1. Economic Growth II: Technology, Empirics, and Policy
Chapter 9 of Macroeconomics, 8th edition, by N. Gregory Mankiw
ECO62 Udayan Roy

2. Recap: Solow-Swan, Ch. 7
L and K are used to produce a final good Y = F(K, L)
k = K/L and y = Y/L = f(k) are per worker capital and output
The population is P, but a fraction u is not engaged in the production of the final good. Therefore, L = (1 – u)P.
Both P and L grow at the rate n.
A fraction s of Y is saved and added to capital
A fraction δ of K depreciates (wears out)

3. Recap: Solow-Swan, Ch. 8
In the long run, the economy reaches a steady state, with constant k and y

4. Recap: Solow-Swan, Ch. 8
In the long run, the economy reaches a steady state, with constant k and y
Like the per-worker variables, k and y, per-capita capital and output are also constant in the long run
Total capital (K) and total output (Y) both increase at the rate n, which is the rate of growth of both the number of workers (L) and the population (P)

5. Recap: Solow-Swan, Ch. 8 Variable Symbol Steady state behavior
Capital per worker k
Constant
Income per worker
y = f(k)
Saving and investment per worker sy
Consumption per worker
c = (1 – s)y
Labor L
Grows at rate n
Capital K
Income
Y = F(K, L)
Saving and investment sY
Population P
Capital per capita
(1 – u)k
Income per capita
(1 – u)y
Saving and investment per capita
(1 – u)sy
Consumption per capita
(1 – u)c

6. The sad lesson of Solow-Swan
It is an undeniable fact that our standards of living increase over time
Yet, Solow-Swan cannot explain this! Why?
Solow-Swan relies on capital accumulation as the only means of progress
Therefore, the model’s failure to show economic progress indicates that we must introduce some means of progress other than capital accumulation

7. Technological Progress
Maybe Solow-Swan fails to show economic progress because there is no technological progress in it
We need to create a theory with technological progress
But how?

8. Technological Progress
A simple way to introduce technological progress into the Solow-Swan model is to think of technological progress as increases in our ability to multitask

9. Technological Progress
Imagine that both population and the number of workers are constant but that steady increases in the workers’ ability to multitask creates an economy that is equivalent to the Solow-Swan economy with steadily increasing population

10. Technological Progress
In such an economy, total output would be increasing—exactly as in the Solow-Swan economy with steady population growth—but without population growth
That is, under multitasking technological progress, per capita and per worker output would be steadily increasing
In this way, a simple re-interpretation of the Solow-Swan economy gives us what we were looking for—steadily increasing income per worker

11. Efficiency of Labor Specifically, section 9−1 defines a new variable
E is the efficiency of labor
Specify some date in the past, say 1984, and arbitrarily set E = 1 for 1984.
Let’s say that technological progress has enabled each worker of 2011 to do the work of 10 workers of 1984.
This implies that E = 10 in 2011.
Therefore, E in any given year is the number of 1984 human workers that one human worker of the given year is equivalent too.

12. Efficiency of Labor
The old production function F(K,L) no longer applies to both 1984 and 2011
Suppose K = 4 in both 1984 and 2011
Suppose L = 10 in 1984 and L = 1 in 2011
The old production function F(K,L) will say that output is larger in 1984
But we know that output is the same in both years because just one worker in 2011 can do the work of 10 workers of 1984
We need a new production function: F(K, E ✕ L)

13. Y = F(K, E ✕ L)
In other words, although the number of human workers is 10 in 1984 and 1 in 2011, the effective number of workers is 10 in both years,
and that’s what matters in determining the level of output
The effective number of workers is E ✕ L

14. Efficiency of Labor
Assumption: the efficiency of labor grows at the constant and exogenous rate g

15. Production
As the production of the final good no longer depends only on the number of workers, but instead depends on the effective number of workers, …
… we replace the production function Y = F(K, L) by the new production function Y = F(K, E ✕ L)

16. From “per worker” to “per effective worker”
Similarly, we will now redefine k, which used to be capital per worker (K/L), as capital per effective worker: k = K/(E ✕ L)
Likewise, we will now redefine y, which used to be output per worker (Y/L), as output per effective worker: y = Y/(E ✕ L)

17. From “per worker” to “per effective worker”
As a result of the redefinition of k and y, we still have y = f(k), except that the definitions of y and k are now in “per effective worker” form
sy = sf(k), is now saving (and investment) per effective worker
Only the growth rate of effective labor is slightly different

18. From “per worker” to “per effective worker”
In Chapter 8, what mattered in production was L, the number of workers, and the growth rate of L was n
Now, however, what matters in production is E ✕ L, the effective number of workers, and the growth rate of E ✕ L = growth rate of E + growth rate of L = g + n

19. From “per worker” to “per effective worker”
Recall from Chapter 8 that the break-even investment per worker was (δ + n)k
This will have to be replaced by the break-even investment per effective worker
We can do this by redefining k as capital per effective worker (which we have already done) and by replacing n by g + n
Therefore, break-even investment per effective worker is now (δ + n + g)k

20. Dynamics: algebra Ch. 8 No technological change
Ch. 9 Technological Progress

21. Dynamics: graph
As in Ch. 8, in the long run, k and y reach a steady state at k = k* and y = y* = f(k*)

22. Describing the Steady State
We just saw that k is constant in the steady state
That is, k = K/(E ✕ L) is constant
Therefore, in terms of growth rates, kg = Kg – (Eg + Lg) = Kg – (g + n) = 0
Therefore, the economy’s total stock of capital grows at the rate Kg = g + n

23. Describing the Steady State
Capital per worker (K/L) grows at the rate Kg –Lg = g + n – n = g
Therefore, the per-worker capital stock, which was constant in Chapter 8, grows at the rate g
As each worker’s ability to multitask increases at the rate g, the capital used by a worker also increases at that rate

24. Describing the Steady State
y = f(k) is constant in the steady state
That is, y = Y/(E ✕ L) is constant
Therefore, in terms of growth rates, yg = Yg – (Eg + Lg) = Yg – (g + n) = 0
Therefore, the economy’s total output grows at the rate Yg = g + n
Recall that this is also the growth rate of the total stock of capital, K.

25. Describing the Steady State
Output per worker (Y/L) grows at the rate Yg – Lg = g + n – n = g
Therefore, the per-worker output, which was constant in Chapter 8, grows at the rate g
Recall that this is also the growth rate of per-worker capital, K/L.

26. Progress, finally!
We have just seen that if we introduce technological progress in the Solow-Swan theory of long-run growth, then in the economy’s steady state
Per-worker output (Y/L) increases at the rate g, which is the rate of technological progress
This is a major triumph for the Solow-Swan theory

27. Solow-Swan Steady State
Table Steady-State Growth Rates in the Solow Model With Technological Progress

28. Solow-Swan Steady State
Remember from Chapter 8 that, when the production function follows the Cobb-Douglas form, the steady state value of k = k* was given by the formula
Now the formula changes to

29. Technological Progress: where does it come from????
But a puzzle remains …
So far, the rate of technological progress, g, has been exogenous
We need to ask, What does g depend on?
We need to make g endogenous

30. Endogenous Technological Progress
Remember that in Chapter 8 we had distinguished between the population (P) and the number of workers (L)
We had defined the exogenous variable u as the fraction of the population that does not produce the final good
Therefore, we had L = (1 – u)P or L/P = 1 – u
In Ch. 8 we had interpreted u as the long-run unemployment rate
Now, we’ll reinterpret u as the fraction of the population that does scientific research

31. Endogenous Technological Progress
Once u is seen as the fraction of the population that is engaged in scientific research, it makes sense to assume that …
Assumption: the rate of technological progress increases if and only if u increases
This assumption is represented by the technology function g(u)
Example: g(u) = g0 + guu

32. Endogenous Technological Progress
We now have a theory that gives an answer to the following question: Why is growth in living standards slow in some cases and fast in others?
Growth in per-worker output is fast when u is high.
That is, our standards of living grow rapidly when we invest more heavily in scientific research

34. Productivity Slowdown
There was a worldwide slowdown in economic growth during Why?
Table Growth Around the World

35. Growth Accounting
Table Accounting for Economic Growth in the United States