Notes on the Solow Model (I) with Answers to

Practice Questions

Pablo Fajgelbaum

Last Updated: April 20, 2015

1 Intro

These notes correspond to the class of 04/08

These notes present the basic version of the Solow model.

Basic idea: differences in capital per worker drive differences in income

The model explains how differences in capital per worker are determined

These notes complement Chapter 3 from Weils textbook

2 Key Assumptions

Key assumptions

Closed economy

Single good used for both consumption and investment

Perfect competition and constant-returns-to-scale technologies

Representative firm

Representative consumer

Discrete time (t = 0, 1, 2, ..) and infinite horizon (the economy exists forever)

3 Technology and Firm Maximization

The production function relates output Y to the factors of production capital K and laborL

Y = AF (K,L) (1)

Technology is free (anyone can use it)

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Neoclassical assumptions:

Output increasing with inputs

F

K FK > 0; F

L FL > 0

Decreasing returns in each individual input

2F

K2 FKK < 0;

2F

L2 FLL < 0

Inada Conditions

limK0

FK = limL0

FL =

limK

FK = limL

FL = 0

Constant returns to scale in every input (CRS)

F (K,L) = F (K, L)

Thanks to CRS, we can define output per worker or per unit of labor as

y =Y

L= Af (k) (2)

where k = K/L is capital per worker and where f (x) F (x, 1) is the intensiveform of the production function

Practice question:

show that

Y

K= Af (k) (3)

Y

L= Af (k) kAf (k) (4)

Answer:

Y

K=AF (K,L)

K= A

LF(KL , 1

)K

= ALf

(KL

)K

= Af (k) (5)

and

Y

L=AF (K,L)

L= A

LF(KL , 1

)L

= ALf

(KL

)L

= Af (k)+ALf

(K

L

)(KL2

)= Af (k)kAf (k)

(6)

show that the assumptions on F imply f (k) > 0 and f (k) < 0

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Answer: we assumed that 0 < FK . And we just proved in (5) that FK =f (k). Therefore, f (k) > 0.

We also assumed that 0 > 2FK2

. Since FK = f (k), then

2FK2

= FKK =

f (k)K =

f (KL )K =

1Lf (k) . Hence 0 <

2FK2

= 1Lf (k) .

show that f(k)k is decreasing with k

Answer: we can write:f (k)

k=Lf (k)

K=F (K,L)

K= F

(1,L

K

)= F

(1,

1

k

)and note that, as k increases, F

(1, 1k

)decreases.

show thatY

KK +

Y

LL = Y (7)

This is called Euler Equation and is an implication of CRS

Answer: we have proven in (5) and (6) that YK = Af (k)and YL = Af (k)kAf (k) . Replacing this in (7),

(Af (k)

)K+

(Af (k) kAf (k))L = Af (k)K+Af (k)LkAf (k)L = Af (k)L = Y

The representative firm solvesmaxK,L

Y rK wL (8)

where r is the cost of capital, and w is the cost of labor

Note that we normalize the price of the final good to 1, so that total output equals total

income in the economy.

The solution to the firm problem gives the first-order conditions (FOC):

Y

K= r (9)

Y

L= w (10)

These conditions say that the marginal product of labor (capital) equals the cost of using

labor (capital)

Using 9 and 10 gives the factor income shares:

wL

Y=Y

L

L

Y(11)

rK

Y=Y

K

K

Y(12)

where wLY is the income share of labor andrKY is the income share of capital

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Note that, using 7, 11, and 12, we obtain:

rK + wL = Y,

i.e., the payments to capital and labor exhaust output (everything produced is dis-

tributed in form payments to capital or labor).

Practice questions

Show the first-order conditions (FOC) of the firm maximization problem (8) are equiv-

alent to a situation where each individual worker hires capital per worker k, so that the

wage solves:

w = maxk

Af (k) rk

Answer: if each worker solves this maximization problem, then

Af (k) = r

From (5), Af (k) = YK . So this impliesYK = r, as in (9). Replacing this

expression (which says that the cost of capital equals the marginal return

to capital) into w = maxk Af (k) rk, we obtain

w = Af (k) rk= Af (k)Af (k) k

And from (6) we have that the last expression is equal to YL . Hence,YK = w, as in (10).

Show that when the production function is Cobb-Douglas, so that F (K,L) = KL1,then the capital share of income is

Answer: Under Cobb-Douglas, we have f (k) = k. Condition (9) implies:

r = Af (k) = Ak1 (13)

In turn, the capital share of income is

rK

Y=rk

y=Ak

y=

where the second equality follows from (13)

Show that when the production function has Constant Elasticity of Substitution

(CES) equal to , so that F (K,L) =(K

1 + (1 )L1

) 1

then the capital

share is increasing or decreasing with the capital-labor ratio(KL

)depending on whether

is greater or less than 1.

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Answer: under CES,

f (k) =F (K,L)

L=

(k

1 + (1 )

) 1

This implies

f (k) =

1(k

1 + (1 )

) 11

1

k1

=(k

1 + (1 )

) 11

k1 = f (k)

1 k

1

=

(f (k)

k

) 1

(14)

Therefore, the capital share of income is

rK

Y=rk

y=Af (k) kAf (k)

=f (k)f (k) /k

=

(f (k)

k

) 1

where the second line follows from using (14). We already know from f(k)kis decreasing with k. Therefore, as k increases, rKY decreases if < 1 andrKY increases if > 1.

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