Exercises, Part IV: THE LONG RUN
4.1
The Solow Growth Model
Consider the Solow growth model without technology progress and with constant population. a) Define the steady state condition and represent it graphically. b) Show the effects of changes in the saving rate and in the depreciation rate of capital on the growth rates of capital and output per capita in the short and long run. 1
1
c) Assume the production function Y = K 2 N 2 and δ = 0.1 . If per capita production is 5,
what is the equilibrium value of s?
4.2
The Solow Model with 2 Countries
Consider two countries Cocoloco (C) and Sambapati (S). They are characterised as follows: Production functions: y C = 20 k C y S = 10 k S Saving rates: s C = 0.1 s S = 0.2 Depreciation of capital: δ C = 0.1 δ S = 0.1 a) Compute the steady state values of capital and output per capita in the two countries. b) Cocoloco’s citizens spend more than those in Sambapati. In fact, they have a lower saving rate. Is it possible that Cocoloco’s citizens have, nevertheless, a higher per capita income? Why? c) Sambapati’s citizens want to have the same per capita income as the one of Cocoloco’s citizens. To this purpose, how much should they change their saving rate? Represent graphically the adjustment process towards the new equilibrium.
4.3
Capital Accumulation and the Marginal Propensity to Save
Consider the following production function: Y = K 3 / 4 N1/ 4
Let the saving rate be s = 0.2 and the depreciation rate of capital be δ =0.1. a) Write down the law of capital accumulation and give its economic interpretation. b) Compute the steady state levels of per capita output and capital per worker. 1
c) Show the effects of an increase in the propensity to save on the level of per capita output graphically and provide some economic intuition for your results.
4.4
The Steady State
In country C the production function per worker is: Y ⎛K⎞ = f⎜ ⎟= N ⎝N⎠
K . N
a) Write down the long run equilibrium condition. Compute the steady state values of capital per worker and output per worker, given that s = 0.5 and δ = 0.1. b) The government of country C decides to give incentives to save. Because of this, the marginal propensity to save increases to s1 = 0.6. Compute the new steady state level of K/N and of Y/N. Show graphically the effects of such a policy and explain the adjustment process towards the new equilibrium. c) Compute the level of consumption per worker before and after the policy described in b). Do you think such a policy is suitable? Give an economic interpretation of your result.
4.5
The Solow Model with Population Growth
Consider the Solow growth model without technology progress ( g A = 0) but with positive population growth, ∆N / N = g N = n. a) Analytically derive the steady state condition and represent it graphically. b) Compute the growth rates of per capita output and aggregate output.
4.6
Growth Rates
Consider the Solow growth model with positive technology progress and positive population growth. a) In steady state, the growth rate of per effective worker output is zero. Compute the growth rates of per capita output and total output. b) Given the production function Y = K 1−α ( NA) α , compute the relation between Solow residual and technology progress. c) Consider the production function at point b) where: α = 0.75, production grows at the annual rate of 7.5%, working force increases at the annual rate of 5.6% and capital grows at the annual rate of 2.4%. Compute the Solow residual (or rate of growth of
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total factor productivity, TFP), the rate of growth of work productivity and the rate of growth of technological progress. d) Why is the rate of growth of the technological progress greater than the Solow residual?
4.7
The Solow Model with Population Growth and Technical Progress
In country A the production function is: Y = K AN
and the rate of technical progress and population growth rate are both positive. a) Derive the expression for the equilibrium level of capital and output per effective worker. b) Derive the expression for the equilibrium level of consumption per effective worker. Which is the value of s that maximises the level of consumption per effective worker? c) Show graphically what happens if there is an increase in the rate of technological progress. What will be the effects on the rates of growth of output, per capita output and output per effective worker?
4.8
The Solow Model with Population Growth and Technical Progress
In country F the production function and the main growth parameters are: 1 2
1 2
Y = K ( NA) , s = 20% , δ = 2%, g A = 3% e g N = 5%.
a)
Compute the steady state values of:
-
capital per effective worker;
-
output per effective worker;
-
growth rate of total product;
-
output per worker.
b)
The government decides to increase taxes. Because of this the saving rate becomes
15%. Compute the effect on output per effective worker and the growth rate of total output in the following cases: -
the government uses the additional resources (assuming that government ‘s balance remains balanced) to finance current expenses;
-
the government uses the additional resources to finance R&D (because of this the rate of technological progress becomes 5%).
c)
After a long period in which the economic growth rate has been positive but small,
country F has experienced a boom (big increase in the economic growth rate). What are the possible causes? How can one discriminate among them? 3
4.9
Growth Factors
Consider the Solow model with technical progress and population growth. Discuss whether the following elements have a transitory or permanent effect on the rate of economic growth, and, if yes, in which direction the effect goes. -
an inflow of new immigrants who become part of the labor force;
-
a one-time exogenous reduction in the level of technology (for example, due to the adoption of a tighter environmental legislation);
-
a birth control program;
-
incentives to research and development which permanently increase the rate of technical progress.
Explain through which mechanisms these elements affect the economy.
4.10
Productivity and the Labor Market
In country A the wage and price equations are respectively Wt = Pt Ate (1 − µ t ) Wt =
At Pt . 1+ µ
a) Assuming that Ate = At , derive the expression for µ t . In which relation are the unemployment rate and productivity? b) Assuming that Ate = At −1 , derive an expression for µ t . What is the relation between the unemployment rate and productivity? What is the difference to your answer given in a)? c) Assume that µ = 25% and At = 10 . Compute the equilibrium levels of unemployment and real wages. d) Assume that Ate = At −1 and that A increases by 10%. Starting from the equilibrium computed in c), compute the equilibrium values of unemployment and real wages in the two following periods. e) If A increases by 20% each year, which is the rate of change of real wages? Explain your answer.
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SOLUTIONS
4.1
The Solow Growth Model
K ⎛K⎞ sf ⎜ ⎟ = δ N ⎝N⎠ In steady state per capita capital is such that per capita savings (left hand side) is equal tto a)
capital depreciation (right hand side).
f(K/N)
Y/N Y/N*
δ (K/N)
sf(K/N)
K/N*
K/N
b) The steady state growth rate does not depend neither on the saving rate nor on the depreciation of capital. Nevertheless, this variables can influence the growth rate in the short run. Reductions in s or increments in δ imply lower steady state levels of per capita capital and of per capita output (see the graph). Hence, during the adjustment process towards the new steady state the growth rate of per capita output has to be negative. The adjustment process is shown in the graph. In the opposite case of increments in s or reductions in δ the adjustment process is the opposite with a temporary positive growth rate.
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Y/N f(K/N) 1
Y/N
δ(K/N)
Y/N2
s1f(K/N) s2f(K/N)
K/N2
K/N1
K/N
2
δ (K/N)
Y/N
f(K/N) 1
Y/N
δ1(K/N) 1
2
sf(K/N)
Y/N
2
K/N
1
K/N
K/N
∆ (Y/N)
t
6
c) Now compute the pre worker production function (divide by N): Y = N
K N
Since:
Y =5 N we get: K = 25 N Substituting in the steady state condition: s ⋅ 5 = 0.1 ⋅ 25
from which: s = 0.5
4.2
The Solow Model with 2 Countries
a) The steady state condition is: K ⎛K⎞ sf ⎜ ⎟ = δ N ⎝N⎠ Cocoloco’s case: 0.1 ⋅ 20 k C = 0.1 ⋅ k C Divide by
kc :
k C = 20 k C = 400 y C = 20 ⋅ k C = 400 Sambapati’s case: 0.2 ⋅ 10 k S = 0.1 ⋅ k S Divide by kS =
kS :
0.2 ⋅ 10 = 20 0.1
k S = 400 y S = 10 ⋅ k S = 200
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b) Cocoloco’s citizens earn a greater per capita income y C > y S . In fact, per capita income depends on 3 factors: the saving rate and the production function (technology) which have a positive effect on it, and the depreciation rate of capital which has a negative effect on it. In the exercise the depreciation rate is the same for both countries. They differ in the saving rate (greater in Sambapati), difference outweighed by the difference in the production function. c) Sambapati’s citizens have to choose a s such that kS satisfies: y S = 10 ⋅ k S = 400 from which: 2
⎛ 400 ⎞ kS = ⎜ ⎟ = 1600 ⎝ 10 ⎠ In order to have kS = 1600 we need: s S ⋅ 10 k S = 0.1 ⋅ k S Substituting: 0,1 ⋅ 1600 sS = = 0.4 400 The increase in s implies an upwards shift in the saving function. Now, in the initial K Y equilibrium point ( = 400, = 200 ) per capita saving exceeds per capita depreciation. N N Hence, per capita capital starts accumulating till we reach the new steady state K Y ( = 1600, = 400 ). During the adjustment process the growth rate of per capita output N N will be positive. Once we are at the new steady state the growth rate will become zero.
Y/Ns
Sambapati
δs(K/N) fs(K/N)
400 200
ss1f(K/N)
ssf(K/N)
400
K/Ns
1600
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4.3
Capital Accumulation and the Marginal Propensity to Save
a) K t +1 K t Y K − = s t −δ t N N N N
Kt +1 K1 K K − = s f ( t ) −δ t N N N N
The capital per worker grows until investment [s ⋅ Y / N ] is greater than the depreciation of capital [δ K / N ]. . Due to decreasing returns to capital the rate of growth of capital per worker is decreasing in time and will converge to a steady state value of zero. b) In steady state we have: K t +1 K t Y K − = s t −δ t = 0 N N N N
Plugging into the production function: K 3 / 4 N1 / 4 K s −δ =0 N N
⎛K⎞ s⎜ ⎟ ⎝N⎠ s
3/ 4
K = δ N
−δ
⎛K⎞ ⎜ ⎟ ⎝N⎠
K =0 N 3/ 4
1/ 4
⎛K⎞ =⎜ ⎟ ⎝N⎠
4
K ⎛s⎞ = ⎜ ⎟ = 2 4 = 16 N ⎝δ ⎠ Y ⎛K⎞ =⎜ ⎟ N ⎝N⎠
3/ 4
=8
c) The increase in the long run of s makes it possible to increase K/N and Y/N. Nevertheless, this will not affect their rates of growth which remain zero. At the beginning, the rate of growth of capital is positive because sY/N > δ K/N. So, the rate of growth of Y/N is positive too. Then, because of the decreasing returns to capital, K/N will grow more than Y/N. Thus, at the end we will have sY/N= δ K/N. The rate of growth converges to zero and the system converges to the steady state.
9
Y/N f(K/N) δ(K/N)
Y/N Y/N’
s’f(K/N) sf(K/N)
’
K/N
K/N
4.4
K/N
The Steady State
a) In steady state we have: s
K K =δ N N
0 .5
K K = 0 .1 N N
K Y = 25 and = N N
K =5 N
b) K/N = 36, Y/N = 6 The increase in the saving rate makes it possible that in period 0 savings per worker are greater than capital demand per worker. This leads to accumulation of capital per worker until we get back to equilibrium. K/N and Y/N increase and the new steady state is 1.
Y/N f(K/N) Y/N’=6
δ(K/N)
Y/N=5
s’f(K/N) 1 sf(K/N) 0
K/N=25
K/N’=36 10
K/N
c) Per capita consumption is: C Y δK s (1 − s ) = − = . δ N N N
With s = 0.5 (golden rule value which maximises per capita consumption) we have C = 2.5 . N
With s = 0.6,
C = 2.4 . N
The action taken by the government leads to a reduction in the level of per capita consumption because the marginal propensity to save in this case is no longer at the golden rule level.
4.5
The Solow Model with Population Growth
a) By definition, in steady state capital and output per capita are constant. If population grows at the rate g N > 0 , in steady state capital and output have to increase at the same rate. In steady state: ∆N ∆Y ∆N = = =n K Y N If this condition is satisfied each new worker will be endowed with an amount of per capita capital equal to that of the old workers. In order to maintain constant per capita ⎛ ⎛ K ⎞⎞ output it is necessary that per capita savings ⎜⎜ sf ⎜ ⎟ ⎟⎟ are equal to per capita ⎝ ⎝ N ⎠⎠ ⎛ K⎞ depreciation of capital ⎜ δ ⎟ plus the per capita capital amount for the new workers ⎝ N⎠
K⎞ ⎛ ⎜ g N ⎟ . Hence, in steady state: N⎠ ⎝ K ⎛K⎞ sf ⎜ ⎟ = ( g N + δ ) N ⎝N⎠
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Y/N
f(K/N)
Y/N*
(gN+δ)(K/N)
s(K/N)
K/N*
K/N
b) In steady state the growth rate of per capita output will be zero while the growth rate of total output will be g N .
4.6
Growth Rates
a) In order for the ratio
Y to grow at a zero rate, denominator and numerator must grow at NA
the same rate: g Y = g A + g N
growth rate of total output
(the growth rate of the denominator can be computed using logarithms). The growth rate of per capita output
Y , is: g Y / N = g Y − g N N
(the growth rate of a ratio is equal to the growth rate of the numerator minus the growth rate of the denominator). Since in steady state g Y = g A + g N , we have:
gY / N = g A + g N − g N = g A
growth rate of per capita output
b) By the production function Y = K 1−α ( NA) α = K 1−α N α Aα , the growth rate of total output is equal to the sum of the growth rates of its components each multiplied by its exponent: g Y = (1 − α ) g K + αg N + αg A
from which:
αg A = g Y − (1 − α ) g K − αg N = residual
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c) The Solow residual can be computed as:
residual = g Y − [αg N + (1 − α ) g K ] Using the given data: residual = 0.075 − [0.75 ⋅ 0.056 + 0.25 ⋅ 0.024] = 0.027 = 2.7%
The growth rate of labor productivity is:
g Y − g N = 7.5% − 5.6% = 1.9% The growth rate of technological progress is
gA =
residual
gA =
0.027 = 0.036 = 3.6% 0.75
α
. Hence:
d) The residual measures the actual contribute of technological progress to economic growth. The residual is smaller than the technological progress because the impact of technological progress on economic growth is limited by its weight in the production function α , which is smaller than 1.
4.7
The Solow Model with Population Growth and Technical Progress
a) Recall that AN is the amount of effective labor, a measure of how technology (A) increases labor’s productivity. The production function per effective worker is Y = AN
K AN
The equilibrium condition in steady state (where output per effective worker and capital per effective worker are constant) is:
K ⎛ K ⎞ sf ⎜ ⎟ = (δ + g A + g N ) AN ⎝ AN ⎠ Substituting the previous production function:
⎛ K s = ⎜⎜ AN ⎝ δ + g A + g N
⎞ ⎟⎟ ⎠
2
Y s = AN δ + g A + g N b) Consumption per effective worker is equal to output minus savings:
C Y Y ⎛ K ⎞ = − sf ⎜ ⎟ = (1 − s ) AN AN AN ⎝ AN ⎠ 13
Substituting:
C s (1 − s ) s = (1 − s ) = AN δ + gA + gN δ + gA + gN In order to compute the golden rule value of s which maximises consumption per effective worker, we derive the expression with respect to s:
∂ C 1 = (1 − 2s ) =0 ∂s AN δ + gA + gN
1 − 2s = 0 1 s= 2 c) Because of an increase in the rate of technological progress the line (δ + g A + g N )
K AN
shifts up. In the new equilibrium output per effective worker and capital per effective worker will be lower.
(gA1+gN+δ)(K/AN)
Y/AN
f(K/AN) Y/A0N Y/AN
(gA+gN+δ)(K/aN)
1
s(K/AN)
K/(A1N)
K/(AN)
K/(A0N)
Due to capital depreciation, the growth of population and the level of technological progress, there is a reduction in capital per effective worker. In the new equilibrium savings are not enough to compensate this reduction. Therefore, the levels of capital and output per effective worker start to decrease until they reach their new steady state values. During the transition, the rate of growth of output per effective worker will be negative. It will be zero again once the new equilibrium is reached (see the graph). The rate of growth of per capita output ( g A ), will increase to the new level g 1A . Analogously the rate of growth of total output increases to the new equilibrium level g 1A + g N .
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This apparent contradiction can be explained as follows. An increase in the rate of technological progress simply implies an increase in the number of units of effective labor. This implies a greater level of total output. But, since, as pointed out before, capital per effective labor unit decreases, output per effective labor unit decreases as well. gY/NA
0 t
4.8
The Solow Model with Population Growth and Technical Progress
a)
See previous exercise:
K ⎛ s = ⎜⎜ NA ⎝ δ + g A + g N
⎞ ⎟⎟ ⎠
2
Y s = NA δ + g A + g N
Substituting: 2
0.2 K ⎛ ⎞ =⎜ ⎟ =4 NA ⎝ 0.02 + 0.03 + 0.05 ⎠ 0.2 Y = =2 NA 0.02 + 0.03 + 0.05 The growth rate per worker is g A = 4% , the growth rate of total output is g A + g N = 8% , while the growth rate of output per effective worker is zero. c) In the first case the only effect is through the change in the saving rate which reduces output per effective worker and, temporarily, the rate of growth of total output: Y 0.15 = = 1.5 NA 0.02 + 0.03 + 0.05 In this case the growth rate of steady state output remains: g A + g N = 8% .
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In the second case we have an additional effect going through technological progress. Output per effective worker will reduce even more: Y 0.15 = = 1.25 NA 0.02 + 0.05 + 0.05 while the steady state growth rate of total output is: g A + g N = 10% . d) The boom can be due to an increment in the growth rate of technological progress. In fact, in steady state the economic growth rate is equal to the technological growth rate. Alternatively, the boom can b due to the adjustment process towards a higher level of per capita output and per capita capital. This can be due to an increase in the saving rate or to a reduction in the depreciation rate. In order to discriminate between the two possible explanations it is necessary to verify whether the growth rate of technological progress has same behaviour of the growth rate of per capita output. If it is so, then the first explanation is the correct one.
4.9
Growth Factors
The rate of growth of steady state is g A + g N Permanent effects
Hence, a permanent change in the rate of growth can be obtained only changing one of these two factors. This can be achieved by births control programs, which reduce g N , and incentives to research and development, which permanently increase g A . Temporary effects
The other two programs have only temporary effects on the rate of growth. The inflow of new immigrants who become part of the labor force is equivalent to an increase in N leading to a reduction in capital per effective worker and output per effective worker. In the new short run ⎛ ⎛ K ⎞1 ⎛ Y ⎞1 ⎞ ⎟ , there will be an excess of savings, growth of population (at rate equilibrium ⎜ ⎜ ⎜ ⎝ AN ⎟⎠ ⎜⎝ AN ⎟⎠ ⎟ ⎝ ⎠ g N ) and technological growth (at rate g A ). Thus, capital and output per effective worker will ⎛ K * Y *⎞ ⎟ . During the increase until they are back to their initial equilibrium levels ⎜⎜ ⎟ AN AN ⎝ ⎠ adjustment/transition process the rate of growth of per capita production (usually equal to zero) is positive and the rate of growth of total production will be greater than g A + g N . In the case of a one-time exogenous reduction in the level of technology (different from a change in the rate of growth of A), the process will be opposite to the one describe above. A 16
reduction in A implies an increase in
K Y and so in . The short run equilibrium levels AN AN
⎛⎛ K ⎞2 ⎛ Y ⎞2 ⎞ ⎟ . Starting from here, capital and output per effective worker will will be ⎜ ⎜ ⎜ ⎝ AN ⎟⎠ ⎜⎝ AN ⎟⎠ ⎟ ⎝ ⎠ ⎛ K * Y *⎞ ⎟ . During the decrease until they are back to their initial equilibrium levels ⎜⎜ ⎟ ⎝ AN AN ⎠ adjustment/transition process the rate of growth of per capita production (usually equal to zero) is negative and the rate of growth of total production will be smaller than g A + g N . Y/(AN)
(Y/AN))2
f(K/(AN))
Y/(AN) (Y/(AN))
(gA+gN+δ)(K/(AN)) 1
s(K/(AN))
(K/(AN))1
4.10
K/(AN)*
(K/(AN))2
K/(AN)
Productivity and the Labor market
a) We can rewrite the wage and price equations as: A W = t Pt 1 + µ W = At (1 − u t ) Pt At = At (1 − u t ) 1+ µ 1 µ = ut = 1 − 1+ µ 1+ µ from which:
The unemployment rate does not depend on the technology level, but only on the markup.
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b) If
A e = At −1 , by the wage and price equations, we have:
At = At −1 (1 − u t ) 1+ µ A 1 ut = 1 − t At −1 1 + µ The unemployment rate depends on productivity. If A decreases (increases), u will be greater (smaller) than its equilibrium level. If A remains constant, u will be equal to its equilibrium level. c) Using the formula in point a), unemployment is: ut =
0.25 = 0.2 = 20% 1 + 0.25
Using the price equation, real wages will be: W A 10 = t = =8 Pt 1 + µ 1.25 d) Starting from a situation of equilibrium (t = 1) where u t = 20% e
W = 8, A increases by P1
10%. Thus: A2 = 11 .
In t = 2, using the price equation: A W 11 = 2 = = 8 .8 P 2 1 + µ 1.25
In t = 3, At = At −1 . The unemployment rate goes back to its natural level because the wage equation shifs up. Using the equation at point a): u3 =
0.25 = 0.2 = 20% 1 + 0.25
Real wages remain at their previous level: W = 8.8 P3
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W/P t3
t2
E3
E2
t2=t3
8,8
8
t1
t1
E1 12%
20%
u
e) From the price equation we can see that real wages are proportional to A. This implies that ⎛ 1 ⎞ real wages will increase at the same growth rate as A ( g A ). Note: ⎜⎜ ⎟⎟ is considered as ⎝1+ µ ⎠ a constant coefficient. This can be proved as follows: In period 1 real wages will be: W 1 = A1 P1 1 + µ Since the growth rate of productivity is g A , in the second period we will have: 1 W 1 = A1 (1 + g A ) = A2 P 2 1+ µ 1+ µ The rate of growth of real wages W P2
− WP1 W P1
=
1 1+ µ
W P
between the two periods is:
A1 (1 + g A ) − 1+1µ A1 1 1+ µ
A1
=
1+ gA −1 = gA 1
Hence, if productivity grows at a rate of g A = 20% , and real wages will grow at the same rate.
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