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Economic Growth II: Technology, Empirics, and Policy

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Economic Growth II: Technology, Empirics, and Policy

Chapter 9 of Macroeconomics, 8thedition, by N. Gregory Mankiw

ECO62UdayanRoy

- 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)

- In the long run, the economy reaches a steady state, with constant k and y

- In the long run, the economy reaches a steady state, with constant k and y
- Like the per-workervariables, 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)

- 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

- 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?

- 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

- 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

- 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

- 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.

- 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)

- 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

- Assumption: the efficiency of labor grows at the constant and exogenous rate g

- 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)

- 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)

- 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

- 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

- 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

Ch. 8 No technological change

Ch. 9 Technological Progress

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

- 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

- 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

- 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.

- 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.

- 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

- Table 9.1 Steady-State Growth Rates in the Solow Model With Technological Progress

- 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

- 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

- 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

- 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

- 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

- There was a worldwide slowdown in economic growth during 1972-1995. Why?

- Table 9.3 Accounting for Economic Growth in the United States