Sachs et al. (2004) re-ignited the arguments that most developing countries (especially in SSA region) are caught up in a “poverty trap,” a geographically rooted vicious circle of poverty driven by low agricultural productivity, heavy disease burdens and state of relative isolation. The notion of poverty traps itself goes as far back to the works of Thomas Malthus in the late 18th century. In the 1950s, the idea was revived by Nurkse (1953) when discussing the ‘vicious circle of poverty’ model and Nelson (1956) in the
2For detailed analysis of the fungibilty of aid, Pack and Pack (1990), Feyzioglu et al. (1996), Feyzioglu et al. (1998) Easterly and Rebelo (1993) are among many scholars that have conducted extensive analysis of the issue of fungibility of aid
‘theory of low-level equilibrium trap.”
Poverty traps are self-reinforcing inefficient steady states at low levels of per capita in- comes resulting from both market and institutional failures. Collier (2007) identified four significant mechanisms that lead to poverty traps in which the world’s poorest societies are caught up; internal conflict traps, natural resources traps, land locked by bad neighbour trap and bad governance traps.
The underlying framework of poverty trap models is the neoclassical growth theory, with threshold effects and non-linearities, suggesting that for some reason an economy may exhibit multiple long run equilibria that are history dependent, depending on the start-off points. Thus countries with similar structural characteristics but different initial condi- tions may face different (good or bad) long run economic growth experiences, and these are particularly bad for poor countries and hence the justification for a ‘big push’ to release them from the bad equilibrium. Key to the transition from the classical to the neoclassical models was the emergence of the concept of the individual consumer who maximizes utility in the same manner that firms seek to maximize profits. Individuals make their choices based on preferences, which can be optimized on rational basis, among possible outcomes of their economic activities.
The Solow-Swan Model
The most commonly used framework in the big push models is the Solow3-Swan4 growth
theory (which has become the traditional starting point for review of modern proximate growth theory, and is most popularly known merely as the ’Solow growth model’). The
model is built on four variables: output (Y) referring to total amount of production of
final good; capital (K) corresponding to ‘machines and structures’ used in production;
labor (L) referring to hours of employment or number of employees. Finally, technology
(A) has no specific measure and is used merely as a shifter of the production function;
it is assumed in this model (and in neoclassical growth models in general) to be a freely
available non-excludable and non-rival good (Acemoglu (2009a)).
3Solow (1956) 4Swan (1956)
Households and Production In its basic framework, the model considers a closed economy in discrete time with an infinite horizon and a unique final good. For simplicity the model assumes a homogeneous representative household/agent so that the demand and labour sector of the economy behave as a single household. The Solow model does not explicitly model the household’s optimisation decision so few assumptions are made
about the households behaviour, other than that they save a constant fraction,s ∈(0,1),
of their disposable income.
Other than households, the other key participants in the Solow model are firms. Again, for simplicity the model assumes firms to be homogeneous, implying a representative firm that faces a representative or aggregate production function for the final good given as:
Y(t) = F(A(t), K(t), L(t)) (2.1.6)
wheret is a time index and the variablesY(t),K(t), L(t) andA(t) are as defined earlier.
The production function F is assumed to be twice differentiable and for all values ofK >0
and L > 0, the function has positive but diminishing marginal products for both factor
inputs K and L i.e.FK ≡ ∂F/∂K > 0; FL ≡ ∂F/∂L > 0 and FKK ≡ ∂
2F
/∂2K < 0; FLL ≡
∂2F
/∂2L < 05 (i.e. the production function satisfies the Inada Conditions). Furthermore,
F is linearly homogeneous i.e. exhibits constant returns to scale inK and L.
Endowments, Market Structure and Market Clearing To be in a position to
determine the allocation of resources within the economy, initial stocks of the factor inputs must be specified and the owners of such endowments. The benchmark model assumes that markets are competitive and both households and firms are price takers, each pursuing their own objectives and prices clear markets. In this economy, households own factors of production; they own all labour which they supply to firms inelastically
(endowment of labour equals population, ¯L(t)) and is supplied at any non-zero rental
price, w(t), hence the labour market clearing condition is expressed as
L(t) = ¯L(t)
Households also own the capital stock of the economy and rent it to firms at a rental
price of capital R(t). The capital market clearing condition is that demand for capital
5To conserve on notation, we refer toF
K to represent the partial derivative of the production function with respect the particular factor input, in this example FK means the partial derivative of F with respect capital input; FLL therefore represents the second partial derivative of the production function with respect to labour input
by firms, represented byK(t) must equal supply of capital by households, represented by ¯
K(t). Thus the market clearing condition in the capital market is expressed as
K(t) = ¯K(t)
The initial endowment of capital to the representative household is therefore K(0) > 0,
which is assumed to be a predetermined amount. As machines used in production lose some of their value to wear and tear, capital is assumed to depreciate at an exponential
rate δ ∈ (0,1), so that out of a unit of capital in each period, only 1−δ remains in the
next period.
Firm Optimisation and Equilibrium The representative firm in the model seeks to
maximise its profits subject to rental prices of the production factors so that for a given
level of technology, A(t), the profit maximization problem of the representative firm is
given by the following equation max
{K(t)>0;L(t)>0}F(K(t), L(t), A(t))−R(t)K(t)−w(t)L(t) (2.1.7)
Households maximize utility from consumption of the final good in the current period, whilst saving some of their present income for future use.
The Solow model assumes competitive markets, hence in the competitive equilibrium the representative firm maximizes profits while factor markets clear. This implies de- mand for the production factors K(t) and L(t) must equal their supply and also that the representative firm makes zero profits. This argument, together with the assump-
tion of differentiability of F, implies that in equilibrium, the factor prices must equal the
marginal products of the factors, i.e.
w(t) =FL(A(t), K(t), L(t))
R(t) = FK(A(t), K(t), L(t))
From the assertion of zero profits for firms, it implies therefore that the equilibrium condition is
The fundamental law of motion of the Solow model
Use of capital in production results in depreciation, at an exponential rate, and therefore over time the accumulation of capital in the Solow model can be expressed as
K(t+ 1) = (1−δ)K(t) +I(t) (2.1.9)
where I(t) is investment in new capital. From the national income accounts (without
government spending), in a closed economy setting, the total amount of final goods is either consumed or saved to finance investment, thus
Y(t) = C(t) +I(t)
whereC(t) is consumption. Households save a predetermined constant fraction, s, of their
income. In the closed economy benchmark, aggregate savings must equal total investment, i.e.:
S(t) =I(t) =Y(t)−C(t);
S(t) =sY(t)
where, as already stated, s∈(0,1) and implying that households consume the remaining
1−s of their income
C(t) = (1−s)Y(t)
Combining the savings equations described above with the production function (equation (2.1.6)) then in the equilibrium state where supply is equal to demand, the law of capital
of motion (equation (2.1.9)) can be re-written to give the fundamental law of motion of
the Solow–Swan growth model, expressed as:
K(t+ 1) =sF(A(t), K(t), L(t) + (1−δ)K(t) (2.1.10)
or derived in continuous time by following similar steps into ˙
K(t) = sF(A(t), K(t), L(t)−δK(t) where ˙K(t) = ∂K(t)/
∂t. Using the property of constant returns to scale of the production
function, and defining k(t) as the capital-labour ratio, the continuous time fundamental
law of the Solow model can be expressed in per capita terms as ˙
k(t) =s[Af(k(t))]−(n+δ)k(t) (2.1.11)
where n= ˙L(t)/L(t); derived from population growth expressed as L(t) = exp (nt)L(0). Equation (2.1.11) illustrates how capital accumulates in the Solow framework: the left
hand side represents capital deepening while (n +δ)k(t) is the capital widening term
ratio constant. Thus, in the steady state, sAf(k(t)) = (n +δ)k(t); and as long as
sAf(k(t))>(n+δ)k(t) in the economy output per worker will grow.
Sachs et al. (2004) (and proponents of the poverty trap models) point to three main arguments why in developing countries savings are always less than capital widening (sAf(k(t))<(n+δ)k(t)), and hence they are caught in a poverty trap of very low level output per worker;
a) Low levels of capital in poor countries may result from low marginal productivity of capital which leads to a disincentive to invest. This may arise from low state of human development and poor infrastructure.
b) With low income per capita, impoverished households on a predominantly subsis- tence living may have little inclination to save since they are already struggling to meet basic needs as such the saving rate is likely to be very low or even negative. c) Fertility rates in poor countries tend to be high with socio-economic motives for child
bearing (parents rely on children to support the family from early age) implying big family sizes relying on low incomes, the rapid population growth works to prevent capital accumulation.
The ’Big Push’ proponents argue that the solution lies in aid to provide a massive initi- ation of a temporary “big push” in investment that will move the countries to a certain threshold of capital at which the underlying productivity of capital is high enough to stimulate investment or where household incomes will be high enough to allow for them to undertake savings or indeed help the economies grow to a critical level of development before mortality rates are low enough (and the opportunity cost of child-bearing is high enough) to induce reduction in family size (Blackburn and Forgues-Puccio (2011)), i.e. a threshold at which standard forces of competitive theory can take hold.
Scholars that lobby for aid using the neoclassical framework thus argue that temporary (though lasting for sufficiently long enough period) but large enough infusion of aid can help developing economies to ‘jump’ from the low income per capita equilibrium level where savings are less than the capital widening to a higher level equilibrium (in terms of per capita incomes) where sustained growth is achievable with improved saving rate allow-
ing for capital accumulation. Sachs (2001) identified three channels into which foreign aid goes: (a) households for emergency situations, (b) government to finance public invest- ment, and (c) private businesses (for example, farmers) through micro-finance programs and other schemes to finance private investment.
In the benchmark closed economy framework, investment is entirely financed by savings through the capital accumulation equation expressed by equation (2.1.11). To allow for
inflow of foreign aid we assume a small open economy that receives some foreign aid,A(t)
which goes directly into helping capital accumulation for the representative agent. The
capital accumulation equation will now be augmented with aid per capita,a(t) =A(t)/
L(t),
and is now expressed as ˙
k(t) =s[Af(k(t))] +a(t)−(n+δ)k(t) (2.1.12)
While the fundamental arguments in the poverty traps models have provided a strong premise for pro-aid lobbyists, many scholars remain unconvinced that aid is necessary or sufficient for sustainable long run economic growth and development. The very notion of poverty traps is a contentious empirical debate, some scholars arguing that their existence is explicit and that in the long run the distribution of the world’s income is distinctly bimodal, characterised by the rich and the poor (see Bloom et al. (2003), Quah (1996) and Quah (1997)). Other authors find little evidence to support the existence of poverty traps caused by low levels of savings or productivity (see Kraay and Raddatz (2007) and Azariadis and Stachurski (2005)).