Stock vs Flow Variables¶
- A stock variable is measured at a specific point in time. It represents a quantity that exists at that instant (i.e., a snapshot or balance).
- A flow variable is measured over an interval of time. It represents a rate of change or movement over time.
| Stock | Inflow | Outflow |
|---|---|---|
| Lake | Rain | Evaporation |
| National debt | Budget deficit | Budget surplus |
| Wealth | Income | Consumption |
| Capital | Investment | Depreciation |
Capital Accumulation¶
| Start of 2025 | Flow | End of 2025 | |
|---|---|---|---|
| Old Capital | $K_{2025}$ | Depreciation | $(1-\delta)K_{2025}$ |
| New Capital | 0 | Investment | + $I_{2025}$ |
| Capital Stock | $= K_{2026}$ |
- $K_{2026} = (1-\delta)K_{2025} + I_{2025}$
- Capital stock available for production in 2026 is the sum of
- the non-depreciated old capital at the start of 2025
- and investment in new capital over the course of 2025.
Capital Law of Motion
- The above accounting of the capital stock should apply in any year
$$ K_{t+1} = (1-\delta)K_{t} + I_{t} $$
- Which can be expressed as a difference equation to track the changes in the capital stock
$$ \Delta K_{t+1} = K_{t+1} - K_t = I_{t} - \delta K_{t} $$
If $I_{t} = \delta K_{t}$, then the capital stock is "steady" and the growth model is in an equilibrium.
Assuming a constant number of workers, this can be written as
$$ \frac{\Delta K_{t+1}}{N} = \frac{I_{t}}{N}- \delta \frac{K_{t}}{N} $$
Savings¶
- What is $I_t$?
- Suppose households either consumer or save their income
$$ Y_t = C_t + S_t $$
- Assume that households save a fixed fraction of their income. Let the savings function be $S_t = sY_t$, where $s$ is the savings rate.
- Assume that there is a financial market equilibrium where $S_t = I_t$.
- All excess wealth from households, $S_t$, gets converted by banks into loans, bonds, or stocks, which are then used to invest in new capital, $I_t$.
- Thus, $I_t = sY_t$.
Solow Growth Model¶
- This Solow Growth Model has 3 variables $\{Y,K,I\}$ (with $N$ fixed) in the following 3 equations
\begin{gather*} \frac{Y_t}{N} = \left(\frac{K_t}{N}\right)^\alpha \\ \frac{\Delta K_{t+1}}{N} = \frac{I_{t}}{N} - \delta \frac{K_{t}}{N} \\ \frac{I_t}{N} = s\frac{Y_t}{N} \end{gather*}
There are also 3 parameters
- The output elasticity of capital, $\alpha$.
- The depreciation rate, $\delta$.
- The savings rate, $s$.
We could add a 4th variable for consumption, $C$, and equation
$$ Y = C + I $$
which looks like the expenditure approach to GDP without the $G$ and $NX$.
- The solid-blue line is the production function.
- The dashed-green line is the investment/savings function.
- The dashed-red line is the depreciation function.
- If investment equal depreciation, then capital is at its long-run equilibrium or steady-state level.
Steady State¶
- Another name for a long-run equilibrium in a dynamic economic model is steady state.
- In this model, the capital stock is a state variable because given its value in any year allows us to determine the level of output, investment, and consumption per person.
- Thus, if we assume capital is at its steady state, then we can calculate the steady-state, or long-run equilibrium, values for the other variables in the model.
If the model starts in steady state, then it stays there until something happens. Until then $K_{t+1} = K_t = K^*$ or $\Delta K_{t+1} = 0$.
The steady-state system of equations is
\begin{gather*} \frac{Y^*}{N} = \left(\frac{K^*}{N}\right)^\alpha \\ 0 = \frac{I^*}{N} - \delta \frac{K^*}{N} \\ \frac{I^*}{N} = s\frac{Y^*}{N} \end{gather*}
Combining the steady-state equations yields
\begin{gather*} s\left(\frac{K^*}{N}\right)^\alpha = \delta \frac{K^*}{N} \end{gather*}
The solution for steady-state capital per worker is
\begin{gather*} \frac{K^*}{N} = \left(\frac{s}{\delta}\right)^{\frac{1}{1-\alpha}} \end{gather*}
Equilibrium¶
Savings¶
Technological Progress¶
We can relax the assumptions
- that $A = 1$ by assuming $A$ grows at rate $g_A$.
- that $N$ is fixed by assuming $N$ grows at rate $g_N$.
However, $A$ increasing over time complicates our illustration of the model since output per worker would always be increasing.
We need to write production in terms of output per effective worker
$$ \left(\frac{Y}{AN}\right)_t = \left(\frac{K}{AN}\right)^\alpha_t $$
Equilibrium
Now the relevant state of the model is how much capital is available to each effective worker, $K/(AN)$ at any time $t$.
Investment is still an inflow and depreciation is still an outflow to the capital stock, as before.
But now there are two new outflows, $g_A$ and $g_N$, because $A$ and $N$ appear in the denominator of the state.
The equilibrium (steady-state) condition is still that inflow equals outflow
$$ \left(\frac{I}{AN}\right)^* = (\delta + g_A + g_N) \left(\frac{K}{AN}\right)^* $$
Dynamically, the capital (per effective worker) accumulation equation becomes
$$ \Delta \left(\frac{K}{AN}\right)_{t+1} = \left(\frac{I}{AN}\right)_t - (\delta + g_A + g_N) \left(\frac{K}{AN}\right)_t $$
Balanced Growth
The following table is based on Blanchard, Macroeconomics (9th Edition, 2025), Chapter 12, Page 247, Table 12-1
| Variable | Growth Rate |
|---|---|
| $\frac{K}{AN}, \frac{Y}{AN}$ | 0 |
| $\frac{K}{N}, \frac{Y}{N}$ | $g_A$ |
| $K, Y, AN$ | $g_A + g_N$ |
- "Because output, capital, and effective labor all grow at the same rate in steady state, the steady state of this economy is also called a state of balanced growth."
- "Changes in the saving rate do not affect the steady-state growth rate. But changes in the saving rate do increase the steady-state level of output per effective worker." (i.e., the same result as the model with $A=1$ and $N$ fixed)