{M,t}(j)^{\frac{\theta -1}{\theta }}dj\right) ^{\frac{ \theta }{\theta -1}};\) \(C_{\textit{OG},t}\equiv \left( \int _{0}^{1}C_{\textit{OG},t}(j)^{\frac{ \theta -1}{\theta }}dj\right) ^{\frac{\theta }{\theta -1}}\) and \( C_{V,t}\equiv \left( \int _{0}^{1}C_{V,t}(j)^{\frac{\theta -1}{\theta }}dj\right) ^{\frac{\theta }{\theta -1}}\), where \(\theta >1\) is the elasticity of substitution between the varieties within each sector and is assumed to be the same in all sectors.

Each household maximizes its lifetime utility given by Eq. (1) subject to an inter-temporal budget constraint

$$\begin{aligned}&\int _{0}^{1}P_{\textit{OG},t}(j)C_{\textit{OG},t}(j)dj+\int _{0}^{1}P_{V,t}(j)C_{V,t}(j)dj+ \int _{0}^{1}P_{M,t}(j)C_{M,t}(j)dj\nonumber \\&\quad +\; E_{t}\{Q_{t,t+1}B_{t+1}\}\leqslant B_{t}+W_{t}N_{t}-T_{t}+Div_{t} \quad \end{aligned}$$
(6)

where \(P_{s,t}(j)\) is the price of variety j in sector \(s=\textit{OG},V,\) and M. \( B_{t+1}\) is the nominal pay-off in period \(t+1\) of the bond held at the end of period t. \(Q_{t,t+1}\) is the stochastic discount factor. The transversality condition, \(\lim _{T\rightarrow \infty }E_{t}\{B_{t}\}\ge 0\) \(\forall \) t, is assumed to be satisfied. \(W_{t}\) is the economy wide nominal wage rate. \(T_{t}\) are lump-sum taxes to the government, and \(\textit{Div}_{t} \) are the dividends or profits distributed to households by monopolistically competitive firms. Money is excluded from both the budget constraint and utility function as the demand for money is endogenized.

Optimal consumption expenditure allocations are given as solutions to maximizing the composite consumption index subject to a given level of expenditure level. For the agricultural and manufacturing goods, the optimal allocations areFootnote 7:

$$\begin{aligned} C_{A,t}= & {} \delta \left( \frac{P_{A,t}}{P_{t}}\right) ^{-1}C_{t} \end{aligned}$$
(7)
$$\begin{aligned} C_{M,t}= & {} (1-\delta )\left( \frac{P_{M,t}}{P_{t}}\right) ^{-1}C_{t} \end{aligned}$$
(8)

where the aggregate price index for the economy, or equivalently the consumer price index (CPI), is \(P_{t}\equiv \left( P_{A,t}\right) ^{\delta }\left( P_{M,t}\right) ^{1-\delta }\) with \(P_{A,t}\) and \(P_{M,t}\) being the prices of the composite agricultural and manufacturing goods, respectively. Similarly, the optimal allocations of open market grain and vegetable are given by,

$$\begin{aligned} C_{\textit{OG},t}= & {} (1-\mu )\left( \frac{P_{\textit{OG},t}}{P_{A,t}}\right) ^{-1}C_{A,t} \end{aligned}$$
(9)
$$\begin{aligned} C_{V,t}= & {} \mu \left( \frac{P_{V,t}}{P_{A,t}}\right) ^{-1}C_{A,t}, \end{aligned}$$
(10)

respectively, where the price of agricultural goods is given by, \( P_{A,t}\equiv \left( P_{V,t}\right) ^{\mu }\left( P_{\textit{OG},t}\right) ^{1-\mu }\). Finally, the optimal allocation within each category of goods give the following demand functions for the jth variety in the sector s:

$$\begin{aligned} C_{s,t}(j)=\left( \frac{P_{s,t}(j)}{P_{s,t}}\right) ^{-\theta }\;C_{s,t}\quad \text {for all} \;j\in [0,1] \end{aligned}$$
(11)

for \(s=\textit{OG},V,\) and M,  and \(P_{s,t}\equiv \left( \int _{0}^{1}P_{s,t}(j)^{1-\theta }dj\right) ^{\frac{1}{1-\theta }}\) is the sector \(^{\prime }s^{\prime }\) specific price index.

Combining Eqs. (7)–(11), it is straightforward to show that \(\int _{0}^{1}P_{\textit{OG},t}(j)C_{\textit{OG},t}(j)dj+ \int _{0}^{1}P_{V,t}(j)C_{V,t}(j)dj+ \int _{0}^{1}P_{M,t}(j)C_{M,t}(j)dj=P_{t}C_{t}\). Therefore, the budget constraint (6) can be rewritten as

$$\begin{aligned} P_{t}C_{t}+E_{t}\{Q_{t,t+1}B_{t+1}\}\leqslant B_{t}+W_{t}N_{t}-T_{t}+Div_{t}. \end{aligned}$$
(12)

The solution to maximizing (1) subject to (12) yields the following optimality conditions:

$$\begin{aligned}&\displaystyle E_{t}\left[ \beta R_{t}\left( \frac{\Gamma _{t+1}}{\Gamma _{t}}\right) ^{1-\sigma }\left( \frac{C_{t+1}}{C_{t}}\right) ^{-\sigma }\frac{P_{t}}{ P_{t+1}}\right] = 1 \end{aligned}$$
(13)
$$\begin{aligned}&\displaystyle \frac{(N_{t})^{\psi }}{(\Gamma _{t})^{1-\sigma }(C_{t})^{-\sigma }} = \frac{W_{t}}{P_{t}} \end{aligned}$$
(14)

where \(R_{t}=\frac{1}{E_{t}\{Q_{t,t+1}\}}\) is the gross nominal return on the riskless one-period bond. Equation (13) is the Euler equation. Equation (14) is the optimal labor supply equation.

2.2 Terms of Trade: Some Useful Identities

Before proceeding further, we introduce several definitions and identities that will be used in the rest of the paper. CPI inflation is the change in the aggregate price index and is given by \(\pi _{t}=\ln P_{t}-\ln P_{t-1}\). Using the definition of the aggregate price index, CPI inflation can be expressed as a weighted average of sectoral inflation rates: \(\pi _{t}=\delta \pi _{A,t}+(1-\delta )\pi _{M,t}\), where \(\pi _{A,t}\) and \(\pi _{M,t}\) are inflation in the agricultural and manufacturing goods prices, respectively. Similarly, inflation in the agricultural goods prices can be further disaggregated as the weighted average of inflation in the grain and vegetable prices (\(\pi _{\textit{OG},t}\) and \(\pi _{V,t}\), respectively): \(\pi _{A,t}=(1-\mu )\pi _{\textit{OG},t}+\mu \pi _{V,t}\). Therefore, CPI inflation can be expressed in terms of sectoral inflation rates as:

$$\begin{aligned} \pi _{t}=\delta (1-\mu )\pi _{\textit{OG},t}+\delta \mu \pi _{V,t}+(1-\delta )\pi _{M,t}. \end{aligned}$$
(15)

Defining the terms of trade (TOT) between agriculture and manufacturing (inter-sectoral), and also between grain and vegetable within the agricultural sector (intra-sectoral) is important because of their role in influencing aggregate output and inflation dynamics. We define the inter-sectoral TOT as

$$\begin{aligned} T_{\textit{AM},t}\equiv \frac{P_{A,t}}{P_{M,t}}, \end{aligned}$$
(16)

and the intra-sectoral TOT as

$$\begin{aligned} T_{\textit{OGV},t}\equiv \frac{P_{\textit{OG},t}}{P_{V,t}}. \end{aligned}$$
(17)

Equations (16) and (17) reveal that changes in the TOT can be expressed in terms of sectoral inflation rates:Footnote 8

$$\begin{aligned} \Delta \widehat{T}_{\textit{AM},t}=\pi _{A,t}-\pi _{M,t} \end{aligned}$$
(18)

and

$$\begin{aligned} \Delta \widehat{T}_{\textit{OGV},t}=\pi _{\textit{OG},t}-\pi _{V,t}. \end{aligned}$$
(19)

Combining Eqs. (15) with (18) and (19), CPI inflation dynamics can be shown to be directly related to the inter-sectoral TOT and intra-sectoral TOT. This is given by

$$\begin{aligned} \pi _{t}=\pi _{\textit{OG},t}-\mu \Delta \widehat{T}_{\textit{OGV},t}-(1-\delta )\Delta \widehat{T}_{\textit{AM},t}. \end{aligned}$$
(20)

The interpretation of Eq. (20) is the same as Eq. (15). Deteriorations of both the intra-sectoral TOT (i.e., higher inflation in vegetable relative to open grain), and inter-sectoral TOT (i.e., higher inflation in manufacturing relative to agriculture) increase CPI inflation. It will be shown later that these changes in the terms of trade alter resource allocation across sectors thus playing a critical role for the sectoral allocation of resources in the economy.

2.3 Firms

In our model, while firms in the three sectors differ only in their price setting behavior, they are similar in terms of their production technology and market structure. All three markets are monopolistically competitive. Prices in both the grain and vegetable sectors are fully flexible, while in the manufacturing sector prices are set in a staggered fashion as outlined below. Crucially, as mentioned in Sect. 1, the grain sector differs from the vegetable sector due to the government procurement of grain. Our model departs crucially from Aoki (2001) in this respect as the agriculture sector in Aoki (2001) is characterized both by flexible prices and perfect competition.

We assume that in each sector, s,  there are a continuum of firms indexed by \(j\in [0,1]\). Each firm produces a differentiated good using, \( N_{s,t}(j)\), units of labor:

$$\begin{aligned} Y_{s,t}(j)=A_{s,t}N_{s,t}(j), \end{aligned}$$
(21)

for \(s=G,V\) and M. Here, \(A_{s,t},\) is the sector-specific level of technology and its (log) first-difference follows an AR(1) process, i.e., \(\Delta \ln A_{s,t}=\rho _{s}\Delta \ln A_{s,t-1}+\epsilon _{s,t}\). The nominal marginal cost of production in sector s is given by,

$$\begin{aligned} \textit{MC}_{s,t}=\frac{W_{t}}{\textit{MPN}_{s,t}}=\frac{W_{t}}{A_{s,t}}, \end{aligned}$$
(22)

where \(\textit{MPN}_{s,t}\) is the marginal product of labor in sector s,  where \( s=G, \) V and M. Using the definitions of the terms of trade, the sectoral real marginal cost \(\left( \textit{mc}_{s,t}=\frac{\textit{MC}_{s,t}}{P_{s,t}}\right) \) for the grain, vegetable and manufacturing sector, respectively, can be rewritten as

$$\begin{aligned} \textit{mc}_{G,t}= & {} \frac{1}{A_{G,t}}\frac{W_{t}}{P_{t}}(T_{\textit{AM},t})^{-(1-\delta )}(T_{\textit{OGV},t})^{-\mu }\end{aligned}$$
(23a)
$$\begin{aligned} \textit{mc}_{V,t}= & {} \frac{1}{A_{V,t}}\frac{W_{t}}{P_{t}}(T_{\textit{AM},t})^{-(1-\delta )}(T_{\textit{OGV},t})^{(1-\mu )}, \quad \text {and}\end{aligned}$$
(23b)
$$\begin{aligned} \textit{mc}_{M,t}= & {} \frac{1}{A_{M,t}}\frac{W_{t}}{P_{t}}(T_{\textit{AM},t})^{\delta }. \end{aligned}$$
(23c)

Let

$$\begin{aligned} Y_{s,t}\equiv \left( \int _{0}^{1}Y_{s,t}(j)^{\frac{\theta -1}{\theta } }dj\right) ^{\frac{\theta }{\theta -1}} \end{aligned}$$
(24)

represent an index for aggregate sectoral output consumed for \(s=\textit{OG},V,\) and M, analogous to the one introduced for consumption.Footnote 9 Output demand is given by

$$\begin{aligned} Y_{s,t}(j)=\left( \frac{P_{s,t}(j)}{P_{s,t}}\right) ^{-\theta }Y_{s,t}. \end{aligned}$$
(25)

The sectoral labor supply allocation is then obtained as:

$$\begin{aligned} N_{s,t}\equiv & {} \int _{0}^{1}N_{s,t}(j)dj=\frac{1}{A_{s,t}} \int _{0}^{1}Y_{s,t}(j)dj=\frac{Y_{s,t}}{A_{s,t}}\int _{0}^{1}\left( \frac{ P_{s,t}(j)}{P_{s,t}}\right) ^{-\theta }dj=\frac{Y_{s,t}Z_{s,t}}{A_{s,t}}\nonumber \\ \text {for}\;s= & {} \textit{OG}, V,\;\text {and}\;M. \end{aligned}$$
(26)

The last equality in Eq. (26) uses the sectoral output demand equation.Footnote 10 Here \(Z_{s,t}=\int _{0}^{1}\left( \frac{Ps,t(j) }{Ps,t}\right) ^{-\theta }dj\) represents the price dispersion term. The price dispersion term would be there only for the sticky price sector i.e., only the manufacturing sector and for the flexible price sectors it would be one.Footnote 11 However, equilibrium variations in the term, \(\ln Z_{M,t},\) around the perfect foresight steady state are of higher order, and therefore, this term drops out for up to a first order approximation (See Appendix C in Galí and Monacelli 2005).

2.3.1 The Grain Sector and Price Setting

To model the institutional environment for price-setting in the grain sector, we assume that total grain produced is the sum of the amount consumed and procured. Let the government procure, \(Y_{\textit{PG},t}(j),\) of each variety, j,  at the market price, \(P_{\textit{OG},t}(j)\). For simplicity and without loss of generality, assume that the government procures an equal amount of each variety so that \(Y_{\textit{PG},t}(j)=Y_{\textit{PG},t}\) \(\forall j\). Therefore, \(Y_{G,t}(j)=Y_{\textit{PG},t}+Y_{\textit{OG},t}(j)\). Our set-up follows Fig. 1 described in Sect. 1, where higher demand for grain due to procurement, \(Y_{\textit{PG},t},\) increases the market price from the market clearing level, \(P^{*}\), to the higher price level, \(P_{\textit{OG}}.\) Note that in our model, the higher price level at time t, \(P_{\textit{OG},t},\) is the same as the minimum support price at time t (MSP\(_{t})\). In other words, the government announces the amount of grain it wants to procure,  \(Y_{\textit{PG},t},\) based on a given MSP\(_{t}\) it wants to set.Footnote 12 The grain sector firms take the announced procurement amount as given and set prices, \(P_{\textit{OG},t},\) optimally.

We assume that prices are flexible in the grain sector so that each firm, j,  sets its price, \(P_{\textit{OG},t}(j),\) to maximize profits, \(\pi _{\textit{OG},t}\left( j\right) ,\) given by

$$\begin{aligned} \pi _{\textit{OG},t}\left( j\right) =P_{\textit{OG},t}(j)[Y_{\textit{OG},t}(j)+Y_{\textit{PG},t}]-\textit{MC}_{G,t}[Y_{\textit{OG},t}(j)+Y_{\textit{PG},t}], \end{aligned}$$

subject to the demand constraint

$$\begin{aligned} Y_{\textit{OG},t}(j)=\left( \frac{P_{\textit{OG},t}(j)}{P_{\textit{OG},t}}\right) ^{-\theta }Y_{\textit{OG},t} \end{aligned}$$

in every period, for each variety j. The downward sloping demand curve for the jth variety reflects the fact that farmers have some monopoly power.Footnote 13 Profit maximization results in the following price setting equation,

$$\begin{aligned} P_{\textit{OG},t}(j)=\frac{\theta }{(\theta -1)-\frac{Y_{\textit{PG},t}}{Y_{\textit{OG},t}\left( j\right) }}{} \textit{MC}_{G,t}. \end{aligned}$$
(27)

Here \(\frac{\theta }{\theta -1}\) is the standard price markup over marginal cost that is due to monopolistic competition. The \(\frac{Y_{\textit{PG},t}}{ Y_{\textit{OG},t}\left( j\right) }\) term in the denominator is the ratio of the amount procured by the government relative to the amount available in the open market. This term is new and appears due to the additional friction in the grain market resulting from the procurement of grain. In the absence of this term, Eq. (27) gives the standard equilibrium price under flexible price setting. A positive shock to procurement raises the term, \(\frac{Y_{\textit{PG},t}}{Y_{\textit{OG},t}\left( j\right) },\) and leads to an increase in the mark-up. Moreover, the procurement shock also acts as a time-varying mark-up shock in the grain sector.

2.3.2 The Vegetable Sector and Price Setting

Prices are also assumed to be flexible in the vegetable sector. Each firm j can revise its price, \(P_{V,t}(j),\) in every period to maximize profits,

$$\begin{aligned} \pi _{V,t}\left( j\right) =P_{V,t}(j)Y_{V,t}(j)-\textit{MC}_{V,t}Y_{V,t}(j), \end{aligned}$$

subject to the demand constraint

$$\begin{aligned} Y_{V,t}(j)=\left( \frac{P_{V,t}(j)}{P_{V,t}}\right) ^{-\theta }Y_{V,t}, \end{aligned}$$

for variety j. Profit maximization results in the following price setting equation,

$$\begin{aligned} P_{V,t}(j)=\frac{\theta }{\theta -1}{} \textit{MC}_{V,t}. \end{aligned}$$
(28)

Equation (28) shows that all firms in the vegetable sector set the same price given the same marginal cost and markup. Note that the only distortion in this sector is this price markup, which is due to monopolistic competition.

2.3.3 The Manufacturing Sector and Price Setting

The manufacturing sector differs from the two other sectors in terms of its price setting behavior. Prices are sticky in this sector and are set a la Calvo (1983). Firms adjust prices with probabilities \((1-\alpha _{M})\) independent of the time passed since the previous adjustment. By the law of large numbers a fraction of \((1-\alpha _{M})\) firms adjust prices while the rest of the firms do not. Price re-setting firm j sets a new price at period t to maximize the current value of all future profits,

$$\begin{aligned} \max _{P_{M,t}^{*}(j)}E_{t}\sum _{k=0}^{\infty }\alpha _{M}^{k}Q_{t,t+k} \left[ P_{M,t}^{*}(j)-\textit{MC}_{M,t+k}\right] Y_{M,t+k}(j) \end{aligned}$$

subject to the demand constraint

$$\begin{aligned} Y_{M,t+k}(j)=\left( \frac{P_{M,t}^{*}(j)}{P_{M,t+k}}\right) ^{-\theta }Y_{M,t+k}. \end{aligned}$$

where \(Q_{t,t+k}=\beta ^{k}\left( \frac{\Gamma _{t+1}}{\Gamma _{t}}\right) ^{1-\sigma }\left( \frac{C_{t+1}}{C_{t}}\right) ^{-\sigma }\left( \frac{P_{t}}{P_{t+1}}\right) \) is the stochastic discount factor for nominal payoffs. Profit maximization results in the following price setting equation,

$$\begin{aligned} P_{M,t}^{*}(j)=\frac{\theta }{\theta -1}\frac{E_{t}\sum _{k=0}^{\infty }\alpha _{M}^{k}Q_{t,t+k}Y_{M,t+k}(j)\textit{MC}_{M,t+k}}{E_{t}\sum _{k=0}^{\infty }\alpha _{M}^{k}Q_{t,t+k}Y_{M,t+k}(j)}. \end{aligned}$$
(29)

The above equation shows that the manufacturing sector price is a markup over weighted current and expected future marginal costs. It is important to mention that under flexible prices, firms change their price whenever they get a chance to do so; therefore, the above optimal dynamic price setting boils down to its static counterpart similar to Eq. (28) as:

$$\begin{aligned} P_{M,t}^{*}(j)=\frac{\theta }{\theta -1}{} \textit{MC}_{M,t}. \end{aligned}$$
(30)

Under sticky price setting, the dynamics of the manufacturing sector price index is given by:

$$\begin{aligned} P_{M,t}^{1-\theta }=\alpha _{M}(P_{M,t-1})^{1-\theta }+(1-\alpha _{M})(P_{M,t}^{*})^{1-\theta }. \end{aligned}$$
(31)

Note that the nominal marginal cost entering Eqs. (27)–(29) are given by Eq. (22).

3 Equilibrium Dynamics

3.1 Market Clearing

Markets clear for each variety j in all three sectors. These can be written as: \(C_{M,t}(j)=Y_{M,t}(j),\) \(C_{\textit{OG},t}(j)+Y_{\textit{PG},t}=Y_{G,t}(j)\) and \( C_{V,t}(j)=Y_{V,t}(j)\). Aggregating over all j, using the CES aggregator on consumption of sectoral goods as assumed in Sect. 2.1, we get

$$\begin{aligned}&\displaystyle C_{M,t} =Y_{M,t}\end{aligned}$$
(32a)
$$\begin{aligned}&\displaystyle C_{V,t} =Y_{V,t}\end{aligned}$$
(32b)
$$\begin{aligned}&\displaystyle C_{\textit{OG},t} =Y_{\textit{OG},t}\end{aligned}$$
(32c)
$$\begin{aligned}&\displaystyle Y_{\textit{OG},t}+Y_{\textit{PG},t} =Y_{G,t}. \end{aligned}$$
(32d)

The government budget constraint is

$$\begin{aligned} G_{t}=T_{t}=\frac{P_{\textit{OG},t}}{P_{t}}Y_{\textit{PG},t}\quad \forall t. \end{aligned}$$
(33)

\(Y_{t}\), or aggregate output, can be written in “consumption-bundle” terms as,

$$\begin{aligned} Y_{t}=C_{t}+\frac{P_{\textit{OG},t}}{P_{t}}Y_{\textit{PG},t}. \end{aligned}$$
(34)

The above equation is the aggregate goods market clearing condition and can be re-written as,

$$\begin{aligned} Y_{t}=C_{t}+(T_{\textit{OGV},t})^{\mu }(T_{\textit{AM},t})^{1-\delta }Y_{\textit{PG},t}. \end{aligned}$$
(35)

Finally, the labor market clearing condition is given by,

$$\begin{aligned} N_{t}=N_{G,t}+N_{V,t}+N_{M,t}. \end{aligned}$$
(36)

3.2 The Steady State

Define X (without t subscript) as the steady state value of the variable, \(X_{t}\). Assuming no trend growth in productivity, the steady state value of \(A_{s}=1\) for \(s=G,V,\) and M. From Eq. (22), we have

$$\begin{aligned} \textit{MC}_{s}=W \end{aligned}$$

for \(s=G,V,\) and M. Steady state sectoral prices can be expressed as,

$$\begin{aligned} P_{M}= & {} P_{V}=\frac{\theta }{(\theta -1)}W, \\ P_{\textit{OG}}= & {} \frac{\theta }{(\theta -1)-\frac{c_{p}}{1-c_{p}}}W, \end{aligned}$$

where \(c_{p}=\frac{Y_{\textit{PG}}}{Y_{G}}\) is the share of grain procured by the government in the steady state. This gives the aggregate price level,

$$\begin{aligned} P=(1/\gamma )^{\delta (1-\mu )}\frac{\theta }{(\theta -1)}W, \end{aligned}$$

where \(\gamma =\frac{\left( \theta -1\right) \left( 1-c_{p}\right) -c_{p}}{ \left( \theta -1\right) \left( 1-c_{p}\right) }\).Footnote 14 Therefore, the above sectoral prices can also be rearranged as,

$$\begin{aligned} P_{M}= & {} P_{V}=\gamma ^{\delta (1-\mu )}P, \\ P_{\textit{OG}}= & {} (1/\gamma )^{1-\delta (1-\mu )}P. \end{aligned}$$

The steady state intra-sectoral and inter-sectoral TOT are,

$$\begin{aligned} T_{\textit{OGV}}= & {} 1/\gamma , \\ T_{\textit{AM}}= & {} \left( 1/\gamma \right) ^{1-\mu }. \end{aligned}$$

respectively. Sectoral steady state consumption demands are:

$$\begin{aligned} C_{M}= & {} (1-\delta )\gamma ^{-\delta (1-\mu )}C,\end{aligned}$$
(37a)
$$\begin{aligned} C_{V}= & {} \mu \delta \gamma ^{-\delta (1-\mu )}C,\end{aligned}$$
(37b)
$$\begin{aligned} C_{\textit{OG}}= & {} (1-\mu )\delta \gamma ^{-\delta (1-\mu )+1}C. \end{aligned}$$
(37c)

Steady state aggregate employment is derived from sectoral employment and market clearing conditions:

$$\begin{aligned} N=N_{G}+N_{V}+N_{M}=\gamma ^{-\delta (1-\mu )}\left[ 1+\left( \gamma -1\right) (1-\mu )\delta \right] C+Y_{\textit{PG}}. \end{aligned}$$
(38)

3.3 The Log-Linearized Model

Given the steady state, we log-linearize the key relationships. Log-linearization of the Euler equation (13) and the labor supply Eq. (14) yields the following two equations:

$$\begin{aligned}&\displaystyle \widehat{C}_{t} =E_{t}\{\widehat{C}_{t+1}\}-\frac{1}{\sigma }[(\widehat{R} _{t}-E_{t}\{\pi _{t+1}\})+(1-\sigma )E_{t}\{\Delta \widehat{\Gamma }_{t+1}\}] \end{aligned}$$
(39)
$$\begin{aligned}&\displaystyle \widehat{W}_{t}-\widehat{P}_{t} =\psi \widehat{N}_{t}+\sigma \widehat{C} _{t}-(1-\sigma )\widehat{\Gamma }_{t} \end{aligned}$$
(40)

where \(\widehat{R}_{t}-E_{t}\{\pi _{t+1}\}\) is the (ex-ante) real interest rate. The sectoral real marginal costs (see Eqs. 23a–23c), expressed in terms of the aggregate real wage, sectoral productivity shocks, and terms of trade terms, are log-linearized to obtain the following expressions:

$$\begin{aligned} \widehat{mc}_{G,t}= & {} \widehat{W}_{t}-\widehat{P}_{t}-\widehat{A} _{G,t}-(1-\delta )\widehat{T}_{\textit{AM},t}-\mu \widehat{T}_{\textit{OGV},t}\end{aligned}$$
(41a)
$$\begin{aligned} \widehat{mc}_{V,t}= & {} \widehat{W}_{t}-\widehat{P}_{t}-\widehat{A} _{V,t}-(1-\delta )\widehat{T}_{\textit{AM},t}+(1-\mu )\widehat{T}_{\textit{OGV},t} \end{aligned}$$
(41b)
$$\begin{aligned} \widehat{mc}_{M,t}= & {} \widehat{W}_{t}-\widehat{P}_{t}-\widehat{A} _{M,t}+\delta \widehat{T}_{\textit{AM},t} \end{aligned}$$
(41c)

The sectoral employment Eq. (26) for the vegetable and manufacturing sectors are log-linearized as

$$\begin{aligned} \widehat{N}_{s,t}=\widehat{Y}_{s,t}-\widehat{A}_{s,t}, \end{aligned}$$
(42)

for \(s=V\) and M. For the grain sector, it is log-linearized as

$$\begin{aligned} \widehat{N}_{G,t}=c_{p}\widehat{Y}_{\textit{PG},t}+(1-c_{p})\widehat{Y}_{\textit{OG},t}- \widehat{A}_{s,t}, \end{aligned}$$

where \(c_{p}\) is the steady state share of grain procured (\(Y_{\textit{PG}}/Y_{G}\)).

Combining the log-linearized sectoral demand Eqs. (7–10) and sectoral market clearing conditions, (32a–32c), sectoral output levels can be expressed in terms of aggregate consumption and terms of trade as:

$$\begin{aligned}&\displaystyle \widehat{Y}_{M,t} =\widehat{C}_{t}+\delta \widehat{T}_{\textit{AM},t}\end{aligned}$$
(43a)
$$\begin{aligned}&\displaystyle \widehat{Y}_{\textit{OG},t} =\widehat{C}_{t}-\mu \widehat{T}_{\textit{OGV},t}-(1-\delta ) \widehat{T}_{\textit{AM},t}\end{aligned}$$
(43b)
$$\begin{aligned}&\displaystyle \widehat{Y}_{V,t} =\widehat{C}_{t}+(1-\mu )\widehat{T}_{\textit{OGV},t}-(1-\delta ) \widehat{T}_{\textit{AM},t}. \end{aligned}$$
(43c)

The aggregate goods market clearing equilibrium, Eq. (35), is log linearized as:

$$\begin{aligned} \widehat{Y}_{t}=\left( 1-\lambda _{c}\right) \widehat{C}_{t}+\lambda _{c}[ \widehat{Y}_{\textit{PG},t}+\mu \widehat{T}_{\textit{OGV},t}+(1-\delta )\widehat{T}_{\textit{AM},t}] \end{aligned}$$
(44)

where \(\lambda _{c}=\gamma ^{\delta (1-\mu )-1}c_{p}s_{g}\) and we define \( s_{g}=\frac{Y_{G}}{Y}=\frac{\delta \left( 1-\mu \right) }{1-c_{p}\left( 1-\delta \left( 1-\mu \right) \right) }\) as the steady state share of grain sector output to total output. As can be seen in Eq. (44), the procurement of grain creates a wedge between aggregate output and aggregate consumption. Log-linearizing the labor market clearing condition (36), and then substituting sectoral employment in terms of sector specific output and productivity levels gives us:

$$\begin{aligned} \widehat{N}_{t}=\Theta _{1}\left[ \widehat{C}_{t}-\widehat{A}_{t}+(1-\mu )(\gamma -1)\delta \left( \widehat{Y}_{\textit{OG},t}-\widehat{A}_{G,t}\right) \right] +\Theta _{2}\left( \widehat{Y}_{\textit{PG},t}-\widehat{A}_{G,t}\right) \qquad \end{aligned}$$
(45)

where

$$\begin{aligned} \widehat{C}_{t}= & {} (1-\mu )\delta \widehat{C}_{\textit{OG},t}+\mu \delta \widehat{C}_{V,t}+(1-\delta )\widehat{C}_{M,t} \qquad \end{aligned}$$
(46a)
$$\begin{aligned} \widehat{A}_{t}= & {} (1-\mu )\delta \widehat{A}_{G,t}+\mu \delta \widehat{A} _{V,t}+(1-\delta )\widehat{A}_{M,t}\qquad \end{aligned}$$
(46b)
$$\begin{aligned} \Theta _{1}= & {} \frac{\left( 1-c_{p}s_{g}\gamma ^{\left[ \delta \left( 1-\mu \right) -1\right] }\right) \gamma ^{-\delta \left( 1-\mu \right) }}{\gamma ^{-\delta \left( 1-\mu \right) }\left[ 1+(1-\mu )(\gamma -1)\delta \right] \left( 1-c_{p}s_{g}\gamma ^{\left[ \delta \left( 1-\mu \right) -1\right] }\right) +c_{p}s_{g}} \end{aligned}$$
(46c)
$$\begin{aligned} \Theta _{2}= & {} \frac{c_{p}s_{g}}{\gamma ^{-\delta \left( 1-\mu \right) } \left[ 1+(1-\mu )(\gamma -1)\delta \right] \left( 1-c_{p}s_{g}\gamma ^{\left[ \delta \left( 1-\mu \right) -1\right] }\right) +c_{p}s_{g}}.\qquad \end{aligned}$$
(46d)

Log-linearizing and combining Eqs. (29) and (31) yields the NKPC (New Keynesian Phillips Curve) in the manufacturing sector (for details, see Galí 2008, Chap. 3)

$$\begin{aligned} \pi _{M,t}= & {} \beta E_{t}\{\pi _{M,t+1}\}+\lambda _{M}\widehat{mc}_{M,t} \nonumber \\ \text {where}\;\lambda _{M}= & {} \frac{(1-\alpha _{M})(1-\alpha _{M}\beta )}{ \alpha _{M}}. \end{aligned}$$
(47)

Note that the above log-linearized expression of the price setting equation in the manufacturing sector is independent of \(\theta ,\) the elasticity of substitution between the varieties within this sector. Similarly, the log linearized expression of the pricing Eq. (48) in the vegetable sector as shown below is independent of \(\theta .\) However, a similar log-linearized price setting Eq. (49) to the grain sector is not independent of \(\theta \) as shown below

$$\begin{aligned}&\displaystyle \widehat{mc}_{V,t}=0, \end{aligned}$$
(48)
$$\begin{aligned}&\displaystyle \widehat{mc}_{G,t}=\left( \frac{c_{p}}{(\theta -1)(1-c_{p})-c_{p}}\right) \left( \widehat{Y}_{\textit{OG},t}-\widehat{Y}_{\textit{PG},t}\right) . \end{aligned}$$
(49)

It should be noted that assuming different values of \(\theta \) for different sectors will not change the dynamics as only \(\theta \) for the grain sector, \(\theta _{G},\) will show up in the log-linearized (up to first order) system of equations of the model. This would be equivalent to assuming the same value of \(\theta \) for different sectors.

3.3.1 Shock Processes

The structural shock processes in log-linearized form are assumed to follow AR(1) processes,

$$\begin{aligned} \Delta \ln A_{G,t}= & {} \rho _{A_{G}}\mathbf {\ }\Delta \ln A_{G,t-1}+\epsilon _{A_{G},t}, \epsilon _{A_{G},t}\thicksim i.i.d.\;\left( 0,\sigma _{A_{G}}\right) \end{aligned}$$
(50a)
$$\begin{aligned} \Delta \ln A_{V,t}= & {} \rho _{A_{V}}\mathbf {\ }\Delta \ln A_{V,t-1}+\epsilon _{A_{V},t},\;\epsilon _{A_{V},t}\thicksim i.i.d.\;\left( 0,\sigma _{A_{V}}\right) \end{aligned}$$
(50b)
$$\begin{aligned} \Delta \ln A_{M,t}= & {} \rho _{A_{M}}\mathbf {\ }\Delta \ln A_{M,t-1}+\epsilon _{A_{M},t },\;\epsilon _{A_{M},t}\thicksim i.i.d. \left( 0,\sigma _{A_{M}}\right) \end{aligned}$$
(50c)
$$\begin{aligned} \ln Y_{\textit{PG},t}-\ln Y_{\textit{PG}}= & {} \rho _{Y_{\textit{PG}}}\left( \ln Y_{\textit{PG},t-1}-\ln Y_{\textit{PG}}\right) +\epsilon _{Y_{\textit{PG}},t },\;\epsilon _{Y_{\textit{PG}},t}\thicksim i.i.d.\;\left( 0,\sigma _{Y_{\textit{PG}}}\right) \nonumber \\ \end{aligned}$$
(50d)

3.3.2 The Flexible-Price Equilibrium and the Natural Level

Under flexible prices, the pricing decisions of firms are synchronized. We have sticky prices only in the manufacturing sector. Under flexible prices, price setting boils down to a static decision and each firm sets price by Eq. (30): \(P_{M,t}^{*}=\frac{\theta }{\theta -1}{} \textit{MC}_{M,t},\) which implies a constant real marginal cost. This in turn implies that the real marginal cost log-deviation is zero. We already have flexible prices in both the agricultural sub-sectors. However, given procurement in the grain sector, the real marginal cost log-deviation is non-zero. This is given by the log-linearization of Eq. (27),

$$\begin{aligned} \widehat{mc}_{G,t}^{n}=\Phi (\widehat{Y}_{\textit{OG},t}^{n}-\widehat{Y}_{\textit{PG},t}). \end{aligned}$$
(51)

where \(\Phi =\frac{c_{p}}{(\theta -1)(1-c_{p})-c_{p}}.\) The superscript, n,  is used to denote the natural level of a variable. Here, it is important to stress that the grain procured by the government will be the same under any pricing assumption, so that \(\widehat{Y}_{\textit{PG},t}=\widehat{Y}_{\textit{PG},t}^{n}\). In the case of the manufacturing and vegetable sectors, \(\widehat{mc}_{V,t}^{n}= \widehat{mc}_{M,t}^{n}=0.\) Using these conditions for the real marginal cost log-deviation, Eqs. (41a–41c) can be expressed as

$$\begin{aligned}&\displaystyle \widehat{T}_{\textit{OGV},t}^{n} =-\Phi (\widehat{Y}_{\textit{OG},t}^{n}-\widehat{Y}_{\textit{PG},t})+ \widehat{A}_{V,t}-\widehat{A}_{G,t} \end{aligned}$$
(52)
$$\begin{aligned}&\displaystyle \widehat{T}_{\textit{AM},t}^{n} =-\Phi \left( 1-\mu \right) (\widehat{Y}_{\textit{OG},t}^{n}- \widehat{Y}_{\textit{PG},t})+\widehat{A}_{M,t}-\left( 1-\mu \right) \widehat{A} _{G,t}-\mu \widehat{A}_{V,t} \qquad \end{aligned}$$
(53)

The Euler equation can be rewritten in the flexible price equilibrium as,

$$\begin{aligned} \widehat{C}_{t}^{n}=E_{t}\left\{ \widehat{C}_{t+1}^{n}\right\} -\frac{1}{\sigma }\left[ \left( \widehat{R}_{t}^{n}-E_{t}\left\{ \pi _{t+1}^{n}\right\} \right) +(1-\sigma )E_{t}\left\{ \Delta \widehat{\Gamma }_{t+1}\right\} \right] , \end{aligned}$$
(54)

where \(\widehat{R}_{t}^{n}\) and \(\pi _{t}^{n}\) denote the nominal interest rate and inflation rate under flexible price setting. At a flexible price equilibrium the real wage equation can be derived as

$$\begin{aligned} \widehat{w}_{t}^{n}=\widehat{A}_{t}+\Phi \left( 1-\mu \right) \delta ( \widehat{Y}_{\textit{OG},t}^{n}-\widehat{Y}_{\textit{PG},t}), \end{aligned}$$
(55)

where \(w=\frac{W}{P}\). Using (55), (40), and (45), at a flexible price equilibrium, the natural level of consumption, \(\widehat{C}_{t}^{n},\) can be expressed as

$$\begin{aligned} \widehat{C}_{t}^{n}= & {} \frac{(\psi \Theta _{1}+1)}{\left( \psi \Theta _{1}+\sigma \right) }\widehat{A}_{t}-\frac{(\Phi \left( 1-\mu \right) \delta +\psi \Theta _{2})}{(\psi \Theta _{1}+\sigma )}\widehat{Y}_{\textit{PG},t}\nonumber \\&+\,\frac{ (\Phi \left( 1-\mu \right) \delta -\psi \Theta _{1}\left( \gamma -1\right) \left( 1-\mu \right) \delta )}{(\psi \Theta _{1}+\sigma )}\widehat{Y} _{\textit{OG},t}^{n}\nonumber \\&+\,\frac{(1-\sigma )}{(\psi \Theta _{1}+\sigma )}\widehat{\Gamma }_{t}+\frac{ (\psi \Theta _{1}\left( \gamma -1\right) \left( 1-\mu \right) \delta +\psi \Theta _{2})}{(\psi \Theta _{1}+\sigma )}\widehat{A}_{G,t}. \end{aligned}$$
(56)

Now using the demand equations in a flexible price equilibrium, the natural levels of output for the grain, vegetable and manufacturing sectors can be expressed, respectively, as

$$\begin{aligned} \widehat{Y}_{\textit{OG},t}^{n}= & {} \widehat{C}_{t}^{n}-\mu \widehat{T} _{\textit{OGV},t}^{n}-\left( 1-\delta \right) \widehat{T}_{\textit{AM},t}^{n},\end{aligned}$$
(57a)
$$\begin{aligned} \widehat{Y}_{V,t}^{n}= & {} \widehat{C}_{t}^{n}+(1-\mu )\widehat{T} _{\textit{OGV},t}^{n}-\left( 1-\delta \right) \widehat{T}_{\textit{AM},t}^{n}, \end{aligned}$$
(57b)
$$\begin{aligned} \widehat{Y}_{M,t}^{n}= & {} \widehat{C}_{t}^{n}+\delta \widehat{T}_{\textit{AM},t}^{n}, \end{aligned}$$
(57c)

where \(\widehat{C}_{t}^{n}\) is given by Eq. (56). The aggregate natural level of output, \(\widehat{Y}_{t}^{n},\) can be expressed as,

$$\begin{aligned} \widehat{Y}_{t}^{n}=\left( 1-\lambda _{c}\right) \widehat{C}_{t}^{n}+\lambda _{c}[\widehat{Y}_{\textit{PG},t}+\mu \widehat{T}_{\textit{OGV},t}^{n}+(1-\delta )\widehat{T} _{\textit{AM},t}^{n}]. \end{aligned}$$
(58)

Equations (51)–(58) show how the presence of procurement affects the natural level of variables in the model. Procurement affects these equations as an additive shock since we assume later that procurement follows an AR(1) process. Procurement also affects these equations through the parameter, \(c_{p}\), which enters into the structural coefficients in front of the variables.

3.3.3 The Sticky Price Equilibrium

We define a variable, \(\widetilde{X}_{t}=\widehat{X}_{t}-\widehat{X}_{t}^{n}\), to be the deviation from the natural level. Using Eqs. (40), (41c) and (45) we can write \(\widetilde{mc}_{M,t}\) in terms of the manufacturing sector output gap, \((\widehat{Y}_{M,t}-\widehat{Y}_{M,t}^{n})\):

$$\begin{aligned} \widetilde{mc}_{M,t}=\widehat{mc}_{M,t}=\left( \psi \Theta _{1}+\sigma \right) \widetilde{Y}_{M,t}-\delta \left( \psi \Theta _{1}+\sigma -1\right) \widetilde{T}_{\textit{AM},t} \end{aligned}$$
(59)

Hence, the NKPC in Eq. (47) for the manufacturing sector becomes

$$\begin{aligned} \pi _{M,t}= & {} \beta E_{t}\{\pi _{M,t+1}\}+\lambda _{M}\left( \psi \Theta _{1}+\sigma \right) \widetilde{Y}_{M,t}-\lambda _{M}\delta \left( \psi \Theta _{1}+\sigma -1\right) \widetilde{T}_{\textit{AM},t}. \end{aligned}$$
(60a)
$$\begin{aligned}= & {} \beta E_{t}\{\pi _{M,t+1}\}+\lambda _{M}\left( \psi \Theta _{1}+\sigma \right) \widetilde{C}_{t}+\lambda _{M}\delta \widetilde{T}_{\textit{AM},t}. \end{aligned}$$
(60b)

Equation (60b) shows that inflation in the manufacturing sector gets affected by terms of trade changes and aggregate consumption demand. This happens because the demand for the manufacturing sector good depends on the terms of trade and the aggregate consumption demand conditions, as shown in Eq. (43a). Also note that the presence of procurement reduces the effect of aggregate consumption on inflation as procurement lowers the consumed part of aggregate output. Since prices are flexible in the vegetable and manufacturing sectors, no such individual NKPC exists in either sector. However, because of procurement there is a static “Phillips curve” type equation in the grain sector as can be seen from Eq. (49). Combining Eqs. (44) and (58), we obtain

$$\begin{aligned} \widetilde{Y_{t}}=\left( 1-\lambda _{c}\right) \widetilde{C}_{t}+\lambda _{c}(1-\delta )\widetilde{T}_{\textit{AM},t}. \end{aligned}$$
(61)

For the aggregate analysis, it is convenient to express the NKPC in terms of CPI inflation. Equations (60a) and (61) with Eqs. (43a–43c), (56) and, \(\pi _{t}-\pi _{M,t}=\delta \Delta \widehat{T}_{\textit{AM},t},\) can be rearranged to get the aggregate NKPC for the economy:

$$\begin{aligned} \pi _{t}= & {} \beta E_{t}\{\pi _{t+1}\}+\lambda _{M}\frac{\left( \psi \Theta _{1}+\sigma \right) }{\left( 1-\lambda _{c}\right) }\widetilde{Y} _{t}+\lambda _{M}\left( \delta -\frac{\lambda _{c}\left( \psi \Theta _{1}+\sigma \right) (1-\delta )}{1-\lambda _{c}}\right) \widetilde{T}_{\textit{AM},t} \nonumber \\&+\,\delta \Delta \widehat{T}_{\textit{AM},t}-\beta \delta E_{t}\left\{ \Delta \widehat{T} _{\textit{AM},t+1}\right\} . \end{aligned}$$
(62)

The right hand side of the Eq. (62) can be consolidated and written in terms of aggregate consumption and terms of trade terms as,

$$\begin{aligned} \pi _{t}= & {} \beta E_{t}\{\pi _{t+1}\}+\lambda _{M}\left( \psi \Theta _{1}+\sigma \right) \widetilde{C}_{t}+\lambda _{M}\delta \widetilde{T}_{\textit{AM},t} \nonumber \\&+\,\delta \Delta \widehat{T}_{\textit{AM},t}-\beta \delta E_{t}\left\{ \Delta \widehat{T} _{\textit{AM},t+1}\right\} . \end{aligned}$$
(63)

Similar to Eq. (60b) aggregate inflation in (63) depends on the terms of trade and aggregate consumption demand. This equation is very similar to the aggregate NKPC derived in Aoki (2001), except that the presence of procurement affects the impact that aggregate consumption has on inflation as procurement lowers the consumed part of aggregate output (as in (44)). Also, the terms of trade terms in (62) shift the Phillips curve. These terms capture the effect of terms of trade shocks on aggregate inflation.

Similarly, we derive the aggregate DIS equation by combining Eqs. (39), (54) and (61):

$$\begin{aligned} \widetilde{Y}_{t}=E_{t}\{\widetilde{Y}_{t+1}\}-\frac{\left( 1-\lambda _{c}\right) }{\sigma }\left[ \left( \widehat{R}_{t}-E_{t}\{\pi _{t+1}\}\right) -\widehat{r} _{t}^{n}\right] -\lambda _{c}(1-\delta )E_{t}\left\{ \Delta \widetilde{T} _{\textit{AM},t+1}\right\} , \nonumber \\ \end{aligned}$$
(64)

where \(\;\widehat{r}_{t}^{n}=\sigma E_{t}\{\Delta \widehat{C} _{t+1}^{n}\}-(1-\sigma )E_{t}\{\Delta \widehat{\Gamma }_{t+1}\},\) is the natural rate of interest.

The NKPC and DIS equations at the aggregate level along with a monetary policy rule constitute the basis of our analysis for output and inflation dynamics.

3.3.4 Monetary Policy Rule

Since monetary policy follows a simple Taylor’s rule with the nominal interest rate as a function of aggregate inflation and the economy wide output gap, monetary policy gets affected with procurement policy. To capture this, we use a simple generalization of Taylor (1993):

$$\begin{aligned} R_{t}=\left( R_{t-1}\right) ^{\phi _{r}}\left( \pi _{t}\right) ^{\phi _{\pi }}\left( \frac{Y_{t}}{Y_{t}^{n}}\right) ^{\phi _{y}} \end{aligned}$$

The log-linearized version of the Taylor-rule shows that:

$$\begin{aligned} \widehat{R}_{t} =\phi _{r}\widehat{R}_{t-1}+\phi _{\pi }\pi _{t}+\phi _{y}(\widehat{Y}_{t}-\widehat{Y}_{t}^{n}) =\phi _{r}\widehat{R}_{t-1}+\phi _{\pi }\pi _{t}+\phi _{y}\widetilde{Y}_{t} \end{aligned}$$
(65)

i.e., the nominal interest rate, \(\widehat{R}_{t},\) depends on its lagged value, aggregate inflation’s deviation from its target, \(\pi _{t},\) and the aggregate output gap, \(\widetilde{Y}_{t}\).Footnote 15 This closes the model.

3.4 Difference Between NKPC and the DIS With and Without Procurement

Without a procurement distortion (\(c_{p}=0,\) \(\lambda _{c}=0\)), the aggregate NKPC and DIS equations in (62) and (64) respectively are:

$$\begin{aligned}&\displaystyle \pi _{t} =\beta E_{t}\{\pi _{t+1}\}+\lambda _{M}\left( \psi +\sigma \right) \widetilde{Y}_{t}+\lambda _{M}\delta \widetilde{T}_{\textit{AM},t}+\delta \Delta \widehat{T}_{\textit{AM},t}-\beta \delta E_{t}\left\{ \Delta \widehat{T}_{\textit{AM},t+1}\right\} , \end{aligned}$$
(66)
$$\begin{aligned}&\displaystyle \widetilde{Y}_{t} =E_{t}\left\{ \widetilde{Y}_{t+1}\right\} -\frac{1}{\sigma }\left[ \left( \widehat{R}_{t}-E_{t}\{\pi _{t+1}\}\right) -\widehat{r}_{t}^{n}\right] . \end{aligned}$$
(67)

Equation (66) above is a standard NKPC for a multi-sector set-up.Footnote 16 As in Aoki (2001) changes in the terms of trade lead to shifts in the NKPC. In contrast, the DIS equation in a multi-sector set-up is not affected by the terms of trade as seen in Eq. (67). On the other hand, the presence of procurement, as can be seen from Eq. (64) adds a terms of trade term which shifts the DIS equation too.Footnote 17 The terms of trade also shifts the NKPC. Since a procurement shock shifts both the NKPC and the DIS curves, it acts as a supply shock as well as a demand shock respectively. Note that, when there is no procurement the NKPC still retains some terms of trade expressions because of the multi-sector set-up.

Moreover, we can show that when, \(0\le \lambda _{c}\le 1,\) the slope of the DIS curve and the NKPC increases monotonically with higher values of the steady state procurement parameter, \(c_{p}.\) Footnote 18

Suppose \(\lambda _{c}>0.\) An increase in the slope of the NKPC means that for a given level of the output gap, \(\widetilde{Y}_{t},\) aggregate inflation, \(\pi _{t},\) is higher. Moreover, in the DIS equation, (64), the response of aggregate output to a change in the real interest depends on the value of, \(\sigma ,\) and, \(\lambda _{c}.\) For positive values of \(c_{p},\) this responsiveness of the output gap to changes in the real interest rate becomes less, making the DIS curve steeper. This implies that to achieve a given output gap, a greater change in the real interest rate is required. The slope changes because procurement creates a wedge between the output produced and the output consumed. The changes in the real rate of interest however affects only output consumed which is a constant proportion of total output. Hence, procurement weakens monetary policy transmission since monetary transmission only applies to consumed output. Moreover, a positive steady state procurement level distorts the steady state level of all variables which makes aggregate inflation higher and the economy-wide output gap also higher.

4 Calibration

In this section, we calibrate the model to the Indian data.Footnote 19 Our goal is to understand the quantitative implications of a positive procurement shock to the economy and compare it with a negative productivity shock. We consider these two cases because they typify the kind of shocks experienced by the Indian agriculture sector. Hence, we give a single period positive procurement shock and analyze its effect on inflation, the output-gap and sectoral labor reallocation. We then contrast this with a single period negative productivity shock. We use the impulse response functions to assess implications for monetary policy set by the Reserve Bank of India, or more generally, emerging market central banks who face terms of trade shocks. In particular, we will see how a single period procurement and productivity shock affects the deviations of various variables from their steady state values.

4.1 Description of Parameters

It is well known that the values of several structural parameters are unknown in developing and emerging market economies. Therefore, while we use some parameter estimates from the literature, we also estimate some parameters from the data. We set the discount factor for India at \(\beta =0.9823\) as calibrated in Levine et al. (2012). We choose the value of the inverse of the Frisch elasticity of substitution, \(\psi =3\) (Anand and Prasad 2010). We fix the value of the inter-temporal elasticity of substitution, \(\sigma =1.99,\) as estimated in Levine et al. (2012).Footnote 20 We calculate the expenditure share on agriculture sector goods and vegetable sector goods to be, \(\delta =0.52,\) \(\mu =0.44,\) using household expenditure data, NSS (National Sample Survey) 68th round (2011–2012).Footnote 21 We fix the elasticity of substitution between varieties of the same sector goods, \(\theta =7.02,\) as estimated by Levine et al. (2012). We set the measure of stickiness for the manufacturing sector, \(\alpha _{M}=0.75,\) as estimated in Levine et al. (2012) for the formal sector in India. We choose the value of AR(1) coefficients in Eqs. (50a–50c) and standard error of these regressions following Anand and Prasad (2010).Footnote 22 Thus, for productivity shocks in the agriculture sector, the AR(1) coefficient for grain and vegetable sector is calibrated to be, \({\rho }_{A_{G}}=\rho _{A_{V}}=0.25\) and for the manufacturing sector, \(\rho _{A_{M}}=0.95.\) The standard error of regression for the grain and the vegetable sector is given by, \(\sigma _{A_{G}}=\sigma _{A_{V}}=0.03,\) and for the manufacturing sector, \(\sigma _{A_{M}}=0.02.\) We estimate an AR(1) process on procurement in the grain sector as described in Eq. (50d) using the procurement data published by the Ministry of Consumer Affairs (MCA), India from 1992 to 2012.Footnote 23 We fix the interest rate smoothening parameter, \(\phi _{R}=0\), initially. We put standard weights on inflation, \(\phi _{\pi }=1.5,\) and the output gap, \(\phi _{y}=0.5,\) in the Taylor Rule (Taylor 1993). We calculate the steady state value of \(c_{p}\) to be 0.08 using the annual grain production data from the RBI Indian database and procurement data from the Ministry of Consumer Affairs from 1992 to 2012.Footnote 24 We get this steady state by taking the average of the ratio of the net procured good to total production of wheat and rice. Finally, we ignore the role of preference induced demand shocks in the model, i.e., \(\Gamma _{t}=1\;\forall t\). Table 1 summarizes the structural parameters used in the calibration exercise in our model and their values.

Table 1 Summary of parameter values
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4.2 Transmission of a Single Period Positive Procurement Shock in the Grain Sector

Figure 2a–d plot the impulse response functions of a single period positive procurement shock, \(\widehat{Y}_{\textit{PG},t}.\)

On impact a positive procurement shock increases the markup over marginal cost, \(\widehat{MC}_{G,t},\) as shown in Eq. (27). This increases the open grain market goods price, leading to inflation in this sector, \(\pi _{\textit{OG},t},\) [see Fig. 2b (row 1, column 1)]. At the same time this increase in the markup reduces real marginal costs in the grain sector (see Eq. 51), making firms produce more grain, \(\widehat{Y}_{G,t}\), which increases the demand for labor, \(\widehat{N}_{G,t},\) [see Fig. 2c (row 2, column 2) and d (row 1, column 1)].Footnote 25 The nominal wage rises in this sector because of higher labor demand and labor gets pulled out from the other two sectors as shown in Fig. 2c (row 3, columns 1 and 2). Labor supply in the manufacturing sector, \(\widehat{N}_{M,t},\) and in the vegetable sector, \(\widehat{N}_{V,t},\) keep on falling till the time nominal wages equalize in all the sectors. The firms in these two sectors revise their prices upward due to higher nominal wages in their sectors leading to positive inflation in, \(\pi _{M,t}\) and \(\pi _{V,t}\), {see Fig. 2b [(row 1, column 2) and (row 2, column 1)]}. This is the mechanism through which the inflationary impact of a positive procurement shock gets transmitted to other sectors and leads to aggregate inflation, \(\pi _{t}\), [see Fig. 2b (row 2, column 2)].

Fig. 2
figure 2figure 2

a Single period positive procurement \(\left( \widehat{Y}_{\textit{PG},t}\right) \) shock, b–d impact of a single period positive procurement \(\left( \widehat{Y}_{\textit{PG},t}\right) \) shock on economy wide variables

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Since a positive procurement shock acts as a negative cost push shock (because of higher nominal wages), output in the manufacturing sector, \( \widehat{Y}_{M,t},\) and the vegetable sector, \(\widehat{Y}_{V,t},\) falls on impact. As, the manufacturing sector is a sticky price sector and thus only a fraction of firms revise their prices, this creates a positive output gap, \(\widetilde{Y}_{M,t},\) in this sector. More specifically, a positive output gap in the manufacturing sector, \(\widehat{Y}_{M,t}-\widehat{Y}_{M,t}^{n},\) results because a positive procurement shock in the grain sector leads to a reduction in manufacturing sector output. Due to price stickiness in the manufacturing sector, actual output, \(\widehat{Y}_{M,t},\) falls by less value than its natural level, \(\widehat{Y}_{M,t}^{n},\) and thus the term, \( \widehat{Y}_{M,t}-\widehat{Y}_{M,t}^{n},\) becomes positive on impact. At the same time the economy wide output gap, \(\widetilde{Y}_{t},\) also rises as shown in Fig. 2d (row 3, column 3). Monetary policy responds to this increase in inflation and the positive output gap by an increase in the nominal interest rate, \(\widehat{R}_{t}\) (see Eq. 65) given the Taylor rule parameters in Table 1. This increase in the nominal interest rate, adjusted for a one period future expected inflation increases the real interest rate, \(\widehat{ r}_{t}\), as shown in Fig. 2c (row 1, column 2).Footnote 26 From the Euler equation (39), a rise in the real interest rate induces current consumption, \(\widehat{C}_{t}\), to fall due to the inter-temporal substitution effect. From the demand function (Eqs. 43a–43c), the sectoral demand for goods will depend upon the income effect from falling consumption, \( \widehat{C}_{t},\) and the inter-good substitution effect due to the changing terms of trade, \(\widehat{T}_{\textit{AM},t}\) and \(\widehat{T}_{\textit{OGV},t}\). As can be seen from Fig. 2d [(row 1, columns 2 and 3) and (row 2, column 2)], the income effect dominates and the quantity demanded falls for all three sectors in the first period using the calibrated parameters from Table 1.Footnote 27 Over time the economy goes back to the steady state.

4.3 Transmission of a Single Period Negative Productivity Shock in the Grain Sector

Figure 3a–c plot the impulse response functions of a single period negative productivity shock, \(\widehat{A}_{G,t}\).Footnote 28

Fig. 3
figure 3

a–c Impact of a single period negative productivity \(\left( \widehat{A}_{G,t}\right) \) shock on economy wide variables

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On impact, a negative productivity shock reduces grain output, \(\widehat{Y} _{G,t},\) and increases the nominal marginal cost, \(\widehat{MC}_{G,t},\) leading to positive inflation in the grain sector, \(\pi _{\textit{OG},t},\) as shown in Fig. 3a (row 2, column 1). A rise in the prices of the grain sector good induces consumers to shift their demand to other sector goods, \(\widehat{Y}_{M,t}\) and \(\widehat{Y}_{V,t},\) [see Fig. 3c (row 1, columns 1 and 3)]. Foreseeing this rise in demand, the manufacturing and vegetable sector firms increase their output by employing more labor, \(\widehat{N}_{M,t}\) and \(\widehat{N}_{V,t}.\) This increase in the labor demand increases the nominal wages across all sectors. The manufacturing and vegetable sector firms revise their prices upward leading to positive inflation in these two sectors, \(\pi _{M,t}\) and \(\pi _{V,t}\), as shown in Fig. 3a [(row 1, column 2) and (row 2, column 1)]. This is the mechanism through which the inflationary impact of a negative productivity shock gets transmitted to other sectors and leads to aggregate inflation, \( \pi _{t}\), [see Fig. 3a (row 2, column 2)].

Since a negative productivity shock acts as a positive demand shock to the other two sectors (for their goods), the output in these two sectors, \( \widehat{Y}_{M,t}\) and \(\widehat{Y}_{V,t},\) rises on impact. As, the manufacturing sector is a sticky price sector and thus only a fraction of firms revise their prices and this creates a negative output gap, \( \widetilde{Y}_{M,t},\) in this sector on impact. More specifically, a negative output gap in the manufacturing sector, \(\widehat{Y}_{M,t}-\widehat{Y} _{M,t}^{n},\) results because a negative productivity shock in the grain sector leads to a rise in the demand for manufacturing sector goods. Due to price stickiness in the manufacturing sector, actual output, \(\widehat{Y} _{M,t},\) rises by less value than its natural level, \(\widehat{Y}_{M,t}^{n},\) and thus the term, \(\widehat{Y}_{M,t}-\widehat{Y}_{M,t}^{n},\) becomes negative on impact. At the same time the economy wide output gap, \( \widetilde{Y}_{t},\) also falls slightly as shown in Fig. 3c (row 3, column 1). Monetary policy responds to this increase in inflation and slightly negative output gap by an increase in the nominal interest rate, \(\widehat{R}_{t}\) (see Eq. 65) given the Taylor rule parameters in Table 1. This increase in the nominal interest rate, adjusted for one period future expected inflation increases the real interest rate, \(\widehat{r}_{t}\), as shown in Fig. 3b (row 1, column 2). From the Euler equation (39 ), a rise in the real interest rate induces current consumption, \( \widehat{C}_{t}\), to fall due to the inter-temporal substitution effect. From the demand function (Eqs. 43a–43c), the sectoral demand for goods will depend upon the income effect from falling consumption, \(\widehat{C}_{t},\) and the inter-good substitution effect due to the changing terms of trade, \(\widehat{T}_{\textit{AM},t}\) and \(\widehat{T} _{\textit{OGV},t} \). As can be seen from Fig. 3c (row1, columns 1 and 3), the substitution effect dominates and the quantity demanded rises for manufacturing and vegetable sector goods in the first period using the calibrated parameters from Table 1. Over time the economy goes back to the steady state. The main differences are summarized below in Table 2.

Table 2 Main differences between a one period positive procurement shock and a one period negative productivity shock
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5 Implications for the Reserve Bank of India

The above calibration exercise suggests that both a positive procurement shock and a negative productivity shock leads to positive aggregate inflation and a qualitatively similar response from the central bank. As discussed above, both differ strikingly from each other in how the shock gets transmitted to the aggregate economy. Figure 4 plots the monetary policy response for a range of values of \(c_{p}\in [0,0.6],\) for a common single period procurement shock, \(\widehat{Y}_{\textit{PG},t},\) on impact.

Fig. 4
figure 4

Monetary policy response \(\left( \widehat{R}_{t}\right) \) and steady state share of procured grain \(\left( c_{p}\right) \)

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Figure 4 shows a non-linear, increasing and monotonic relation between \( \widehat{R}_{t}\) and \(c_{p}.\) From Eq. (65), the nominal interest rate \(\widehat{R}_{t}\) depends on aggregate inflation, \(\pi _{t},\) and the aggregate output gap, \(\widetilde{Y}_{t}.\) A higher interest rate response of the monetary authority on impact for higher values of \( c_{p} \) is thus possible if and only if higher values of \(c_{p}\) lead to higher aggregate inflation or a higher aggregate output gap or both. To understand this it is important to see how \(c_{p}\) changes the aggregate NKPC and DIS curves. From Eq. (62), and under the sufficient condition, \(0<\lambda c\le 1,\) a higher value of \(c_{p}\) makes the aggregate NKPC steeper which means a given output gap is now associated with higher inflation. Moreover, according to the DIS equation, (64), the response of the real economy to changes in the real interest rate, \(\widehat{r}_{t},\) decreases with higher values of \(c_{p}\), thus requiring a stronger monetary response for a given output gap. Hence the monetary policy response for a procurement shock should depend on the steady state value of \(c_{p}.\) This figure implies that central banks in EMDEs like the Reserve Bank of India should respond to changes in the terms of trade over time in a systematic way as outlined in our model, especially since the importance of food inflation in monetary policy setting over the last several years has become increasingly important (Reserve Bank of India 2015).

6 Conclusion

Central banks in EMDEs such as India often grapple with understanding the inflationary impact of a shock from the agriculture sector because the precise relationship between aggregate inflation and the terms of trade may be unknown. To address this, we develop a three-sector (grain, vegetable, and manufacturing) closed economy NK-DSGE model for the Indian economy to understand how one major distortion—the procurement of grain by the government—affects overall inflationary pressures in the economy via changes in the sectoral terms of trade. Our main contribution is to identify the mechanism through which changes in the terms of trade—because of changes in procurement—leads to aggregate inflation, changes in sectoral output gaps, sectoral resource allocation, and the economy wide output gap. We then calibrate the model to India to discuss the role of monetary policy in such a set-up. We show that a positive procurement shock to grain leads to higher inflation, a change in the sectoral terms of trade, and a positive output gap because of a change in the sectoral allocation of labor. We also compare the transmission of a single period positive procurement shock with a single period negative productivity shock. We consider these two cases because they typify the kind of shocks experienced by the Indian agriculture sector (upward increase in procurement, a poor monsoon). For a positive productivity shock, we show that on impact, the economy experiences higher inflation, and a slightly negative output gap. Under a positive procurement shock, labor reallocates away from the manufacturing and the vegetable sector. Under a negative productivity shock, labor reallocates towards the manufacturing and vegetable sectors. In addition, the presence of procurement changes the standard NKPC and DIS curves of the aggregate economy. Under a sufficient condition, we show that the NKPC and DIS curves become steeper suggesting that the central bank’s response to a terms of trade shock needs to be stronger. We also show that procurement weakens monetary policy transmission.

Our paper contributes to a growing literature on monetary policy in India and other emerging market economies. Future work will characterize how the terms of trade influences optimal monetary policy.