Download Farms, Fertiliser, and Financial Frictions: Yields

Farms, Fertiliser, and Financial Frictions:
Yields from a DSGE Model
Sébastien E. J. Walker∗
CSAE and Department of Economics,
University of Oxford
March 22, 2013
Abstract
This paper elaborates a DSGE (Dynamic Stochastic General Equilibrium) model
which seeks to capture key features of a typical low-income country, where the agricultural sector often dominates the economy. It does so by giving a central role
to farmers who must borrow to buy fertiliser (which could also be interpreted as
a composite which includes fertiliser and oil). The farmers are constrained by financial frictions, and are subject to shocks from variable weather conditions and a
volatile local-currency price of imported fertiliser. The shocks are exacerbated by
the financial frictions which become more severe as a result of the shocks, resulting
in a so-called financial accelerator. These elements provide a realistic characterisation of low-income countries where supply-side shocks tend to predominate over
demand-side shocks; this predominance poses distinct challenges for policymakers
given the negative correlation between inflation and the output gap in such cases.
We will use this model to carry out simulations that yield policy rules for managing
shocks to which low-income countries are prone.
Keywords: Monetary policy, Africa, financial frictions, foreign capital flows, DSGE
models, supply shocks.
∗
Email: [email protected] or [email protected]. This paper is work in
progress, and therefore preliminary and incomplete; comments and suggestions are most welcome. This
is a chapter of the doctoral thesis I am writing under the supervision of Christopher Adam, whom I thank
for his invaluable help. Thanks are due to John Muellbauer, to participants at the 2013 CSAE Conference,
and to participants at the Gorman Workshop in the University of Oxford’s Economics Department for
helpful comments. The usual disclaimer applies. Financial support from the UK Economic and Social
Research Council (ESRC) is gratefully acknowledged.
1
Contents
1 Introduction
2 The
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
3
model
Households . . . . . . . . . . . . . . . . . .
Foreign behaviour . . . . . . . . . . . . . .
Farms . . . . . . . . . . . . . . . . . . . . .
2.3.1 Agricultural production . . . . . . .
2.3.2 Working capital . . . . . . . . . . . .
2.3.3 Financial Frictions . . . . . . . . . .
2.3.4 Net worth and demand for fertiliser
Retailers, price setting, and inflation . . . .
Resource constraint . . . . . . . . . . . . . .
Government budget constraint . . . . . . .
Balance of payments . . . . . . . . . . . . .
Policy rules . . . . . . . . . . . . . . . . . .
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4
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3 Concluding remarks
16
Appendix: Equations for Dynare
17
References
19
2
1
Introduction
This paper aims to develop a model which yields (1) improved understanding of agricultural supply shocks and their amplification by financial frictions in the context
of low-income countries (LICs); and (2) monetary policy rules useful for managing
such shocks. Our Dynamic Stochastic General Equilibrium (DSGE) builds upon
the ‘financial accelerator’ elaborated in Bernanke et al. (1999) [BGG] and Gertler
et al. (2007) [GGN] while borrowing from the framework for “managing aid surges
in Africa” developed in Adam et al. (2008, 2009). This paper’s contribution to its
field of research lies in its adapting the mechanism of the financial accelerator to
agricultural production as well as taking into account the specific constraints faced
by macroeconomic-policy makers in LICs.
A critical feature of many LICs is that agriculture accounts for a relatively large
proportion of GDP, and a larger proportion of employment; in 2008 in Kenya,
agricultural value added represented 23% of GDP, while 79% of the population
was rural (Karugia et al., 2010). As a result, volatile weather conditions can be a
source of substantial shocks to the economies of LICs and to the livelihoods of their
populations. Additional shocks that can affect agricultural production, as well as
the economy more generally, come from the volatile cost of imported inputs, such as
fertiliser and oil (the cost of the two being closely linked in practice); this volatility
arises from fluctuations of the exchange rate as well as of the world cost of inputs.
These stylised facts help to explain the predominance of supply-side shocks over
demand-side shocks notable of LICs (Adam et al., 2010), which presents challenges
for monetary policy. Indeed, supply shocks lead to negatively correlated changes in
the output gap and in inflation, which means that inflation targeting frameworks
attractive to advanced economies can exacerbate output volatility. This is our
rationale for developing a model which captures the details of the supply side of
LICs to generate useful policy response functions.
The central premise on which our model is built is that farmers must borrow
from domestic households to purchase fertiliser in advance of receiving revenue from
selling their production, and therefore face a ‘cash-in-advance’ constraint. This
borrowing is subject to an external finance premium (EFP), i.e., the rate of interest
paid on borrowing exceeds the opportunity cost of funds raised internally (such
as retained earnings). The financial frictions caused by the EFP are amplified by
shocks to agricultural production from volatile weather conditions and imported
fertiliser costs, which in turn further constrains production – this is the financial
accelerator expounded below.
On the one hand, the fact that the supply shocks discussed above are exacerbated
by the financial accelerator worsens the trade-off between moderating inflation and
stabilising output. On the other hand, there is a potential for monetary policy to
ease borrowing costs for farmers, through a reduction in the ‘risk-free interest rate’
to compensate for a higher EFP; this suggests a less-unfavourable trade-off than
might otherwise be the case. We will use the model developed below to conduct
simulations, in a bid to assess the extent of this impact.
3
2
2.1
The model
Households
Let Ct be a composite of home- and foreign-produced consumption goods, CtH and
CtF , respectively. The following constant elasticity of substitution (CES) function
index then defines household preferences over consumption:
h
(ρ−1)/ρ
(ρ−1)/ρ iρ/(ρ−1)
Ct = (γ)1/ρ CtH
+ (1 − γ)1/ρ CtF
,
(1)
with γ ∈ (0, 1) a parameter reflecting the relative share of home- and foreignproduced goods in domestic consumption. The corresponding consumer price index
(CPI), Pt is
h
1−ρ
1−ρ i1/(1−ρ)
Pt = (γ) PtH
+ (1 − γ) PtF
,
(2)
where PtH and PtF are the (domestic-currency) prices of the domestically and
foreign-produced consumption goods, respectively. Household intertemporal utility is given by
h
i1−ν
∞
(Ct )1−ς (1 − Lt )ς
X
(3)
U0 = E0
βt
1−ν
t=0
with Lt the quantity of household labour supplied, β ∈ (0, 1) the discount factor,
ν ≥ 0, and ς ∈ (0, 1).
Denote Wt the nominal wage; Πt real dividends (paid to domestic households
by domestic retail firms); Tt lump-sum real tax payments; ΩH
t+1 nominal household
holdings of debt emitted by domestic farms or the domestic government1 ; and it the
domestic gross nominal interest rate. The household budget constraint is then
Ct =
ΩH − (1 + it−1 )ΩH
Wt
t
Lt + Πt − Tt − t+1
.
Pt
Pt
(4)
The (first-order) optimal consumption allocation condition, which is obtained by
maximising Ct subject to PtH CtH + PtF CtF being equal to a given constant, is
H −ρ
γ
Pt
CtH
=
.
(5)
F
1 − γ PtF
Ct
Substituting the expression for Ct into the instantaneous utility function, and maximising with respect to Lt , yields the following (first-order) optimality condition for
labour supply:
(1 − ς)
1
1 Wt
=ς
.
Ct Pt
1 − Lt
(6)
Finally, maximising the household’s intertemporal utility, subject to the summation
over t of the household’s budget constraints, and with respect to Ct and Bt+1 , yields
the following (first-order) optimality condition for consumption and saving:
Pt
λt = βEt λt+1 (1 + it )
,
(7)
Pt+1
1
We assume that domestic households can only buy domestic debt; details to follow below.
4
where λt , the Lagrangian multiplier on the budget constraint at time t, which is
also the marginal utility of the consumption index at t, is given by
λt = (1 − ς) (Ct )(ν−1)(ς−1)−1 (1 − Lt )ς(1−ν) .
The ex-ante real interest rate, rt+1 , is defined as2 :
Pt
1 + rt+1 = Et (1 + it )
.
Pt+1
2.2
(8)
(9)
Foreign behaviour
Let St be the nominal exchange rate defined in units of domestic currency per unit of
foreign currency, so that an increase in St is a depreciation of the domestic currency.
The law of one price holds for wholesale goods (but not for the retail goods which
are ultimately consumed, as these are differentiated by monopolistically competitive
F and P F ∗ the price of foreign wholesale goods in
retail firms). Therefore, with PW,t
W,t
domestic and foreign currency respectively, it follows that
F
F∗
PW,t
= St PW,t
.
(10)
F ∗ is determined exogenously, and that foreign demand for the
We assume that PW,t
home tradeable good CtH∗ is given by
CtH∗ =
"
PtH∗
Pt∗
−
#ζ
Yt∗
H∗
Ct−1
1−ζ
,
0 ≤ ζ ≤ 1, > 0
(11)
where Yt∗ is real foreign output, Pt∗ is the price of real foreign output (that is, the
foreign CPI), PtH∗ is the foreign-currency price of domestic consumption goods, and
H∗ 1−ζ term indicates inertia in foreign demand for domestic products. The
the Ct−1
fact that the domestic economy is small (in the sense that it cannot affect foreign
variables) helps to motivate an empirically reasonable reduced-form export demand
curve. We also assume balanced trade in the steady state, and normalise the steadyF /P H = 1, where X
b denotes the steady-state
state terms of trade to unity, so that P\
value of X.
2.3
2.3.1
Farms
Agricultural production
There is a continuum of farms3 over [0, 1], each of which produces homogeneous
wholesale consumption goods with a CES production function Yt , given by
h
i1/δ
Yt = ωt At αMtδ + (1 − α)Lδt
(12)
where ωt is a firm-specific productivity shock; At is a common productivity factor; Mt is an imported intermediate good (e.g., fertiliser4 ); Lt is the labour input;
2
Note that rt+1 is defined in terms of it rather than it+1 .
I will use the terms ‘farm’ and ‘farmer’ interchangeably.
4
This could also be fuel, which is necessary to take production to market; in any case the cost of fuel
and fertiliser should be highly correlated in practice.
3
5
α ∈ (0, 1) is the share parameter; and δ ∈ (−∞, 0) is a parameter related to the
elasticity of substitution Σ by δ = (Σ − 1)/Σ ⇐⇒ Σ = 1/(1 − δ). As δ → −∞
the production function becomes Leontief (fixed proportions), while as δ → 0 the
production function becomes Cobb-Douglas. BGG and the literature that has followed use the Cobb-Douglas production function; in contrast, we will use the more
general CES production function, which allows us to impose a relatively low degree of substitutability between labour and fertiliser (while stopping short of the
zero-substitutability Leontief function).
The arguments of the production function Yt are omitted for notational convenience, and Yt exhibits constant returns to scale (CRS). We assume that production
of wholesale goods does not require any physical capital. We further assume that
all farms have the same fixed amount of land available, and that land does not need
purchasing or hiring (because it is conserved within infinitely lived households).
Farmers’ production is subject to the idiosyncratic shock ωt which is i.i.d. across
time and farms, with a continuous and once-differentiable c.d.f., F(ω), over a nonnegative support. Specifically, we assume that ωt is log-normally distributed, such
that
1 2 2
ln ωt ∼ N − σω , σω , ∀t.
(13)
2
It follows that Et−1 (ωt ) = 1, ∀t.
We interpret the common productivity factor, At , as reflecting weather conditions; we will therefore assume that changes in At between periods will be large
enough to trigger substantial output fluctuations, as will be reflected in our parameterisation. Higher values of our ‘weather variable’ can be interpreted as more
favourable weather conditions. Let At be determined, relative to its steady-state
b by the stationary AR(1) process
value A,
h
i
b + ηA,
b = ρA ln (1 + At−1 ) − ln(1 + A)
(14)
ln (1 + At ) − ln(1 + A)
t
2 5.
where ρA ∈ (0, 1) and ηtA ∼ N 0, σA
At time t, the farmer purchases fertiliser for use at time t+16 . The ex-post gross
return to using a unit of fertiliser from t to t + 1 (‘return on fertiliser’) for a farm is
M ), where (1 + r M ) is the ex-post gross return to fertiliser averaged over
ωt+1 (1 + rt+1
t+1
farms (which does not depend on realisations of the idiosyncratic shock because ωt
is i.i.d. across farms, and the number of farms is large).
2.3.2
Working capital
Farmers need ‘working capital’ to finance their expenditure on fertiliser. We assume
that labour is only required at the harvesting stage and does not need paying for in
advance, so that no borrowing is required to purchase labour. At the end of period
t (going into period t + 1), the farmer has net worth of Nt+1 available, which is
defined as liquid assets plus collateral value of illiquid assets net of liabilities. In the
present model, farmers start off with some exogenously given (perhaps inherited) net
5
The weather variable could be given a seasonal component if periods are interpreted as quarters.
A possible interpretation could be that the fertiliser must be purchased and dissolved into the ground
in period t as the crops are planted, but production only occurs when the crops are harvested in period
t + 1.
6
6
worth, and subsequent additions to their net worth come from retained earnings.
We assume that land cannot be offered as collateral because of a failure in the
market for land7 . Fertiliser attracts an ad valorem susbsidy at a fixed rate. To be
able to purchase fertiliser net of the subsidy in excess of the value of her net worth,
the farmer needs (uncollateralised) nominal debt Bt+1 such that
Bt+1
= κmt Mt+1 − Nt+1 ,
Pt
(15)
where mt is the real local-currency price of fertiliser purchased at time t, which
depends on the world price of fertiliser and the exchange rate (as specified later);
(1 − κ) ∈ [0, 1) is the rate at which fertiliser is subsidised, so that κ ∈ (0, 1] is the
proportion of the value of fertiliser not covered by the government subsidy. The
shocks in our model are calibrated so that farmers cannot acquire enough net worth
to be fully self financing and therefore always need to borrow to purchase fertiliser.
Working capital is provided by a financial intermediary which obtains its funds
exclusively from domestic households. The relevant opportunity cost of funds between periods t and t + 1 is the economy’s riskless gross rate of return (1 + rt+1 ).
This is the relevant opportunity cost because idiosyncratic credit risk is perfectly
diversified away by the intermediary, and the farmers, whom we assume to be risk
neutral, absorb any aggregate risk to relieve risk-averse savers from potential losses.
In the presence of aggregate risk (as is the case here due to stochastic weather
conditions), the borrower guarantees the lender a return that is free of any systemM ), the borrower offers a
atic risk: conditional on the ex-post realisation of (1 + rt+1
(state-contingent) non-default payment that guarantees the lender a return equal
in expected value to the riskless rate.
2.3.3
Financial Frictions
We introduce a costly state-verification problem8 wherein the financial intermediary
must pay an ‘auditing cost’ to observe an individual farmer’s realised production
(the farmer observes her own production costlessly), which can be interpreted as
reflecting the cost of bankruptcy; these costs are likely to be all the more significant
given the weak insolvency frameworks, inefficient courts, etc., which one is likely to
encounter in low-income countries. The monitoring cost is assumed to be a fixed
proportion τ ∈ (0, 1) of the realised gross payoff to the farm’s fertiliser, so that the
cost of monitoring a farm at time t + 1 is
M
τ ωt+1 (1 + rt+1
)κmt Mt+1 .
(16)
M
The optimal contract features a gross non-default loan rate, Rt+1 , and a threshold
value of the idiosyncratic shock, ω t+1 , such that for ωt+1 ≥ ω t+1 the farmer is able
M
to repay the loan at the rate of interest Rt+1 ; therefore, ω t+1 is given by
M
M
ω t+1 (1 + rt+1
)κmt Mt+1 = Rt+1
Bt+1
.
Pt
(17)
M
Thus, if ωt+1 ≥ ω t+1 , the farmer pays the lender Rt+1 BPt+1
, and keeps
t
M
M
ωt+1 (1 + rt+1
)κmt Mt+1 − Rt+1
7
8
Bt+1
≥0
Pt
This could notably arise from poorly defined and enforced property rights.
As first studied by Townsend (1979).
7
(18)
in profit. If ωt+1 < ω t+1 , the farmer declares bankruptcy and receives nothing, and
the lender pays the auditing cost and keeps what is left. Therefore, the intermediary’s net recepits in this case are
M
(1 − τ )ωt+1 (1 + rt+1
)κmt Mt+1 .
(19)
The expected gross return to the lender must at least equal the intermediary’s
opportunity cost of lending. Therefore, assuming that this constraint binds, the
M
values of ω t+1 and Rt+1 under the optimal contract are given by
Z ωt+1
M Bt+1
M
[1 − F(ω t+1 )] Rt+1
+ (1 − τ )
ωt+1 (1 + rt+1
)κmt Mt+1 dF(ωt+1 )
Pt
0
Bt+1
= (1 + rt+1 )
, (20)
Pt
where F(ω t+1 ) is the probability of default. Combining equations (15) and (17)
with equation (20) gives
Z ωt+1
M
[1 − F(ω t+1 )] ω t+1 + (1 − τ )
ωt+1 dF(ωt+1 ) (1 + rt+1
)κmt Mt+1
0
= (1 + rt+1 ) (κmt Mt+1 − Nt+1 ) .
(21)
M
Eliminating Rt+1 allows us to express the lender’s expected return simply as a
function of the cutoff value of the farm’s idiosyncratic productivity shock, ω t+1 .
Under the assumed distribution of the idiosyncratic shock, the expected return
reaches a maximum at an unique interior value of ω t+1 . For simplicity, we restrict
attention to equilibria where lending is not rationed, that is, where the equilibrium
value of ω t+1 is always less than its maximum feasible value.
Given aggregate risk due to stochastic weather conditions, equation (21) implies
a set of restrictions, with one for each realisation of At+1 , resulting in a schedule for
M
ω t+1 , contingent on realised weather. The loan rate Rt+1 is countercyclical: worsethan-expected weather leads to a higher non-default interest rate to compensate for
the greater probability of a default occurring; this in turn implies a higher cutoff
value of the idosyncratic shock, ω t+1 . Thus, default probabilities and risk premia
rise when lower-than-expected weather leads production to fall.
The farmer’s expected return is the expectation (with respect to the weather)
of profits in states of the world where the realised idiosyncratic shock is above its
cutoff value ω t+1 , net of loan repayments:
(Z
)
∞
B
M
t+1
M
(22)
Et
ωt+1 (1 + rt+1
)κmt Mt+1 dF(ωt+1 ) − [1 − F(ω t+1 )] Rt+1
Pt
ω t+1
which equals
(Z
)
∞
M
M
Et
ωt+1 (1 + rt+1
)κmt Mt+1 dF(ωt+1 ) − [1 − F(ω t+1 )] ω t+1 (1 + rt+1
)κmt Mt+1 ,(23)
ω t+1
and where it is understood that ω t+1 may be made contingent on the realisation of
the weather at t + 1. It follows from equation (21) that
Z ωt+1
M
M
)κmt Mt+1 = −
)κmt Mt+1 dF(ωt+1 )
[1 − F(ω t+1 )] ω t+1 (1 + rt+1
ωt+1 (1 + rt+1
0
Z ωt+1
M
+τ
ωt+1 (1 + rt+1
)κmt Mt+1 dF(ωt+1 ) + (1 + rt+1 ) (κmt Mt+1 − Nt+1 )(24)
0
8
which we can substitute into expression (23) to give the following expression for the
farmer’s expected return:
"Z
#
∞
M
ωt+1 (1 + rt+1
)κmt Mt+1 dF(ωt+1 ) − Et [·]
Et
(25)
ω t+1
where Et [·] is the expected value of the right-hand side of equation (24). Rearranging
expression (25) gives
Z ωt+1
rM
M
Et 1 − τ
ωt+1 dF(ωt+1 ) Ut+1 Et (1 + rt+1
) κmt Mt+1
0
− (1 + rt+1 ) (κmt Mt+1 − Nt+1 ) ,
as the expression which the farmer seeks to maximise, where
rM
M
M
Ut+1
≡ (1 + rt+1
)/Et (1 + rt+1
) ,
(26)
(27)
and where the ‘−τ ...’ term in the large square brackets indicates that the farmer
internalises the expected default cost. The problem is then to choose Mt+1 and a
schedule for ω t+1 (as a function of realised weather) to maximise expression (26),
subject to the set of state-contingent constraints implied by equation (21).
2.3.4
Net worth and demand for fertiliser
For farmers to be willing to purchase fertiliser in the competitive equilibrium, a
necessary condition is that
"
#
M
1 + rt+1
st ≡ Et
≥ 1,
(28)
(1 + rt+1 )
that is, the expected discounted return to fertiliser must be at least unity. The
variable st can also be interpreted as the EFP. Indeed, in equilibrium farmers will
equate the discounted marginal return to fertiliser to the marginal cost of external
finance; if st is strictly greater than unity, there is a ‘wedge’ between the cost of
external finance and the risk-free rate. Given st ≥ 1, the first-order conditions imply
that optimal fertiliser purchases satisfy
κmt Mt+1 = χ (st ) Nt+1 ,
(29)
where χ(·) is a strictly increasing function with χ(1) = 1. This relation shows
that fertiliser purchases by each farm are proportional to the farmer’s net worth,
and the proportionality factor is increasing in the expected discounted return to
fertiliser, st . Ceteris paribus, a rise in the expected discounted return to fertiliser
reduces the expected default probability; therefore, the farmer can borrow more
and increase the size of her farm, within the constraints imposed by the fact that
expected default costs increase as leverage (the ratio of total assets to net worth)
increases. Rearranging equation (29) as
κmt Mt+1
= χ (st )
Nt+1
(30)
shows the positive relation between leverage, which is the left-hand side of equation
(30), and the EFP.
9
Higher leverage drives further apart the interests of the borrower and the lender.
For a given amount of total assets, a lower proportion of equity (net worth) and a
correspondingly higher proportion of debt mean that the farmer has less to lose if
her venture turns out to be unprofitable. Therefore, at least from the point of view
of the lender, a more highly leveraged borrower carries a higher risk of default amid
information asymmetries between lender and borrower, other things being equal. In
the context of a high- or middle-income country, a possible explanation would be
that higher leverage leads borrowers to take on higher-risk projects, since they would
stand to make a higher return on equity on the upside, while having less to lose on
the downside. In a low-income country, it seems more plausible to assume that a
more highly leveraged farmer has a lesser incentive to exercise ‘due care’ or exert
‘appropriate effort’. This explains the negative relationship between net worth and
the EFP. Procyclical net worth thus leads to a countercyclical EFP, so that financial
frictions magnify economic downturns by increasing the cost of external funds: this
is the ‘financial accelerator’ introduced by Bernanke et al. (1996).
The farmer’s demand for fertiliser satisfies the optimality condition
M
Et 1 + rt+1
= [1 + χt (·)] Et [1 + rt+1 ]
(31)
where Et [1 + rt+1 ] would be the (expected) gross cost of funds in the absence of
financial frictions.
Given that the production function Yt is CRS, we can write the production
function as an aggregate relationship. Let PW,t denote the nominal price of wholesale
output, so that PW,t /Pt is the real price of wholesale output. We continue to assume
that fertiliser must be purchased at the end of one period for use in the following
period, so that κmt−1 is the cost of fertiliser that is used in period t. The (riskneutral) farmers choose fertiliser and labour inputs, for given production and factor
prices, to maximise the expected value of
i1/δ
PW,t h
Wt
At αMtδ + (1 − α)Lδt
− κmt−1 Mt −
Lt .
(32)
Pt
Pt
The first-order condition with respect to fertiliser is
h
i(1−δ)/δ
PW,t 1
− κmt−1 = 0,
At δαMtδ−1 αMtδ + (1 − α)Lδt
Pt
δ
(33)
which gives
PW,t
1/(1−δ) 1−δ
αMtδ−1 At
Yt
= κmt−1 ,
Pt
(34)
and therefore
Mt =
αPW,t
Pt κmt−1
1/(1−δ)
1/(1−δ)2
At
Yt ,
(35)
or equivalently
Mt =
αPW,t
Pt κmt−1
Σ
2
AΣ
t Yt ,
(36)
as the amount of fertiliser used for a given desired production level Yt , since Σ =
1/(1 − δ) and δ < 0 so 1/(1 − δ)2 = Σ2 . Similarly,
[(1 − α)PW,t ] /Pt Σ Σ2
Lt =
At Yt ,
(37)
Wt /Pt
10
giving
(1 − α)PW,t
Lt =
Wt
Σ
2
AΣ
t Yt .
(38)
Farmers are assumed to produce in a perfectly competitive market for wholesale
goods, to facilitate aggregation. We introduce ‘nominal stickiness’ by assuming
that farmers sell their (wholesale) production to retail firms which operate in a
monopolistically competitive market for final goods. Retailers transform wholesale
goods into differentiated final goods, and rebate their profits to households as a
lump sum. Rearranging equation (36) gives
PW,t
Yt 1/Σ
Σ
κmt−1 =
αAt
,
(39)
Pt
Mt
which is the usual condition that the farmer equates the value of the marginal
product of fertiliser to the cost of the factor. It then follows that
Yt 1/Σ
Pt κmt−1
Σ
= αAt
.
(40)
PW,t
Mt
The rent paid to a unit of fertiliser, in terms of retail prices, is
PW,t
Yt 1/Σ
Σ
αAt
,
Pt
Mt
(41)
and the expected gross return to using a unit of fertiliser from t to t + 1 can then
be written as

1/Σ 
"
PW,t+1
Yt+1
#
Σ
PW,t+1
1
Yt+1 1/Σ

 Pt+1 αAt Mt+1
Σ
M
Et
αAt
Et 1 + rt+1 = Et 
,(42)
=
κmt
κmt
Pt+1
Mt+1
noting that this is both the marginal and the average return to capital, given constant returns to scale. Substituting in the formula for Yt+1 gives

( 1/δ )1/Σ 
δ + (1 − α)Lδ
A
αM
PW,t+1
1
t
t+1
t+1
M
 ; (43)
Et 1 + rt+1
=
Et 
αAΣ
t
κmt
Pt+1
Mt+1
which simplifies to
M
Et 1 + rt+1
=

P
"
1
W,t+1
Σ+1/Σ
Et
αAt
α + (1 − α)
κmt  Pt+1
Lt+1
Mt+1

δ #(1−δ)/δ 
, (44)

yielding a demand function for fertiliser. To obtain a supply function for finance,
we aggregate equation (29) over farms and invert χ(·), which yields
Nt+1
M
Et 1 + rt+1 = s
(1 + rt+1 )
(45)
κmt Mt+1
where s(·) is a decreasing function for Nt+1 < κmt Mt+1 ; it is also the ratio of the
costs of external and internal finance, and this ratio is clearly increasing in the
farmer’s leverage.
11
We assume that farmers start off with an amount of net worth N0 > 0. Farmers’
net worth is thereafter determined by
)
(
Rω
τ 0 t ωt (1 + rtM )κmt−1 Mt dF(ωt )
Bt
M
.(46)
Nt+1 = (1 + rt )κmt−1 Mt − (1 + rt ) +
Bt
P
t−1
P
t−1
That is, net worth at the end of period t, Nt+1 , equals gross earnings on fertiliser
used from t − 1 to t less borrowing repayments; the rate of borrowing repayments
is the term in square brackets, which is the risk-free rate plus an external finance
premium given by the ratio of default costs to quantity borrowed. Substituting in
the expression for the amount borrowed yields a difference equation for farmers’ net
worth:
Nt+1 = (1 + rtM )κmt−1 Mt
(
)
Rω
τ 0 t ωt (1 + rtM )κmt−1 Mt dF(ωt )
− (1 + rt ) +
(κmt−1 Mt − Nt ) .
κmt−1 Mt − Nt
(47)
This difference equation and the supply function for finance, equation (45), are the
two ‘basic ingredients’ of the financial accelerator.
2.4
Retailers, price setting, and inflation
There is a continuum of retail firms of measure one, with each such firm indexed by
z ∈ (0, 1). Retail firms purchase wholesale goods in a perfectly competitive market,
and differentiate them at a fixed resource cost ξ which represents distribution and
selling costs (assumed to be proportional to the steady-state value of wholesale
output). Let YtH (z) be the good sold be retailer z, and total final goods YtH be
given by a CES composite of individual retail goods net of fixed resource costs:
Z 1
ϑ/(ϑ−1)
H (ϑ−1)/ϑ
H
Yt =
Yt (z)
dz
−ξ
(48)
0
where ϑ > 1. The corresponding price index is
Z 1
1/(1−ϑ)
H 1−ϑ
H
Pt =
Pt (z)
dz
.
(49)
0
It follows that the demand curve facing retailer z is given by
H −ϑ
Pt (z)
H
Yt (z) =
Yt .
PtH
(50)
The marginal cost to retailers of producing a unit of output is the relative wholesale
price, PW,t /PtH . We assume Calvo (1983) pricing, with each retailer able to change
her price with constant probability (1 − θ) in any period, so that in each period a
fraction (1 − θ) of retailers reset their price while a fraction θ ∈ (0, 1) keep their
prices unchanged. The average time that a price remains fixed is then 1/(1 − θ).
Since there are no firm-specific state variables, all retailers choose the same optimal
H
price P t . It can be shown that, in the neighbourhood of the steady sate, the
domestic price index evolves according to
H 1−θ
H θ
PtH = Pt−1
Pt
.
(51)
12
Retailers able to set their prices in a given period do so to maximise expected
discounted profits, subject to the constraint on the frequency of price adjustments.
It can be shown that, within a local neighbourhood of the steady state, the optimal
price is
H
Pt = µ
∞
Y
i
(PW,t+i )(1−βθ)(βθ) ,
(52)
i=0
where
µ=
1
1 − 1/ϑ
(53)
is the retailers’ desired gross mark-up over wholesale prices. Because prices may be
fixed for some time, retailers set prices based on the expected future path of marginal
cost. Combining equations (51) and (52) above gives the following expression for
domestic inflation (within the neighbourhood of a zero-inflation steady state):
!β
H
Pt+1
PW,t λ
PtH
Et
,
(54)
= µ H
H
Pt−1
Pt
PtH
with λ = (1−θ)(1−βθ)
a parameter decreasing in θ, the measure of price rigidity.
θ
This is a ‘Phillips curve’ (for domestically produced goods alone) closely related
to the ‘standard’ New-Keynesian Phillips curve (NKPC) which describes a positive
relationship between current inflation on the one hand and expected future inflation
and the output gap on the other hand. The ‘Phillips curve’ given by equation (54)
replaces the output gap term with a marginal cost term, given by the relative
wholesale price, as suggested by Galı́ and Gertler (1999). These authors consider
measures of marginal costs as the relevant determinant of inflation to be more
consistent with economic theory than an ad-hoc output gap. They also find this
version of the NKPC to be a good first-approximation to the dynamics of inflation
(for US data), unlike the ‘standard’ NKPC.
F , expressed in curRetailers of foreign goods face the wholesale good price PW,t
rency units of the domestic country, and satisfying. Retailers of foreign goods set
prices in Calvo fashion, with probability of price change 1 − θf . The inflation rate
for foreign goods is then given by
PtF
=
F
Pt−1
µ
PF
f W,t
! λf
Et
PtF
F
Pt+1
PtF
!β
,
(55)
with µf and λf defined analogously to the domestic case, and given that
F = S P F ∗ it follows that
PW,t
t W,t
PtF
=
F
Pt−1
µ
S PF∗
f t W,t
!λf
Et
PtF
F
Pt+1
PtF
!β
.
(56)
This specification of the pricing process for domestically sold foreign goods implies
temporary deviations from the law of one price due to delayed exchange rate passthrough, with θf capturing the degree of this delay9 .
9
GGN set θ = θf in their calibration.
13
Domestic CPI inflation is a composite of domestic and foreign good price inflation, and is given by (within a local region of the steady state):
!γ
!1−γ
PtF
Pt
PtH
.
(57)
=
H
F
Pt−1
Pt−1
Pt−1
2.5
Resource constraint
The resource constraint for the domestic tradeable good sector is
YtH = CtH + CtH∗ .
2.6
(58)
Government budget constraint
The government only spends on fertiliser subsidies and debt (principal and interest)
repayments. Government expenditure is financed by real lump-sum taxes Tt , which
adjust as appropriate to satisfy the budget constraint; new borrowing; and real
(foreign-currency) net budgetary aid at . The government borrows by issuing oneperiod domestic-currency nominal bonds Dt on which it pays the nominal interest
rate it in arrears. In real terms, the government pays an amount of interest it−1 Dt /Pt
on its borrowings. Indexing by H and F respectively the domestic government bonds
held by domestic and foreign investors, we have:
Dt = DtH + DtF ,
(59)
and analogously for farm bonds:
Bt = BtH + BtF .
(60)
Total nominal domestic debt held by foreign investors, in domestic currency, is given
by:
ΩFt = DtF + BtF ,
(61)
and ΩH
t is defined analogously. Foreign investors can buy both foreign and domestic
debt, but we assume for simplicity that domestic investors can only buy domestic
debt.
Let PtF ∗ be the price of foreign-produced consumption goods in foreign currency.
Then the real exchange rate, in units of foreign-produced consumption goods per
unit of domestically produced consumption goods, is given by
et =
St PtF ∗
.
PtH
(62)
The budget constraint faced by the domestic government, in real terms, is therefore:
(1 − κ)et m∗t Mt+1 + it−1
Dt
∆Dt+1
= Tt + et at +
Pt
Pt
(63)
where m∗t is the real foreign currency price of fertiliser, so mt = et m∗t ; the first term
on the left-hand side (LHS) represents real expenditure on fertiliser subsidies; the
second term on the LHS represents real interest payments on government debt; and
14
the final term on the right-hand side represents new government borrowing at the
end of period t to for the budget constraint to be satisfied.
The real price of fertiliser in foreign currency, m∗t , is determined, relative to its
steady-state value m
b ∗ , by the stationary AR(1) process
∗ ∗
ln (1 + m∗t ) − ln (1 + m
b ∗ ) = ρm ln 1 + m∗t−1 − ln (1 + m
b ∗ ) + ηtm ,
(64)
∗
∗
2
where ρm ∈ (0, 1) and ηtm ∼ N 0, σm
∗ .
2.7
Balance of payments
Supply of funds by the foreign investor is given implicitly by the arbitrage condition
F
Et [et+1 ] −γ
Ωt /Pt
∗
φ
(65)
1 + rt+1 = 1 + rt+1
et
YtH
∗
where rt+1
is the exogenous world real interest rate, and φ > 0 is a parameter
denoting the slope of the schedule for the supply of funds from abroad. The φ(·)
term corresponds to the risk premium, since the (foreign) investors need a higher
yield to compensate them for the greater risk which comes with an increased debt-toGDP ratio. As φ −→ 0, the arbitrage condition reduces to the uncovered interest
parity condition which obtains amid perfect asset substitutability. As φ −→ ∞
foreign investors demand an infinite risk premium, and the government can no longer
borrow from abroad (or indeed at all, under our assumptions), resulting in a closed
capital account. The intermediate case we consider, where φ is strictly positive and
finite, corresponds to the case in which the capital account is ‘imperfectly open’,
as is common in much of Sub-Saharan Africa (Adam et al., 2008). The world real
interest rate rt∗ is determined, relative to its steady-state value rb∗ , by
∗ ∗
∗
ln (1 + rt∗ ) − ln (1 + rb∗ ) = ρr ln 1 + rt−1
− ln (1 + rb∗ ) + ηtr ,
(66)
∗
∗
where ρr ∈ (0, 1) and ηtr ∼ N 0, σr2∗ .
The current account identity, in real terms, is
et m∗t Mt+1 + it−1
∆ΩFt+1
ΩFt
= YtH − CtH − CtF + et at +
Pt
Pt
(67)
so that payments to abroad, for importing fertiliser and servicing debt, are equal to
GDP net of consumption plus net budgetary aid from abroad and the decrease in
net foreign assets (that is, the increase in net debt owed to the foreign country).
2.8
Policy rules
We will consider shocks to the economy under two different frameworks: a fixed
exchange rate regime, and a floating exchange rate regime in which monetary policy
is conducted through the nominal interest rate.
Under the fixed exchange rate, the nominal exchange rate is set at a constant
level:
St = S, ∀t.
(68)
The nominal interest rate is then set to satisfy the arbitrage condition given by
equation (65) above.
15
Under the flexible exchange rate, the nominal interest rate is determined by a
interest rate rule under which it reacts to deviations of CPI inflation and domestic
output from their target values. We assume that the target for the gross inflation
rate is unity; and that the output target level, Yt0 , is the level of output that would
obtain under fully flexible prices. The interest rate rule is then given by
γ
Pt γπ YtH y
1 + it = (1 + rb)
,
(69)
Pt−1
Yt0
where γπ > 1 and γy > 0, and where rb is the steady-state real interest rate. We
take this rule to be a form of flexible inflation targeting, or ‘constrained discretion’,
so that the central bank adjusts the interest rate to achieve the inflation target in
the medium run, while enjoying flexibility in the short run to meet output stabilisation objectives. We assume that the central bank is fully independent of elected
politicians, and is able credibly to commit to following the interest rate rule.
3
Concluding remarks
This paper has developed a model which seeks to provide an adequate characterisation of the economy of a typical LIC. We next intend to use this model to conduct
simulations which will shed light on optimal policy responses to shocks from variable
weather conditions and from the volatile local-currency cost of fertiliser. Specifically, we intend to calibrate the model to the Kenya’s economy, to yield answers to
questions that have arisen in the context of that country.
A natural extension to the model as it stands, and which we intend to include
in due course, is to bring ‘financial frictions’ to bear on households as well as farms.
We intend to do so by restricting the ability of a proportion of households to save
and borrow, so that these ‘hand-to-mouth’ consumers are restricted to consuming
exclusively current income. This would add to the realism of the model, given that
in LICs, and notably in Kenya, there is typically a significant proportion of the
population with limited access to financial services.
16
Appendix: Equations for Dynare
Demand side
YtH = CtH + CtH∗
(70)
λt = (1 − ς) (Ct )(ν−1)(ς−1)−1 (1 − Lt )ς(1−ν)
(71)
Pt
λt = βEt λt+1 (1 + it )
Pt+1
(72)
h
(ρ−1)/ρ
(ρ−1)/ρ iρ/(ρ−1)
Ct = (γ)1/ρ CtH
+ (1 − γ)1/ρ CtF
(73)
γ
CtH
=
F
1
−
γ
Ct
PtH
PtF
−ρ
(74)
h
1−ρ
1−ρ i1/(1−ρ)
Pt = (γ) PtH
+ (1 − γ) PtF
CtH∗ =
"
Lt =
M
Et 1 + rt+1
PtH∗
Pt∗
−
#ζ
Yt∗
(1 − α)PW,t
Wt
H∗
Ct−1
Σ
1−ζ
2
AΣ
t Yt


"
δ #(1−δ)/δ 
P
L
1
W,t+1
Σ+1/Σ
t+1
=
Et
αAt
α + (1 − α)

κmt  Pt+1
Mt+1
(75)
(76)
(77)
(78)
Supply side
h
i1/δ
Yt = ωt At αMtδ + (1 − α)Lδt
(1 − ς)
Et 1 +
M
rt+1
1
1 Wt
=ς
Ct P t
1 − Lt
=s
Nt+1
κmt Mt+1
17
(79)
(80)
(1 + rt+1 )
(81)
Pricing, interest rates, exchange rates
PtH
=
H
Pt−1
PW,t λ
µ H
Et
Pt
1 + rt = (1 + it )
PtF
=
F
Pt−1
µ
Pt
=
Pt−1
S PF∗
f t W,t
Et
et =
!γ
!β
(82)
Pt
Pt+1
! λf
PtF
PtH
H
Pt−1
H
Pt+1
PtH
PtF
F
Pt−1
(83)
F
Pt+1
PtF
!β
(84)
!1−γ
(85)
St PtF ∗
.
PtH
(86)
Budget constraints and other identities
Bt+1
= κmt Mt+1 − Nt+1
Pt
(87)
Nt+1 = (1 + rtM )κmt−1 Mt
(
)
Rω
τ 0 t ωt (1 + rtM )κmt−1 Mt dF(ωt )
(κmt−1 Mt − Nt )
− (1 + rt ) +
κmt−1 Mt − Nt
(1 − κ)et m∗t Mt+1 + it−1
1 + rt+1 = 1 +
et m∗t Mt+1 + it−1
∗
rt+1
Dt
∆Dt+1
= Tt + et at +
Pt
Pt
Et [et+1 ]
et
−γ
φ
ΩFt /Pt
YtH
(88)
(89)
∆ΩFt+1
ΩFt
= YtH − CtH − CtF + et at +
Pt
Pt
(90)
(91)
Exogenous processes and policy rules
h
i
b = ρA ln (1 + At−1 ) − ln(1 + A)
b + ηtA
ln (1 + At ) − ln(1 + A)
∗
∗
∗
∗
ln 1 + rt−1
− ln (1 + rb∗ ) + ηtr
ln (1 + m∗t ) − ln (1 + m
b ∗ ) = ρm
ln (1 + rt∗ ) − ln (1 + rb∗ ) = ρr
1 + it = (1 + rb)
Pt
Pt−1
γπ ∗
ln 1 + m∗t−1 − ln (1 + m
b ∗ ) + ηtm
YtH
Yt0
(92)
(93)
(94)
γy
...
18
or
St = S, ∀t
(95)
(96)
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19