Chapter 4 Note

8/20/2019 Chapter 4 Note

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1 Supplement to Chapter 4 in Galor.

This note goes through the mathematical models of the chapter. These relate to three

of the main hypotheses of the demographic transition. These are (1) The demographic

transition was caused by increases in income; (2) the demographic transition was caused

by declines in mortality; (3) the increased demand for human capital. These three are

covered in the following. The note is best read in conjunction with Galor’s book.

1.1 The rise in income per capita: non-robustness result

The central hypothesis is that the decline in fertility observed in the late 19th century

was caused by increased income due to the industrialization. The following model shows

that this result is not theoretically robust. Consider a consumer who obtains utility from

consumption (c) and surving children (n):

u = ln n + (1 ) ln c

The constraint is:

c = (1 n) y;

where is the time cost associated with raising a child and y is the income earned if

all time is spent on working.

The utility function can be written:

u = ln n + (1 )ln((1 n) y)

u = ln n + (1 )ln(1 n) + (1 ) ln y

Di¤erentiating the utility function wrt to n gives:

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1

n + (1 )

1

1 n ( ) = 0

1

n

= (1 )

1 n (1 n) = (1 ) n

n = n n

n =

The number of children is unrelated to income in this model. Thus, theoretically

we would not expect that the opportunity cost of children (related to income growth)

necessarily plays a role in the timing of the fertility decline.

1.2 The decline in Infant and Child Mortality

This hypothesis suggests that the decline in infant and child mortality that preceded the

fertility decline is a plausible explanation of the onset of the demographic transition. The

hypothesis can be illustrated in the framework of the previous sections by making the

number of surviving children a function of the probability of surviving infancy and the

number of children the household will bear nb

:

n = nb:

Let utility maximization problem be the same as in the previous subsection. Therefore,

the solution for n is:

n =

:

The actual number of children is then:

nb =

:

It is seen that an increase in leads to a lower number of children being born. In the

absence of uncertainty, there is, however, no e¤ect on the surviving number of children.

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Thus, the central prediction is that higher mortality rates (lower ) leads to higher fertility

rates, whereas the number of surviving children is unchanged.

1.3 The rise in demand for Human Capital

The second phase of industrialization witnessed a signi…cant increase in the demand for

human capital. The increased demand for human capital led families to invest in the

education of their children according to this hypothesis. Thus, the argument is that the

increased demand for human capital led to a quantity-quality trade-o¤.

The idea can be illustrated by considering the following utility maximization problem:

u = (1 ) ln c + ln n + ln h;

where h is the level of human capital of each child, and 0 < 0): The following

assumptions are made:

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lime!0

he(e; g) = 1:

lime!1

he(e; g) = 0

h(0; g) > 0:

The e¤ect of g is an obsolence is e¤ect. Technological change means that some education

is made obsolete. Further heg(e; g) > 0.

The utility maximization problem can be rewritten as:

maxe;n u = (1 )ln((1 n (

q

+

e

e) y) + ln n + ln h (e; g) =(1 ) ln y + (1 ) ln(1 n ( q + ee)) + ln n + ln h (e; g) :

The …rst order conditions become:

@u

@n = (1 )

1

1 n ( q + ee) ( ( q + ee)) +

1

n = 0:

@u

@e =

1

h (e; g)he(e; g)

1

1 n ( q + ee)n e = 0:

The solution for n is:

n =

q + ee:

Now the second condition can be rewritten:

1

h (e; g)he(e; g) =

1

1 n ( q + ee)n e:

By substituting the solution for n into this expression, we get:

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1

h (e; g)he(e; g) =

1

1 q+ ee

( q + ee)

q + ee e

1

h (e; g)

he(e; g) = 1

1

q

+ e

e

e

1

h (e; g)he(e; g) =

1

q + ee e

eh (e; g) = he(e; g) ( q + ee) :

It can be established generally that an optimal solution for e and n exists:

e = e(g ; ; q; e):

n = q + ee(g ; ; e; q)

:

To establish this we de…ne the implicit function as determined by the necessary …rst

order condition:

G (e; g ; ; q; e) eh (e; g) he(e; g) ( q + ee) = 0

The assumptions on he(e; g) implies that:

lime!0

G (e; g ; ; q; e) < 0

lime!1

G (e; g ; ; q; e) > 0

This follows from the fact that as e ! 0,he ! 1 (and h(0; g) > 0). Further, as e!1,

h (e; g) will converge at a positive value (if strictly increasing, it will go towards in…nity)

and he(e; g) converges to 0.

Further, we can establish that:

@G (e; g ; ; q; e)

@e = ehe (e; g) hee(e; g) (

q + ee) ehe(e; g)

= (1 ) ehe (e; g) hee (e; g) ( q + ee) > 0:

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Since 0 < 0 will make G(:) = 0:

Given this we can now sign how this value of e responds to the other parameters in

the model by simply using the implicit function theorem. Recall that the average product

of a strictly, increasing and concave function decreases with the value of the independent

variable. In our case, this means:

@ he

@e =

ehe h (e; g)

e2 ehe:

This is used to establish the sign of the derivative wrt e below:

We next obtain the following derivates and their signs:

@G (e; g ; ; q; e)

@g

= ehg (e; g) heg (e; g) ( q + ee) < 0

Since hg 0:

@G (e; g ; ; q; e)

@ = he(e; g) (

q + ee) < 0

@G (e; g ; ; q; e)

@ q = he(e; g) < 0:

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Notice that there is a q too much in Galor’s formula on page 138 compared to the

above formula for @G(g;; q ; e)

@ q :

@G (e; g ; ; q; e)

@ e = h (e; g) he (e; g) e > 0:

Which uses h(e; g) > ehe: > ehe since 0 < 0

@e

@ =

@G (e; g ; ; q; e)

@ =

@G (e; g ; ; q; e)

@e > 0

@e@ q

= @G (e; g ; ; q; e)

@ q =@G (e; g ; ;

q; e)@e

> 0

@e

@ e =

@G (e; g ; ; q; e)

@ e =

@G (e; g ; ; q; e)

@e


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n =

q + ee(g ; ; e; q)

This can be di¤erentiated partially with respect to the four variables:

@n

@g =

e @e@g

[ q + ee(g ; ; e; q)]2


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Where we make the assumption that g ĝ = 2

(1) which ensures e 0: It is easily

seen that this function is increasing in g the rate of technological growth, and that n is

decreasing in e and hence g :

References

[1] Lagerlöf, N.-P., 2006. The Galor-Weil model revisited: a quantitative exercise. Review

of Economic Dynamics 9(1), 116-142.

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Chapter 4 Note

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