ECON6034 (S2 2022) Global Economic History
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ECON6034 (S2 2022) Global Economic History
Notes on Week 2: The Malthusian Model
1 Malthusian growth model
This is just a simplification of [Galor and Weil, 2000] that was included in the presentation [Galor, ]. I almost literally transcribed the explanations in the paper so you can understand step by step the Malthusian Growth model. Later on [Galor and Weil, 2000] has become a central part of [Galor, 2011], what is called Unified Growth Theory - book available at the library.
The reason for the simplification is to easily understand the main features of a Malthusian economy: stagna- tion of income per capita levels in the long-run, population increases after positive shocks (land expansion, technological advances). One of the central elements being a decrease of the Marginal Product of Labour when population increases, which leads to a decrease of of income per capita.
Consider an overlapping generations model with infinite discrete periods of time, t = 0, 1, 2, 3, .... There is only one good produced in the economy, using labor and land as factors of production. Land is fixed and determined exogenously. Labor is chosen endogenously.
Production occurs following a constant returns to scale technology, which is also exogenous, here. The output produced at time t, yt , is:
yt = (Ax)α (Lt )1 −α
a ÷ (0, 1)
(1)
where Lt is the labor input at time t; determine the effective resources of the
Output per worker at time t, yt is
x is the fixed amount of land; A is the technology level and Ax economy.
yt = = = / 、α a ÷ (0, 1) (2)
In each period t a generation that consists of Lt identical individuals joins the labor force. Members of generation t live for two periods. In the first period they consume a fraction of their parents’ budget. In the second period (parenthood), they allocate their budget between their own consumption and the cost of raising children. They choose the optimal mixture of number of children they will raise (and the cost associated) and their own consumption.
Preferences of generation t are represented by the utility function
ut = nt(γ)ct(1) −γ
A ÷ (0, 1)
(3)
where nt is the number of children of an individual at t and ct is the consumption of an individual at time t.
The budget constraint will be determined by the level of income of each individual yt . They have to allocate yt among the costs of raising a child pnt , where p is fixed and exogenous here for simplicity and their own consumption ct . That is,
pnt + ct 之 yt (4)
Each individual maximise his/her utility subject to his/her budget constraint. Therefore, the Lagrangian is defined as
e(nt , cT , 入) = nt(γ)c 入(pnt + ct _ yt )
First order conditions are:
= V / 、1 −γ _ 入p
= (1 _ V) / 、γ _ 入
ae
From 8 we know that ct = yt _ pnt . Equalise 6 and 7 to 0 and divide them. You will find that
V c n(c)t(t) 、1 −γ
(1 _ V) c c(n)t(t) 、γ = p
Rearrange 9
V ct 1 _ V
1 _ V nt V
Substitute 10 into 8
pnt + pnt = yt = yt = nt
V V
Use 11 in 10 to find that ct = (1 _ V)yt . Therefore nt = c 、α
The development of the economy is characterised by the evolution of output per worker and population. The
size of the population evolves as follows Lt+1 = nt Lt where nt = c 、α . Therefore
Lt+1 = / 、α Lt = Lt+1 = (Ax)α Lt(1) −α _ φ(Lt , A)
In steady state Lt+1 = Lt =
l
= (Ax)α 1 −α = = / 、 a Ax = (A)
The population level in steady state will depend on the technology level of the economy. That is, a positive technological shock will have permanent effects on the size of the population. In particular the size of the population will be larger after a positive technological shock. See graph below.
t +1
L = L
Lt +1 =φ(Lt ; Ah )
Lt +1 =φ(Lt ; Al )
450
L(Ah )
Galor(2014)
The size of income per worker evolves as follows
yt+1 = / 、α = / 、α =
where nt = yt . Substitute the latter expression into 14 and the dynamics of income per worker is
yt+1 = / 、 α yt(1) −α _ w(yt )
In steady state yt+1 = yt =
y¯ = / 、 α y¯1 −α = y¯ =
Income per capita level in steady state does not depend on the technology level of the economy. That is, a positive technological shock will have positive temporary effects on the level of income per capita, but in the long-run the economy will return to the steady state of the economy, income per capita will decrease until reaching . See graph below.
yt +1
yt +1 = yt
yt +1 =ψ(yt )
450
~
t
Galor(2014)
References
[Galor, ] Galor, O. The malthusian theory.
[Galor, 2011] Galor, O. (2011). Unified Growth Theory. Princeton University Press.
[Galor and Weil, 2000] Galor, O. and Weil, D. (2000). Population, technology, and growth: From malthusian stagnation to the demographic transition and beyond. The American Economic Review, 90(4):806–828.
2022-09-14