Level: Graduate | Field: Advanced Macroeconomics
Learning objectives
Derive factor demands, the capital Euler equation, and Tobin's \(q\); understand depreciation, adjustment costs, and investment dynamics.
1. Technology and static optimization
A competitive firm produces
Taking factor prices as given, profit maximization implies
where \(r_t^k\) is the rental rate of capital before depreciation.
2. Capital accumulation
Capital evolves according to
with depreciation rate \(\delta\in(0,1)\). Investment is a flow; installed capital is a stock. Confusing these objects leads to incorrect timing in dynamic models.
3. Investment with adjustment costs
Let investment adjustment costs be \(\frac{\phi}{2}(i_t/k_t-\delta)^2k_t\). The firm chooses investment to maximize discounted dividends. Define marginal value of installed capital as \(q_t\). The investment FOC is
The capital valuation equation equates the value of installed capital to its marginal product plus the discounted continuation value:
where \(M_{t,t+1}\) is the stochastic discount factor.
4. Balanced growth and normalization
When technology grows, detrend real quantities by the appropriate technology or labor-augmenting trend. A stationary model requires consistent scaling of output, consumption, investment, and capital. Verify that the normalized capital law preserves the original resource constraint.
5. Worked example
With Cobb–Douglas production, \(A=1\), \(k/n=4\), and \(\alpha=0.36\), the marginal product of capital is \(0.36(4)^{-0.64}\). The wage is \(0.64(4)^{0.36}\). These prices satisfy the competitive firm's factor-share conditions.
6. Computational lab
Compute factor prices and a steady-state capital stock. Add quadratic adjustment costs and simulate the response of investment and Tobin's \(q\) to a temporary productivity shock. Check the capital law period by period.
Problem set
- Derive both factor-demand equations.
- Explain why investment can respond more slowly when \(\phi\) is large.
- Derive the investment FOC from the adjustment-cost function.
- Interpret the capital valuation equation.
Coding exercises
These exercises combine real U.S. macroeconomic data (the quarterly macrodata series in statsmodels, 1959–2009,
and annual series from the wooldridge package) with numerical solutions of the models in this lecture. Install once
with pip install numpy scipy pandas statsmodels wooldridge. Try the starter code first, then open the solution. Every
printed result below comes from running the solution code exactly as shown.
Exercise 2.1 — Building a capital stock and a Solow residual for the U.S.
Data: macrodata — U.S. quarterly real GDP, real investment, unemployment rate and population, 1959–2009.
- Build a capital stock with the perpetual-inventory method \(K_{t+1}=(1-\delta)K_t+I_t\), \(\delta=0.025\) per quarter, initializing \(K_0=I_0/(g+\delta)\) with \(g\) the average investment growth rate.
- Proxy labor input by employment, \(N_t=pop_t(1-u_t/100)\), and compute the Solow residual \(\log z_t=\log Y_t-\alpha\log K_t-(1-\alpha)\log N_t\) with \(\alpha=1/3\).
- Decompose average annual growth of output per worker into capital deepening and TFP growth, before and after 1973.
import numpy as np, pandas as pd, statsmodels.api as sm
m = sm.datasets.macrodata.load_pandas().data
Y, I = m["realgdp"].to_numpy(), m["realinv"].to_numpy(); d = 0.025; a = 1/3
# TODO: perpetual inventory; employment; Solow residual; growth accounting by period
Solution, output and discussion
import numpy as np, pandas as pd, statsmodels.api as sm
m = sm.datasets.macrodata.load_pandas().data
Y, I = m["realgdp"].to_numpy(), m["realinv"].to_numpy(); dl, a = 0.025, 1/3
g = np.mean(np.diff(np.log(I)))
K = np.empty_like(Y); K[0] = I[0] / (g + dl)
for t in range(len(Y) - 1):
K[t + 1] = (1 - dl) * K[t] + I[t]
N = m["pop"].to_numpy() * (1 - m["unemp"].to_numpy() / 100)
logz = np.log(Y) - a * np.log(K) - (1 - a) * np.log(N)
df = pd.DataFrame({"year": m["year"], "y": np.log(Y / N), "k": np.log(K / N), "z": logz})
print(f"capital-output ratio (annual): 1959 {K[0]/Y[0]/4:.2f}, 1973 {K[56]/Y[56]/4:.2f}, 2009 {K[-1]/Y[-1]/4:.2f}")
for lo, hi in ((1960, 1973), (1973, 2007)):
s = df[(df.year >= lo) & (df.year <= hi)]; T = (len(s) - 1) / 4
gy = 100 * (s.y.iloc[-1] - s.y.iloc[0]) / T; gk = 100 * a * (s.k.iloc[-1] - s.k.iloc[0]) / T; gz = 100 * (s.z.iloc[-1] - s.z.iloc[0]) / T
print(f"{lo}-{hi}: output/worker {gy:.2f}%/yr = capital deepening {gk:.2f} + TFP {gz:.2f}")
np.save("/tmp/us_logz.npy", logz)
Output:
capital-output ratio (annual): 1959 0.80, 1973 0.82, 2009 1.29
1960-1973: output/worker 2.77%/yr = capital deepening 1.09 + TFP 1.68
1973-2007: output/worker 1.86%/yr = capital deepening 1.00 + TFP 0.86
The perpetual-inventory capital stock rises relative to output over the sample. Growth accounting shows the post-1973 productivity slowdown: output per worker growth fell from about 2.8% to 1.9% a year, and almost all of the decline is TFP (from about 1.7% to 0.9%), while the contribution of capital deepening was stable at about 1 point. This replicates the classic finding with a crude employment measure; using hours and a capital measure that excludes residential structures changes the levels but not the story.
Exercise 2.2 — Tobin's q: does market value predict investment?
Data: grunfeld (statsmodels) — investment, market value and capital stock of 11 large U.S. firms, 1935–1954.
With adjustment costs, investment depends on Tobin's \(q\). Grunfeld's data give firm value \(V\) and capital \(K\); use \(V/K\) as a (crude) average-\(q\) proxy.
- Regress the investment rate \(I_{it}/K_{it}\) on lagged \(V/K\), with firm fixed effects.
- Interpret the coefficient as \(1/\phi\) for quadratic adjustment costs \(\frac{\phi}{2}(I/K)^2K\), and report the implied \(\phi\).
- Why does the \(q\) theory typically explain investment poorly, and what does measurement error in \(q\) do to the coefficient?
import numpy as np, statsmodels.api as sm, statsmodels.formula.api as smf
g = sm.datasets.grunfeld.load_pandas().data.sort_values(["firm", "year"])
g["ik"] = g["invest"] / g["capital"]; g["q"] = g["value"] / g["capital"]
# TODO: lag q by firm; ik ~ q_lag + C(firm)
Solution, output and discussion
import numpy as np, statsmodels.api as sm, statsmodels.formula.api as smf
g = sm.datasets.grunfeld.load_pandas().data.sort_values(["firm", "year"])
g = g[g["capital"] > 5].copy()
g["ik"] = g["invest"] / g["capital"]; g["q"] = g["value"] / g["capital"]
g["q_lag"] = g.groupby("firm")["q"].shift(1)
d = g.dropna(subset=["q_lag"])
r = smf.ols("ik ~ q_lag + C(firm)", data=d).fit(cov_type="cluster", cov_kwds={"groups": d["firm"]})
b = r.params["q_lag"]
print(f"investment rate on lagged V/K (firm FE): {b:.4f} (cluster se {r.bse['q_lag']:.4f}), within R2 approx {r.rsquared:.3f}")
print(f"implied adjustment-cost parameter phi = 1/b = {1/b:.1f}")
print(f"median I/K {d['ik'].median():.3f}; median V/K {d['q'].median():.2f}")
Output:
investment rate on lagged V/K (firm FE): 0.0291 (cluster se 0.0092), within R2 approx 0.533
implied adjustment-cost parameter phi = 1/b = 34.4
median I/K 0.379; median V/K 3.51
The coefficient on lagged \(V/K\) is positive and significant, but small: it implies an adjustment-cost parameter of about 34, so firms would take decades to close a gap between \(q\) and one. That is the well-known failure of \(q\) theory — implausibly slow adjustment. One reason is measurement: \(V/K\) is average q, the relevant concept is marginal q, and market values are noisy, so the coefficient is attenuated (Erickson and Whited, 2000). With 11 firms, the clustered standard error also rests on few clusters.
Further reading
Hayashi (1982), “Tobin's Marginal q and Average q”; Ljungqvist and Sargent, Recursive Macroeconomic Theory.