The Solow model in discrete time

In continuous time, the accumulation of capital per head obeys the differential equation established here:

\[ \dot k(t) = sf\bigl(k(t)\bigr) - (n+\delta)k(t) \]

In discrete time, the capital stock installed at the beginning of period \(t+1\) is what remains of the previous stock, plus the investment of the period:

\[ K_{t+1} = (1-\delta)K_t + I_t \]

With a labour force growing at rate \(n\), \(H_{t+1} = (1+n)H_t\), a constant saving rate, \(I_t = sY_t\), and a technology \(Y_t = A_tF(K_t,H_t)\) homogeneous of degree one, dividing by \(H_{t+1}\) yields the law of motion of capital per head:

\[ (1+n)\,k_{t+1} = (1-\delta)k_t + sA_tf(k_t) \]

Three remarks before going further.

First, capital is predetermined. The stock \(k_t\) results from decisions taken at \(t-1\); it does not react to what happens at \(t\). This timing convention is inconsequential in the deterministic model, but it becomes decisive as soon as randomness is introduced, and it is a classic source of error in simulations.

Second, the law of motion is explicit: the right-hand side depends only on \(k_t\). Simulating the nonlinear model therefore requires no solver, a simple loop suffices. We shall see in the section From discrete to continuous time that this convenience comes from the way we discretised, not from the model itself.

Finally, why discrete time. Data are dated quarterly or annually, simulations proceed by steps, and dynamic general equilibrium models are generally written in discrete time. The Solow model is the simplest one on which the difficulties specific to this setting can be seen at work.

Throughout the note we adopt the following calibration, that of McCandless: \(\alpha = 0.36\), \(s = 0.20\), \(n = 0.02\) and \(\delta = 0.10\), the period being the year. Technical progress is nil unless otherwise stated.

The steady state \(\bar k\) satisfies \(\bar k = g(\bar k)\), that is:

\[ (1+n)\bar k = (1-\delta)\bar k + sAf(\bar k) \]

The existence and uniqueness of a non-trivial steady state rest on the same arguments as in continuous time, the average product \(f(k)/k\) being monotonically decreasing from infinity to zero under the Inada conditions. We refer to the note on the Solow model for the proof, which does not depend on the timing convention.

Stability, on the other hand, is stated differently. In continuous time it depends on the sign of the derivative; in discrete time, on the modulus of the slope of \(g\) at the fixed point (which must be less than 1 in modulus to ensure local convergence).

This point is worth stressing, because discrete time is often expected to generate oscillations: the discrete-time Solow model never oscillates for admissible values of depreciation. The section From discrete to continuous time will say why, and under what condition it could.

For our calibration, \(\bar k = 2.2215\) and \(g'(\bar k) = 0.9247\).

Let us approximate the law of motion in a neighbourhood of the steady state. Denoting by \(\tilde k_t = \log(k_t/\bar k)\) the logarithmic deviation, we obtain at first order:

\[ \tilde k_{t+1} = B\,\tilde k_t, \qquad B = g'(\bar k) \]

Capital therefore follows a first-order autoregressive process, whose coefficient is the slope computed in the previous section. This dynamics can also be read as a geometric sequence with ratio \(B\). This expression calls for a comparison with continuous time, where the speed of convergence is \(\beta^{\star} = (1-\alpha)(n+\delta)\), as established here.

Moving to discrete time divides the speed of convergence by the demographic factor. For our calibration, \(B = 0.9247\) and the half-life of the deviation from the steady state is \(\log 2/(-\log B) = 8.85\) years.

This identity holds under the assumption of no technical progress. The section From discrete to continuous time shows what becomes of it without this assumption.

Suppose now that technology grows at a constant rate, \(A_t = (1+a)^tA_0\). Capital per head no longer has a steady state: it grows indefinitely along a balanced growth path, where the growth rate is constant.

The exponent \(\frac{1}{1-\alpha}\) deserves a word, since it does not appear in our continuous-time notes. It comes from the way technology has been introduced. Here it is Hicks-neutral, it multiplies the production function; in the note on the simulation of the Solow model it is labour-augmenting, \(F(K_t,A_tH_t)\). In the Cobb-Douglas case the two conventions are reconciled:

\[ A_tK_t^{\alpha}H_t^{1-\alpha} = K_t^{\alpha}\left(A_t^{\frac{1}{1-\alpha}}H_t\right)^{1-\alpha} \]

so that neutral technical progress at rate \(a\) is equivalent to labour-augmenting progress at the rate \(x\) defined by \(1+x = (1+a)^{\frac{1}{1-\alpha}}\). With \(a = 0.02\) and \(\alpha = 0.36\), this gives \(x = 3.14\,\%\). Confusing the two conventions leads to over- or underestimating growth by a factor \(1/(1-\alpha)\), more than half for a usual calibration.

To reason about a steady state one has to move to efficiency units, \(\hat k_t = K_t/(A_tH_t)\), which gives:

\[ (1+n)(1+x)\,\hat k_{t+1} = (1-\delta)\hat k_t + sf(\hat k_t) \]

Let us step back and explain in what sense the discrete-time model and the continuous-time model answer each other, where they part ways, and why it would be unwise to regard one as the approximate version of the other.

The model has four parameters, \(\alpha\), \(s\), \(n\) and \(\delta\), to which \(x\) and the variance of the shock are added in the stochastic version. Setting them requires distinguishing two categories.

What does not depend on the period. The saving rate \(s\) and the capital share \(\alpha\) are ratios, dimensionless. They are the same at the annual and at the quarterly frequency. The capital share can be read in the national accounts, around \(0.30\) to \(0.36\) depending on the treatment of mixed income; the saving rate can be read in the investment rate, around \(0.20\) to \(0.22\).

What does depend on it. The rates \(n\), \(x\) and \(\delta\) are expressed per period. Moving from the year to the quarter is done through \((1+r)^{1/4}-1\), and not \(r/4\): for \(\delta = 0.10\), one gets \(0.0241\) against \(0.0250\), an approximation off by nearly \(4\,\%\). The discrepancy is negligible for \(n\) and \(x\), where the rates are small, and not for \(\delta\).

There remains the most important point, and the one most often passed over in silence.

This identity links five quantities, four of which are observable. They cannot therefore be chosen freely. With McCandless's calibration, \(s = 0.20\), \(n = 0.02\) and \(\delta = 0.10\), it imposes \(K/Y = 1.42\), whereas the data suggest a value between \(2.5\) and \(3\). Conversely, targeting \(K/Y = 3\) with \(s = 0.20\) and \(n+x = 0.04\) imposes \(\delta = 0.027\), far from the usual ten per cent.

There is no error here, but a trade-off. Calibrating means choosing which moments one agrees to miss. A calibration retaining \(\delta = 0.10\) targets the short-run dynamics, where depreciation governs the speed of adjustment; a calibration retaining \(K/Y = 3\) targets the long-run quantities. The Solow model does not have enough degrees of freedom to hit both at once, and this is information about the model, not about the method.

Let us make technology random. Among the possible formulations, the most convenient retains a log-normal level of technology,

\[ A_t = \bar Ae^{\varepsilon_t} \]

where \(\varepsilon_t\) is a Gaussian white noise with zero mean and standard deviation \(\sigma_{\varepsilon}\). This formulation guarantees the positivity of \(A_t\), which an additive shock would not. The law of motion becomes

\[ k_{t+1} = \frac{(1-\delta)k_t + s\bar Ae^{\varepsilon_t}f(k_t)}{1+n} \]

It remains explicit: simulating the nonlinear model requires nothing more than a loop. This is the convenience of the explicit scheme, and it is also what distinguishes the Solow model from models with expectations, where the path has to be solved globally.

def simule_exact(T, sigma, graine=0):
    """Simulation of the nonlinear model. Capital is predetermined:
    the shock of period t only affects capital at t+1."""
    alea = np.random.default_rng(graine)
    eps = sigma*alea.standard_normal(T)
    k = np.empty(T)
    k[0] = kstar()
    for t in range(T-1):
        k[t+1] = ((1-delta)*k[t] + s*A*np.exp(eps[t])*f(k[t]))/(1+n)
    y = A*np.exp(eps)*f(k)
    return k, y, eps

Timing is the only delicate point. Capital \(k_t\) is decided at \(t-1\): it does not depend on \(\varepsilon_t\). Output \(y_t\), on the other hand, is hit by the contemporaneous shock. Swapping these two lines leads to overestimating the variance of output by nearly ten per cent, without anything signalling it.

The model being nonlinear, the variance of capital cannot be expressed simply in terms of that of the shock. We therefore approximate the model in a neighbourhood of the steady state. Denoting \(\tilde X_t = \log X_t - \log\bar X\), and using \(e^{\tilde X}\simeq 1+\tilde X\) as well as \(\tilde X\tilde Y\simeq 0\), the law of motion becomes

\[ \tilde k_{t+1} = B\tilde k_t + C\varepsilon_t, \qquad B = \frac{(1-\delta)+\alpha(n+\delta)}{1+n}, \qquad C = \frac{n+\delta}{1+n} \]

Two remarks. First, the saving rate appears neither in \(B\) nor in \(C\): it drops out through the steady-state condition. A country that saves more is richer, but converges neither faster nor slower, and does not react differently to shocks. Second, \(\tilde k\) is a first-order autoregressive process; its moving average representation is therefore obtained by the method presented in the note on the MA representation of an AR(2), of which it is the single-root special case:

\[ \tilde k_{t+1} = C\sum_{i=0}^{\infty}B^i\varepsilon_{t-i} \]

The shocks being independent, the variance follows immediately:

\[ \mathrm{var}\bigl(\tilde k\bigr) = \frac{C^2}{1-B^2}\,\mathrm{var}(\varepsilon) \]

Output, for its part, is hit by the contemporaneous shock and inherits predetermined capital, \(\tilde y_t = \varepsilon_t+\alpha\tilde k_t\), hence

\[ \mathrm{var}\bigl(\tilde y\bigr) = \mathrm{var}(\varepsilon)+\alpha^2\,\mathrm{var}\bigl(\tilde k\bigr) \]

For our calibration, \(\mathrm{var}(\tilde k) = 0.0955\,\mathrm{var} (\varepsilon)\) and \(\mathrm{var}(\tilde y) = 1.0124\,\mathrm{var} (\varepsilon)\).

def simule_loglineaire(T, sigma, graine=0):
    """Simulation of the log-linear version, with the same shocks."""
    alea = np.random.default_rng(graine)
    eps = sigma*alea.standard_normal(T)
    B = ((1-delta) + alpha*(n+delta))/(1+n)
    C = (n+delta)/(1+n)
    kt = np.zeros(T)
    for t in range(T-1):
        kt[t+1] = B*kt[t] + C*eps[t]
    yt = eps + alpha*kt
    return kt, yt, eps

It remains to assess the approximation. The figure below compares, on the left, the two paths for the same draw of shocks and, on the right, the mean gap between them as a function of the size of the shock.

The relative error grows proportionally to \(\sigma_{\varepsilon}\), and the absolute error as its square. Relative to the standard deviation of \(\tilde k\), it is \(4.6\,\%\) for \(\sigma_{\varepsilon} = 0.02\), \(23\,\%\) for \(\sigma_{\varepsilon} = 0.10\) and \(46\,\%\) for \(\sigma_{\varepsilon} = 0.20\). This is worth keeping in mind: for a technology shock of realistic size the log-linearisation is excellent, but it degrades quickly. McCandless's figures retain \(\sigma_{\varepsilon} = 0.20\), which makes them legible at the cost of an approximation pushed well beyond its domain of validity.

The previous computation contains an admission. The variance of output is \(1.0124\) times that of the shock: the model generates almost no propagation. All the observed dynamics of output is that of the shock itself, and capital adds only \(1.24\,\%\) to it. With independent shocks, output is practically a white noise, which the data flatly contradict.

One answer is to make the shocks persistent, but it shifts the question without answering it: persistence becomes an assumption instead of a result. Let us rather see what an internal mechanism brings.

The thread of this note follows the first chapter of McCandless, G. (2008), The ABCs of RBCs: An Introduction to Dynamic Macroeconomic Models, Harvard University Press, where the discrete-time Solow model serves as an introduction to real business cycle models. The sections on discretisation, calibration and the rigidity on investment are extensions of it.

On the continuous-time Solow model, see the notes on the Solow model and on its simulation. On the moving average representations used in the last two sections, see the note on the MA representation of an AR(2). The classic reference on growth theory remains Barro, R. and Sala-i-Martin, X. (2004), Economic Growth, MIT Press.