Economic growth: Introduction
EC 235 | Fall 2023
Materials
Required readings:
- Blanchard, ch. 10.
Prologue
Prologue
So far, our lectures focued on the short- and medium-run features of the macroeconomy.
In both time frames, economic fluctuations dominate the picture.
However, when looking at the behavior of aggregate output/income over time, fluctuations become less apparent and economic growth dominates.
Thus, we now turn our attention to the long-run, with the purpose of understanding what determines economic growth.
The standard of living
The standard of living
Thinking about growth
Thinking about growth
The conventional approach to economic growth is due to the work of Robert M. Solow.
The starting point of such approach is through an aggregate production function:
\[ Y = F(K, N) \]
where Y is aggregate output; K is the capital stock; and N, the number of employed workers.
What are some of the limitations of such modeling approach?
Thinking about growth
Given an aggregate production function, how much output (Y) can be produced for given quantities of the capital and labor inputs, K and N, respectively?
The answer lies on technology.
- Countries with more advanced technology will produce more output from the same quantities of K and N than will an economy with less advanced production methods.
Thinking about growth
Now, time to think about some restrictions we may impose on the aggregate production function.
The first is thinking about what happens to F(K, N) when we, for instance, double both the number of workers and the amount of capital in the economy.
We’ll assume constant returns to scale (CRS):
\[ F(2K, 2N) = 2Y \]
More generally:
\[ F(xK, xN) = xY \]
Thinking about growth
What if we assume that only one factor of production increases?
Even under constant returns to scale, there are decreasing returns to each factor.
There are decreasing returns to capital:
- Given labor, increases in capital lead to smaller and smaller increases in output.
There are decreasing returns to labor:
- Given capital, increases in labor lead to smaller and smaller increases in output.
Thinking about growth
From the aggregate production function, we can specify it in terms of the labor input, N:
\[ \dfrac{Y}{N} = F(\dfrac{K}{N}, \dfrac{N}{N}) = F(\dfrac{K}{N}, 1) \]
Thinking about growth
Thinking about growth
Thinking about growth
Two key things from the previous charts:
- Capital accumulation by itself cannot sustain growth;
- Sustained growth requires sustained technological progress.
Interactions between output and capital
Interactions between output and capital
The determination of output over the long-run depends on two relations between output (Y) and capital (K):
The amount of capital determines the amount of output being produced;
The amount of output being produced determines the amount of saving and, in turn, the amount of capital being accumulated over time.
From the aggregate production function normalized by labor, we may simplify things by writing:
\[ \dfrac{Y}{N} = F\bigg(\dfrac{K}{N}, 1\bigg) = f\bigg(\dfrac{K}{N}\bigg) \]
Interactions between output and capital
Again:
\[ \dfrac{Y}{N} = F\bigg(\dfrac{K}{N}, 1\bigg) = f\bigg(\dfrac{K}{N}\bigg) \]
This relation implies that we assume employment N to be constant over the long-run.
This way, we are able to focus on the process of capital accumulation over time and its effects on growth.
Interactions between output and capital
If we introduce time indexes, we may write:
\[ \dfrac{Y_t}{N} = f\bigg(\dfrac{K_t}{N}\bigg) \]
Interactions between output and capital
Next, we move on to how output and capital accumulation are related over time.
We will keep assuming a closed economy, with investment being equal as the sum of private and public savings in equilibrium:
\[ I = S + (T-G) \]
For simplicity, we will assume a balanced budget (i.e., public savings are equal to zero), so investment equals private savings:
\[ I = S \]
Interactions between output and capital
Private savings are proportional to aggregate income:
\[ S = sY \]
where the parameter s is the saving rate (\(0 < s < 1\)).
Combining what we have so far, we can write:
\[ I_t = sY_t \]
The above relation states that investment is proportional to aggregate income/output.
Interactions between output and capital
Turning to the capital stock, K, we will assume that it depreciates at a rate δ per year.
Equivalently, a proportion (1 - δ) remains intact from one year to the next.
The evolution of the capital stock is then given by:
\[ K_{t+1} = (1 - \delta)K_t + I_t \]
Interactions between output and capital
Normalizing the previous relation by the number of employed workers, N:
\[ \dfrac{K_{t+1}}{N} = (1 - \delta)\dfrac{K_t}{N} + \dfrac{I_t}{N} \]
Rearranging…
\[ \dfrac{K_{t+1}}{N} - \dfrac{K_t}{N} = s\dfrac{Y_t}{N} - \delta\dfrac{K_t}{N} \]
Interactions between output and capital
Again:
\[ \dfrac{K_{t+1}}{N} - \dfrac{K_t}{N} = s\dfrac{Y_t}{N} - \delta\dfrac{K_t}{N} \]
This relation implies that the change in the capital stock per worker, represented by the difference between the two terms on the left, is equal to savings per worker, represented by the first term on the right, minus depreciation, represented by the second term on the right.
EC 235 - Prof. Santetti