Chapter Malthusian Model

8/18/2019 Chapter Malthusian Model

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The Malthusian Model

After reviewing the key development and growth facts, it is clear that we need a theory

that can generate a period of constant living standards, followed by a transition period

with modest increases in the living standard, followed by a period of modern economic

growth. We already have a model that can account for the period of modern economic

growth; the Solow Model with technological change generates constant growth of per

capita output. It is true that the Solow Model can generate a steady state with a constant

level of per capita output. This is the Solow Model absent technological change. One

possibility is to interpret the pre-1700 era of constant livings standards as the steady state

of the Solow Model absent technological change. The problem with this interpretation isthat technology was not stagnant before 1700. Joel Mokyr a noted economic historian at

Northwestern University has documented in his book The Lever of Riches that numerous

and important technological innovations occurred well before 1700.

In light of the historical record on technological change, we proceed to alternative

theory and model of this pre-1700 era. This is the Malthusian model that goes back to

David Ricardo and the classical economists. There are two key components of the

model. The first is a production function with a fixed factor of production. By fixed, we

mean that its supply cannot be changed over time. Labor and capital are not fixed

factors as both can be increased over time. In the Malthusian model, the fixed factor is

land. The second key component is a population growth function that is an increasing

function of per capita consumption. These two elements ensure that the steady state is

characterized by a constant living standard even when there is technological change.

We first proceed by studying the Malthusian model with no capital and absent

technological change to help develop intuition for the model. We solve the equilibrium

of the model graphically. We then follow this up with an algebraic study of the

Malthusian model with capital accumulation and exogenous technological change.


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Figure (1) below is a plot of per capita income from 3000 BC to 2000 AD. The

feature of the plot is that the standard of living displayed no trend for the first 4800 years

(this period includes Malthusian and Post-Malthusian regimes) and then exploded

subsequent to 1800 (Modern regime). These notes are concerned with a theory of the

period prior to 1800.

Figure 1

I. Model with No Capital or Technological Change

People. Initially, there are N0 people alive. We use Nt to denote the number of people in

the economy at date t. People prefer more consumption to less.

Demographics: Population growth is determined by the death rate and birth rate of the

population. Thomas Malthus proposed a theory of population dynamics in which the birth

rate was independent of peoples’ living standard. The death rate however was a

decreasing function of the amount people consumed. This assumption followed from the

idea that if people had more to eat they would be stronger and thus their bodies would be

more able to successfully fight off disease. Graphically, we have

Income per Capita

0

10

20

30

year

1

3000 BC 2000 BC 1000 BC 1 AD 1000 2000


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The population growth rate for a given level of consumption is the difference between the

birth rate and the death rate. We denote this function by g(c). When the birth rate equals

the death rate, the population does not change, namely Nt+1=Nt. More generally,

Nt+1=Nt[1+g(c)].

consumption

Birth rate

death rate

Birth and Death Rates as functions

of per capita consumption


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In what follows we will use the gross population growth rate function G(c) which is equal

to 1+g(c). Consequently,

)(1 t t t cG N N =+ .

Graphically, we have

consumption

11 −+

t

t

N

N

Population Growth Rate Function (Net)

0

g(c)


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Endowments: Each person in the economy is endowed with one unit of time each period

which he or she can use to work. All individuals own an equal share of the land and the

total amount of land is fixed at L. Land does not depreciate.

Production Function: The economy produces a single final good using labor and land.

The production function is given by

α α −=

1

t t t N L AY

The letter A is again the Total Factor Productivity (TFP).

consumption

Gross Population Growth Rate Function

1

G(c)

t

t

N

N 1+


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Notice that this production function looks the same as the production function in the

Solow model except that (1-α) is labor’s share and there is land and not capital. The

production function is still characterized by constant returns to scale, and is increasing in

each of its two input. Moreover, the law of diminishing returns applies to each factor

separately. The increases in output associated with an additional unit of labor input

decrease as labor increases holding the land input fixed.

The key feature is that land is essential in production. (Note the last point means that

doubling population while keeping land fixed less than doubles the output).

General Equilibrium:

There are three markets that must clear in each period for the economy to be equilibrium:

the labor market, the land rental market, and the goods market. Equilibrium quantities

are trivially determined in this model, as they were in the Solow Growth model. This is

because the supply of labor and the supply of land are both vertical. People supply their

entire time endowment to the market and their entire land endowment to the market. As

Nt and L

are the equilibrium inputs, it is trivial to determine the equilibrium amount ofoutput. This is α α −= 1t t t N L AY . For the goods market to clear, Ntct=Yt.

The equilibrium prices are the rental price of land rLt and the real wage rate, wt. To

determine these we need to use the firm’s labor demand and its demand for land services.

Once again, these are the marginal product of land and the marginal product of labor.

These follow from the profit maximization problem of the firm. This is

Profits: t Lt t t t t Lr N w N L Aofits −−=

−α ε 1

Pr

Labor Demand:t

t

t t t N

Y N L Aw )1()1( α α α α −=−= −

Capital Demandt

t t t Lt

L

Y N L Ar α α α α == −− 11


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Steady State Analysis

We begin with characterizing the steady state equilibrium. First note that there can not be

any sustained growth in per capita output or consumption in this model. The reason for

this is that the only way a person’s output could be increased in this world is by

increasing the amount of land he uses. But land is fixed and so cannot be increased on a

per person basis. Thus, a steady state equilibrium is characterized by a path of per capita

consumption and per capita output that are constant, i.e. zero growth. Once we recognize

this fact, it must be the case that the population is constant in a steady state. Otherwise,

with diminishing returns, per capita output and per capita consumption would have to fall

as we add more people to work the land.

Now, that we recognize this we can be more formal in solving out the steady state

equilibrium. First, we can exploit the result that Nt+1 = Nt = Nss

in the steady state with

the population growth rate function to solve for the steady state living standard, css

.

Graphically this is just


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Algebraically, what we have done in solving for the steady state is first use the population

growth rate function to solve for css

when Nt+1 /Nt=1. This is )(1 ss

cG= . The next step,

which uses the hands and mouth graph, is to solve for Nss using the goods market clearing

condition and the production function equation. Namely, α α −= 1 N AL Ncss . The left hand

side of the goods market clearing condition is the mouth curve while the right hand side

is the hands.

Comparitive Statics

We can use the diagrams for the population growth function and the hand-mouth to show

how the steady state of the economy is affected by various factors. As the birth and death

rates determine the function G(c), anything that affects either the death rate or the birth

α α −=

1 N ALY

N

Determining the Steady State Population

N t css

Nss


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rate will affect the steady state consumption level and population by changing the

position of the mouths curve. Anything that changes the production function will change

the steady state population through its affect of the Hands curve. The production

function change will not have any effect on the steady state consumption level, however.

Transitional Dynamics

For an economy to be on its steady state it must start out with the right initial population

size, namely, N0=Nss. Suppose and economy fails to start with such a population. Will it

converge to the steady state both in terms of consumption and population? The answer is

yes, it will. This can be seen by using the Hands and Mouth Diagram. Suppose an

economy starts with a population below the steady state level. According to the Hands

equation, output is at Y0. The Mouth equation tells us the amount of output needed to

give each person the steady state consumption level. It is clear that total output exceeds

the required amount. Hence, c0 > css

. Now consider what happens to population in

period 1. Here we use the population growth function. As c0 > css, the population growth

function implies that N1 /N0 > 1, so population expands. Now at N1, it is still the case that

the Hands output exceeds the Mouth output, so that the living standard, c1>css. However,

c1 is smaller than c0. This can be seen in the Hands and Mouth diagram by noting that the

ray from the origin to any point on the hands curve is equal to per capita output, Y/N,

which is equal to per capita consumption. As N increases, the slope of this ray declines.


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We thus have the following time paths for population and consumption. The

convergence property is again the result of the law of diminishing returns and the

increasing nature of the population growth function. If population is low, the marginal

product of labor is high. People will therefore

II. Adding Technological change and Capital

Thus far we have shown that the steady state of the Malthusian model is characterized by

a constant living standard and zero population growth. How do these properties change if

we allow for capital accumulation and technological change? Recall, that we are after a

model of the pre-1700 era that generates a constant living standard and population

growth.

PopulationConsumption


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With the addition of capital it is no longer possible to characterize the properties

of the model graphically. For this reason we will proceed by deriving algebraically the

steady state of the model. It is possible to give some intuition for the algebraic results

that will be derived, however. First, let us think back to the Solow model without

technological change. Was there sustained growth in the per capita consumption and

output in that model without technological change? We showed that sustained growth

was not possible on account of the law of diminishing returns. In the Malthus model the

law of diminishing returns still holds so adding capital to the Malthus model will do

nothing to change the result of a constant living standard. How about the affect of adding

capital accumulation on population growth: Would it change the result of zero population

growth in the steady state? Again, it would not; you could double all the people and

machines and output would less than double because land is fixed. Here we see the

importance of the fixed factor property of land.

What happens when we add technological change? To gain some intuition here,

let us return to the Malthus model without technological change or capital accumulation

and ask what happens to the steady state following an increase in TFP. This is shown

below. As can be seen, the population is higher. The living standard, however, does not

change since that is determined by the population dynamics of the model. We can think

of technological change therefore, as a sequence of increases in TFP. Consequently, in

the case of technological change we will have a steady state with constant population

growth and a constant living standard- the pre-1700 facts.


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General Equilibrium:

The set of conditions that an equilibrium satisfies are listed below. In addition to the land

and labor market, there is now a capital rental market. Thus, there is its rental price and

the firm’s demand for capital services given by equation (7). As there is now capital,

there is also savings (as seen in equation 1) and the law of motion for capital (equation 3).

(1) t t t Y sc N )1( −=

(2) φ α α φ γ −−+= 1])1[( t t

mt t t N LK AY

(3) t t t sY K K +−=+ )1(1 δ

(4) )(1 t t t cG N N =+

(5) t t t t t t

mt N Y N LK Aw / )1()1)(1()1( φ α γ φ α φ α α φ φ α −−=+−−= −−−−

(6) t t t t

mt t Lt LY N LK Ar / ])1[(11 α γ α φ α α φ =+= −−−

α α −=

1 N ALY

N

Increase in TFP

N t c

ss

Nss


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(7) t t t t

mt t kt K Y N LK Ar / ])1[(11 φ γ φ φ α α φ =+= −−−

Balananced Growth Path Equilibrium. A Balanced Growth path equilibrium is an

equilibrium such that for the right initial population and capital stock endowment, all

variables grow at constant rates, with the possibility that this rate is zero for some

variables. Notationally, given K0 and N0, ct+1 /ct=1+gc, yt+1 /yt=1+gy, Yt+1 /Yt=1+gY,

Kt+1 /Kt=1+gK, Nt+1 /Nt= 1+gn, rLt+1 /rLt=1+grL, wt+1 /wt=1+gw, rkt+1 /rke=1+grk .

We divide the solution of the balanced growth path into two parts. In the first, we derive

the growth rates of each of the key variables along the balanced growth path. This is Part

I. In the second part, we actually solve for the paths of the population and the total capital

stock, as well as their initial values.

Part 1. Solving for the growth rates along the balanced growth

path.

Step 1. Use equation (4) to conclude that gc= 0 and ct=css.

Step 2: Use equation (1) to conclude that gn=gY.

Step 3. Use equation (3) to conclude that gK=gY. First divide both sides by Kt. This is

t t t t K sY K K / )1( / 1 +−=+ δ .

Next invoke the steady state condition that Kt+1 /Kt=1+gK. This is

t

t

K K

Y sg +−=+ )1(1 δ


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As the left hand side of the equation is constant, it follows that Y and K must grow at the

same rate along the balanced growth path.

Step 4. We now use the above results, that g K= gn = gY ≡ g with equation (2) to solve for

g. First take the date t+1 output. This is

φ α α φ γ −−+

+

+++ +=

1

1

1

111 ])1[( t t

mt t t N LK AY

Next take the date t+1 output as a ratio of the date t output. This is

φ α α φ

φ α α φ

γ

γ −−

−−

+

+

+++

+

+=

1

1

1

1

111

])1[(

])1[(

t

t

mt t

t

t

mt t

t

t

N L AK

N L AK

Y

Y

Rearranging terms, we arrive atφ α φ

φ α γ

−−

++−−+

+=

1

1111 )1(t

t

t

t m

t

t

N

N

K

K

Y

Y

Now invoke the steady state growth rates

φ α φ γ −−+++=+ 1)]1)(1[()1()1( ggg m

Lastly, solve for (1+g)

(8) α φ α γ / )1(, )1(1 −−

+=+ mg

Part II. Solving for the Balanced growth path of K and N.

Step 1. Now that we have solved for the growth rate of the population, we can solve for

css using from the population growth rate function,

Step 2. Use equation (1) to solve for Y as a function of Nt and css

. This is

(9)s

c N Y

ss

t

t −

=

1.

Step 3. Substitute for Yt in equation (3) using equation (9).


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t

ss

t t N s

scK K

−

+−=+

1)1(1 δ .

Step 4. Use the BGP condition that K t+1=(1+g)K t to solve for Kt. This is

(10) t ss

t N gs

scK

))(1( δ +−=

Step 5. Next take equation (3) substituting for Yt using equation (2). This is

(11) φ α α φ γ δ −−+

++−=1

1 ])1[()1( t t

mt t t t N LK sAK K .

Step 6: We again invoke the BGP condition K t+1=(1+g)K t and use equation (10) in (11).

φ α α φ

φ

γ δ

δ −−−−

+

+−

=+11

1

])1[())(1(

)( t t

mt t

ss

N L N gs

scsAg

This gives us a single equation in a single unkown, Nt. Solving for Nt yields

(12)

α φ

φ α

δ δ γ

/ 11

)1(

))(1()1(

+−+

+=

−

−−

gs

sc

g

sA L N

sst

m

ss

t .

Using equation (12) with (10), gives us the solution for sst K

III. Computing the Equilibrium path when ss N N 00 ≠ orssK K 00 ≠ .

Convergence

If ss N N 00 ≠ orss

K K 00 ≠ , the economy will not be on its balanced growth path. We

showed graphically for the Malthusian model without capital accumulation and

technological change that an economy which does not begin on its steady will converge

to its steady state. Although we cannot show it graphically for this more complex version

of the model, the convergence result holds.


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It is easy to compute the transitional path for the model economy via a

spreadsheet once we assign parameter values to the model. This is discussed in the last

section. The equilibrium consists of a path for the following set of variables:

),,,,,,,( Kt Lt t t t t t t r r wK LY N c .

The equilibrium quantities are trivial to find because the supply of land, labor and

capital are all vertical. That is to say in period t, the amount of capital used is Kt; the

amount of labor used is the population, Nt; and the amount of land used is just the

endowment L. We therefore know what Yt and ct are. We also know what the margina

products of all the inputs are, so we know all of the prices. We can then figure out next

period’s population and capital stock using the population growth rate function and the

capital stock law of motion. The period t=0 quantities of Land, capital and people are

given.

More specifically, we compute the equilibrium path as follows. , K0 , and an

initial population N0, we find the equilibrium path in each period as follows:

1. Determine Yt from the aggregate production function, Equation(2).

2. Determine ct from the

IV. Calibration to pre-1700 Observations

We can calibrate the model so that it matches the pre-1700 growth facts. We need to

assign values to the following list of parameters: Asm ,,,,, δ γ φ α . In addition we need to

specify the population growth function and assign parameter values to that function. In

what follows, we assign parameter values so that the model matches the following pre-

1700 observations associated with England’s experience.


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1. an average annual population growth rate of .003 per year.

2. Historians’ estimates of labor’s share of income equal to 2/3.

3. Historians’ estimates of Land’s share of income at 3/12.

4. Historians’ estimates of capital’s share of income at 1/12.

We continue to set the savings rates, s= .20 that we used for the Solow model.

Additionally, we continue to use the depreciation rate that we used for the Solow model,

namely, δ = .05. Just as in the Solow model, an input’s share of income equals its

coefficient in the production function. For land, this coefficient is α; for capital, this

coefficient is φ ; and for labor this coefficient is just φ α −−

1 . Thus, 12 / 3=

α and

12 / 1=φ . We can determine the value of γm using the balanced growth relation between

population growth and the exogenous growth rate of technological change. This is

α φ α γ / )1(, )1(1 −−

+=+ mg .

Using the observation that g=.003 and the values for 12 / 3=α and 12 / 1=φ , we can

solve for the value of γm. This is 001.=

mγ . The last technology parameter, A, is just

normalized for one as it determines the units in which output is measured. The

parameterized production function is thus

3 / 212 / 312 / 1 ])001.1[( t t

t t t N L AK Y = .

The Population growth function

For the predictions of Malthus to hold, we need a population growth function that is

increasing in the living standards that governed the pre-1700 era. As the model predicts a

constant population growth along the balanced growth path, it is clear that we need an

episode where the English economy was not on its balanced growth path. Such an

episode is the Black Death that occurred between 1347 and 1350. The effect of the Black


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Death on Europe’s population was dramatic; historians estimate that one half to one third

of Europe’s population died in this three year period. While deaths were concentrated in

this three year period, it took nearly 200 years for the population to return to its pre 1346

level in Europe.

Our strategy is thus to use the data on living standards (actually real wages) and

population growth for available years in the 1350 to 1550 period to estimate a linear

equation for population growth. Namely,

t t t c N N 101 / β β +=+ .

β0 is the y-intercept of this linear relation and β1 is the slope. The observations for the

English economy for living standards and population growth are shown in the following

figure:


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Population Growth Rate for England between 1350 and 11550

0.99

0.995

1

1.005

1.01

1.015

80 90 100 110 120 130 140 150 160 170

real wage

The estimated slope line is also depicted. This line is t t t c N N 00004.999. / 1 +=+ . This is

the function G(c).

V. Empirical Support for the Malthusian TheoryWe conclude this chapter by presenting some evidence in support of the Malthusian

theory.

The English Economy From 1250 to the Present

A. The Period 1275–1800

The behavior of the English economy from the second half of the 13th century

until nearly 1800 is described well by the Malthusian model. Real wages and, more

generally, the standard of living display little or no trend. This is illustrated in Figure 1,


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which shows the real farm wage and population for the period 1275–1800.i During this

period, there was a large exogenous shock, the Black Death, which reduced the

population significantly below trend for an extended period of time. This dip in

population, which bottoms out sometime during the century surrounding 1500, is

accompanied by an increase in the real wage. Once population begins to recover, the real

wage falls. This observation is in conformity with the Malthusian theory, which predicts

that a drop in the population due to factors such as plague will result in a high labor

marginal product, and therefore real wage, until the population recovers.

Population and Real Farm Wage

0

50

100

150

200

250

300

1275 1350 1425 1500 1575 1650 1725 1800

Population

Wage

Figure 1

Another prediction of Malthusian theory is that land rents rise and fall with

population. Figure 2 plots real land rents and population for England over the same

1275–1800 period as in Figure 1.ii Consistent with the theory, when population was

falling in the first half of the sample, land rents fell. When population increased, land

rents also increased until near the end of the sample when the industrial revolution had

already begun.


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Population and Real Land Rent

0

50

100

150

200

250

300

1275 1350 1425 1500 1575 1650 1725 1800

Population

Rent

Figure 2

i The English population series is from Clark (1998a) for 1265–1535 (data from parish records in 1405-

1535 is unavailable, so we use Clark’s estimate that population remained roughly constant during this

period) and from Wrigley et al. (1997) for 1545–1800. The nominal farm wage series is from Clark

(1998b), and the price index used to construct the real wage series is from Phelps-Brown and Hopkins

(1956). We have chosen units for the population and real wage data so that two series can be shown on the

same plot.

ii The English population series and the price index used to construct the real land rent series are the same

as in Figure 1. The nominal land rent series is from Clark (1998a).

Chapter Malthusian Model

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