L14 - The Neo-Classical Model of Economic Growth

Question 1

Why did economic growth dramatic change after the war?

Answer 1

In 1952 Moses Abramowitz surveying the theory of economic growth concluded that little progress had been made since the Classical period
This changed dramatically in the in post-war years, partly due to:
- the theoretical stimulus of Keynesian economics
- the western world embarked on a period of sustained growth
- economists increased knowledge of mathematics

Question 2

What are some Key National Accounts details?

Answer 2

• Total Output (Q) consists of consumption goods (C) and capital goods (I) so: Q = C + I
• National Income (Y) earned from productive activity is either spent on consumption goods (C) or saved (S) so: Y = C + I
• In equilibrium Y= Q and thus S = I

Question 3

What is the Production Function?

Answer 3

• A production function can be written as: Y=F(K, N) where Y is output, K is the stock of capital (machines) and N is the supply of labour. The production function defines the technology that translates inputs into output
• Constant returns to scale is when if all inputs are doubled output doubles.
• Generally: zY=F(zK, zN), where z can be any positive number. If K and N are both increased by 5% z =1.05 and Y also rises by 5%.
• If z =1/N then the production function becomes: Y/N=F(K/N, 1) This says that output per worker increases as capital per worker increases, which can be written more easily as y=f(k)

Question 4

What is the Marginal Product of Capital?

Answer 4

• The Neo-Classical model, like the Classical model before it, focuses on the role of capital – specifically a diminishing marginal product of capital.
• This says that as more and more capital is employed for a given labour force, the marginal product of the capital (MPK) will eventually decline
• In other words the production function is non-linear and becomes flatter as more of the variable input in added – as shown by the following diagram

Question 5

What does the Basic Neo-Classical Model of Economic Growth look like?

Answer 5

• With Output per head y=Y/N on the y-axis and Capital per head k=K/N on the x-axis.
• With a positive gradient curve of y=f(k)
• the MPK given by slope of production function. Tangents get flatter as k rises showing declining MPK

Question 6

How is Savings defined in the Neo-classical Growth model?

Answer 6

• It is assumed that savings depends positively on income (Y) then S= sY where s is the average (and marginal) propensity to save and 0 < s < 1
  Dividing through by N gives the expression in per capita terms and replacing y with its equal f(k) gives:
• S/N = s(Y/N) = sy = sf(k)
• which says that savings per worker will be fixed proportion of output per worker

Question 7

How is Investment defined in the Neo-classical Growth model?

Answer 7

• Investment is defined as I = ∆K+dK – new investment (∆K) plus depreciation (dK).
• To determine the equilibrium value of K/N, the question is how much investment is needed to keep K/N constant? There are 2 factors: the rate of depreciation, d; and the rate of growth of the labour, n
• If capital depreciates at rate d per period then investment per head must be d x k to stop the K/N from falling
• If labour grows at rate n per period, then an additional investment of n x k will be needed to keep K/N constant

Question 8

What is Equilibrium Savings and Investment?

Answer 8

• Hence (d+n)k is called the required investment to keep K/N constant
  The long-run equilibrium condition is:
• Δk= i – (d+n)k = 0
• In equilibrium S = I (and so s = i) which means for a constant K/N we have:
• sf(k)= (d+n)k
• This is the equilibrium growth relation where savings and investment are equal. It says that output, capital and labour all grow at rate n as illustrated below

Question 9

What does the Neo-Classical Growth model look like Including Saving and Investment?

Answer 9

• With Output per head y=Y/N on the y-axis and Capital per head k=K/N on the x-axis.
• With a positive gradient curve of y=f(k)
• Another positive gradient curve of savings y=sf(k) but it is smaller than the curve y=f(k)
• a straight diagonal line of Investment that maintains the capital labour ratio, (d+n)k
• Equilibrium it at the point where y=sf(k) and (d+n)k intercept and the corresponding value for y for the function y=f(k) given the value of k of the equilibrium point

Question 10

How can the Neo-Classical Growth model including Saving and Investment be interpreted on a graph?

Answer 10

• The point E is the steady-state level of output and capital per head. At levels of k to the left of k{1} k is rising; at points to the right of k{1}, k is falling.
  Although output per head is constant the economy’s total output,Y, will be growing at the same rate as the total population, n. This is an important conclusion for the Neo-classical model.

Question 11

How can you calculate rates of change of y?

Answer 11

equilibrium in the Neo-classical model is a dynamic meaning it is a growth rate not a stationary point:
Y = Y/N –> Output per head
y(hat) = Y(hat)-N(hat)

• where (hat) is percentage rate of change
• in the diagram at point y, y(hat) is 0 therefore Y(hat)=N(hat)=n
• where the rate of growth of output is the same as the rate of growth of the labour force
• so at equilibrium y is growth at rate n

Question 12

How can calculate the rate of change on k?

Answer 12

k=K/N –> capital stock per head
k(hat)=K(hat)-N(hat)
- where (hat) is percentage rate of change
- in the diagram k(hat)=0 therefore K(hat)=N(hat)=n
- where the rate of growth of capital stock is the same as the rate of growth of the labour force
- so at equilibrium k is growth at rate n

Question 13

What happens when there is a rise in the saving rate?

Answer 13

• This is captured on the chart by s rise in the sf(k) line, as s rises from s{1} to s{2}.
• Gross investment will now exceed required investment and the capital stock will increase until k reaches k{2}. - While k is between k{1} and k{2} average labour productivity will be increasing, so the growth of Y will be exceeding that of N. Once equilibrium is reached Y/N is again constant
• Thus savings alone cannot permanently raise the growth rate, because of the diminishing marginal product of capital

Question 14

What happens to the Growth Rate of Labour during a increase in the Savings Rate?

Answer 14

• growth rate has not permanently increased only temporarily
• during the time between the original capital stock per head k{1} to the new capital stock per head k{2} there will be a jump in the growth rate because savings is higher than investment so as investment rises to meet it, but it will come back down once we reach k{2}

Question 15

What is the summary of the Neo-Classical Growth model including Saving and Investment?

Answer 15

• The Neo-classical growth model says that economic growth is dependent on technical progress – which raises the productivity
• A rise in the savings rate will only lead to a temporary rise in the growth rate as the K/N ratio rises to the new equilibrium – where production is more capital intensive. Once this level is reached the economy grows at rate n, as before
• The model suggests convergence in Y/N over time with poor countries (or regions) growing faster than rich countries (or regions)

Question 16

What are two shocks that could affect the Neo-Classical growth Model?

Answer 16

• savings shock

- technical process shock

Question 17

What is a Technical Progress Shock?

Answer 17

• Technical progress is an improvement in knowledge that enables a higher output to be produced from existing resources
• A technical improvement shifts up the whole production function (and hence also the savings function) to give a higher level of average labour productivity (Y/N)
• If technology continues to improve the production function will continue to shift out and aggregate output will continue to grow over time

Question 18

What does a Technical Progress Shock look like on the Neo-Classical Growth model?

Answer 18

• This has caused a change in the production function
• A technical Progress Shock move y=f{1}(k) up to y=f{2}(k) which also increased the saving function from i=sf{1}(k) to i=sf{2}(k)
• both having at equilibrium was at k{2} and to y{2}
• the increase in Output head is much more than with a shock to savings
• technical shock is said to be continuous so y with be rising continuously as well as k

Question 19

What is a Summary of the shocks that can occur in the Neo-Classical growth model?

Answer 19

• The Neo-classical growth model says that economic growth is dependent on technical progress – which raises the productivity –> but become more capital intensive
• A rise in the savings rate will only lead to a temporary rise in the growth rate as the K/N ratio rises to the new equilibrium – where production is more capital intensive. Once this level is reached the economy grows at rate n, as before
• The model suggests convergence in Y/N over time with poor countries (or regions) growing faster than rich countries (or regions)

Question 20

What is the Framework for the Sources of Growth?

Answer 20

-The production function: Y=AF(K, N), where Y is output, A is technical progress, K is the capital stock and N the level of labour inputs.
- In terms of changes this can be written as:
ΔY = F(K, N)ΔA+F{K}ΔK+F{N}ΔN
-where F{K} and F{N} are the marginal product of capital (MPK) and the marginal product of labour (MPN) respectively
Divide through by Y: ΔY/Y = F(K, N)ΔA/Y+F{K}ΔK/Y+F{N}ΔN/Y
Multiply the first RHS term top and bottom by K and the second RHS term top and bottom by N, which gives:

ΔY/Y = ΔA/A+((F{K}xK)/Y)ΔK/K+(F{N}xN/Y)ΔN/N

Question 21

What do the terms in the framework for the sources of growth mean?

Answer 21

The term (F{K} x K) equals total capital income (income per unit times the number of units) and the term(F{K}xK)/Y equals capital share of total output
Similarly for labour. If labour is paid a real wage equal to its marginal product then total labour income is   (F{N} x N) and (F{N} x N) /Y equals labour share of total income 
-As there are only two terms in the production function then the shares of capital and labour income must sum to one. Let capital share be α and labour’s share be (1-α)

Question 22

What can the framework for the sources of growth be simplified to?

Answer 22

ΔY/Y = ΔA/A+αΔK/K+(1-α)ΔN/N

This says that the growth of the economy depends upon

• the rate of technical progress
• the rate of growth of the capital stock weighted by the share of capital in income and
• the rate of growth of the labour force weighted by the share of labour in total income

Question 23

What is Total Factor Productivity?

Answer 23

• Rewriting the equation in terms of ΔA/A we can redefine ΔA/A as total factor productivity (TFP). That is, all growth not accounted for by labour and capital, such as entrepreneurship, or the legal environment
• ΔA/A = ΔY/Y-αΔK/K-(1-α)ΔN/N

Question 24

What is some Empirical Analysis of Factor Shares?

Answer 24

• Factors shares have remained largely constant over a long period of time in both the UK and the US (and indeed in other developed countries) so from the data α = 0.3 and hence (1 - α ) = 0.7.
  A table can be drawn up showing the contributions to UK economic growth from Angus Maddison (1991)

Question 25

What are some comments that can be made about the table showing the contributions to UK economic growth from Angus Maddison (1991)?

Answer 25

• The negative value of labour inputs reflects the shorter working hours in the post-war world
• Madison’s calculations assume constant returns to scale, but economic growth is most likely characterised by increasing returns to scale
• The TFP calculation is not possible to do if there are increasing returns to scale – so Madison may have underestimated the effect of TFP on growth