Solow: Model and Measurement - Econ 204

From first order equation

\begin{aligned} f(k_t,n_t) &= r_t k_t + w_t n_t \\ &= \alpha \frac{f(k_t,n_t)}{k_t}k_t + (1-\alpha)\frac{f(k_t,n_t)}{n_t}n_t \end{aligned}

Hence

\alpha = \frac{r_t k_t}{f(k_t,n_t)} = 1 - \frac{w_t n_t}{f(k_t,n_t)}.

With Cobb–Douglas production, capital’s and labor’s shares of output are constant and equal to $\alpha$ .

Summary¶

\begin{aligned} \text{Product approach:}\quad & y_t = k_t^\alpha n_t^{1-\alpha} \\ \text{Expenditure approach:}\quad & y_t = c_t + i_t + g_t + (x_t - m_t) \\ \text{Income approach:}\quad & y_t = r_t k_t + w_t n_t \\ \text{Law of motion:}\quad & k_{t+1} = (1-\delta)k_t + i_t \end{aligned}

Match the model to the data¶

Choose parameters so model behavior matches empirical moments
Long-run ratios (e.g. factor shares) are relatively stable
Parameters are calibrated to match means of these ratios

Model and measurement¶

Income decomposition:

y_t = r_t k_t + w_t n_t

With Cobb–Douglas:

\alpha = \frac{r_t k_t}{y_t} = 1 - \frac{w_t n_t}{y_t}

Set $\alpha$ to match the average capital share of income

Model and measurement (continued)¶

Along a detrended balanced growth path:

\delta k = i \qquad \Rightarrow \qquad \delta = \frac{i}{k}

Use observable ratios:

\delta = \frac{i}{y}\left(\frac{k}{y}\right)^{-1}

Simplifying assumptions¶

Closed economy

x_t = m_t = 0

Fixed savings rate

i_t = s y_t

Reduced model¶

\begin{aligned} y_t &= f(k_t,n_t) \\ y_t &= c_t + i_t \\ y_t &= r_t k_t + w_t n_t \\ k_{t+1} &= (1-\delta)k_t + i_t \\ i_t &= s y_t \end{aligned}

Per-capita formulation¶

\begin{aligned} y_t &= f(k_t) \\ y_t &= c_t + i_t \\ y_t &= r_t k_t + w_t \\ k_{t+1} &= (1-\delta)k_t + i_t \\ i_t &= s y_t \end{aligned}

(All variables are per capita.)

The Solow model¶

Mechanics of national accounts
Strong behavioral assumption: constant savings rate
For any $s \in [0,1]$ , a unique steady state exists

\delta k = s y

y = k^\alpha

k^* = \left(\frac{s}{\delta}\right)^{\frac{1}{1-\alpha}}

Steady-state values¶

\begin{aligned} y^* &= \left(\frac{s}{\delta}\right)^{\frac{\alpha}{1-\alpha}} \\ i^* &= s y^* \\ c^* &= (1-s)y^* \\ r^* &= \alpha \frac{\delta}{s} \end{aligned}

Studying the model numerically¶

Simple law of motion
Low-dimensional dynamics
Naturally suited for computational implementation

Intertemporal choices¶

Returns now vs. later:
- consume or invest?
- school or work?
- rebuild or replace?
- eat the cake now?

Question 1¶

$\alpha = 0.35$ , $\delta = 0.06$ , $s = 0.20$
50% capital destruction
How much capital is rebuilt after 50 years?

Question 2¶

Write pseudo-code and program
Track $y_t$ , $i_t$ , $c_t$
Compute growth rates
Plot levels and growth rates

Question 3¶

How long until output is within 0.5% of its pre-shock level?

Pseudo-code (Question 1)¶

α ← 0.35
δ ← 0.06
s ← 0.20
k* ← (s/δ)^(1/(1-α))

k1 ← 0.5 · k*
for t = 1,…,50:
    y_t ← k_t^α
    i_t ← s y_t
    k_{t+1} ← (1-δ)k_t + i_t

return k_51 / k*