Macro-Economic Modelling 1

Long Term Dynamics: Growth Theory

For many centuries before the 18th century economic growth seemed to be inexistent: most people lived in the same conditions as their parents and grandparents with little to no hopes of improving their standards of living. Then quite suddenly in the 18th century, something happened that generated unprecedented growth, ushering in the modern era.

In 1957, Nicolas Kaldor listed [1] some empirical stylised facts that economic growth seemed to satisfy. These became known as the Kaldor facts:

Output per worker growth
Constant capital-to-output ratio
Capital per worker growth
Constant rate of return on capital
Labour and capital capture a contant shares of total income
The growth rate of output per worker is vastly heterogeneous across countries

Most of these have passed the test of time, with perhaps the exception of (4) and (5) - interest rates have been on a downward trend for centuries[2] and labour share has been falling since the early 1980s[3]. Jones and Romer (2010) have more recently compiled a more up to date list of "new Kaldor facts"[4], but we shall focus on the original list above as historically it provided the motivational foundation of the first working model of economic growth.

The Solow model

Let us consider an economy[5] with only physical capital \(K\) and labour \(L\) as inputs, producing output \(Y\) according to the production function \begin{equation} Y = F(K, L, t), \end{equation} which is assumed to be neoclassical.

Definition 1 — Neoclassical production function

A production function \(F\) is called a neoclassical production function if it has the following properties

Positive and diminishing marginal products with respect to each input, that is \(\partial_x F>0\), \(\partial_x^2 F < 0\), for \(x \in \{K,L\}\);

Constant return to scale, that is \(F(\lambda K, \lambda L) = \lambda F(K, L)\) for any \(\lambda >0\);

Respect the following Inada conditions \begin{equation*} \begin{split} &\lim_{K\rightarrow 0}\partial_K F = \lim_{L\rightarrow 0}\partial_L F = \infty,\\ &\lim_{K\rightarrow\infty}\partial_K F = \lim_{L\rightarrow\infty}\partial_L F = 0. \end{split} \end{equation*}

Definition 2 — Intensive production function

Because of the constant return to scale condition assumed for the production function one has \(F(K,L) = L\cdot F(K/L, 1)\). It is therefore possible to define output-per-capita \(y = Y/L\), and capital-to-labour ratio \(k=K/L\), and write the production function in its intensive form \begin{equation} f(k):=F(k,1). \end{equation}

Definition 3 — Marginal products

Under perfectly competitive markets, capital and labour are payed their marginal products. Specifically, the return on capital is rental rate \begin{equation} \label{eq:def_return_on_capital} r_K(t) = \frac{\partial F}{\partial K} = f'(k(t)). \end{equation} Then, if capital depreciates at a rate \(\delta\), the real rate of interest of the economy is \begin{equation} r(t) = r_K(t) - \delta. \end{equation} The return on labour is instead the real wage \begin{equation} w(t) = \frac{\partial F}{\partial L} = f(k)-k(t)f'(k(t)). \end{equation}

In such an economy, output is a good which can either be consumed or invested to create new capital. Denoting by \(s\) the fraction of output saved (saving rate), and by \(\delta>0\) the constant depreciation rate of capital, one can write the following equation of motion for the amount of capital in the economy \begin{equation} \dot{K} = s\cdot F(K,L,t) - \delta K. \end{equation} The equation of motion for capital can therefore be rewritten in intensive form as \begin{equation} \frac{\dot{K}}{L} = s\cdot f(k,t) - \delta k. \end{equation} Further assuming a constant exogenous growth rate of labour (assumed equivalent to the population) of \( n = \dot{L}/L > 0\) (that is \(L(t) = e^{nt}\)) we can further find \(\dot{K}/L=\dot{k}+nk\), so that the equation of motion finally turns into \begin{equation} \label{eq:solow_ode} \boxed{\dot{k} = s\cdot f(k,t) - (n+\delta) k.} \end{equation} Definition 4 — Balanced growth path

An economy in which both \(L\) and \(K\) (and consequently \(Y\)) grow at a constant rate, is said to be on a balanced growth path or in the steady state: thus, one has \(\dot{k}=0\).

In the steady state one has capital per capita \(k^*\) such that \begin{equation} \label{eq:steady_state_condition_k} s f(k^*) = (n+\delta)k^*. \end{equation} Notice that the notion of balanced growth path is relevant because the ODE \eqref{eq:solow_ode} has asymptotically stable solutions. Writing \(g(k) = s f(k)-(n+\delta)k\), the system is asymptotically stable[6] if \(\left.\frac{d g(k)}{dk}\right|_{k=k^*} < 0 \) or unstable when \(\left.\frac{d g(k)}{dk}\right|_{k=k^*} > 0 \): \begin{equation} \label{eq:solow_stability_condition} \begin{split} \left.\frac{d g(k)}{dk}\right|_{k=k^*} &= sf'(k^*)-(n+\delta)\\ &=\frac{s}{k^*}f(k^*)\left(\frac{k^*f'(k^*)}{f(k^*)}-1\right). \end{split} \end{equation} Therefore, stability hinges on the value of the output elasticity of capital \(\frac{k^*f'(k^*)}{f(k^*)}\), which being always bound in the unit interval[7] guarantees that \eqref{eq:solow_stability_condition} is negative, and hence that the system is asymptotically stable. This can also be seen graphically by the dynamics depicted in Fig. 2.

Output per capita is constant through time \(y^*=f(k^*)\) unless the production function changes. Clearly this does not mean that there is no growth in the levels of capital \(K\) and output \(Y\): their growth is however merely driven by population growth.

Let us now introduce exogenous technological change. Without much loss of generality and for sake of tractability let us consider the special case of the Cobb-Douglas production function \begin{equation} \label{eq:cobb_douglas} F(K,L, t) = A(t) K^\alpha(t) L ^{1-\alpha}(t), \end{equation} or \begin{equation} f(k, t) = A(t) k^\alpha(t), \end{equation} with \(A(t)\) the level of technology and \(0 < \alpha < 1 \). It is easy to verify that \eqref{eq:cobb_douglas} respects the conditions of neoclassical production functions stated above, and that \(\alpha\) is the output elasticity of capital \(\frac{k^*f'(k^*)}{f(k^*)}\).
Let us denote the growth rate of technological level by \(a = \dot{A}/A\). This implies that \(\dot{y}/y=a\), that is per capita rate of growth of output is driven exclusively by technological change. A different way to look at changes in technology is through labour efficiency \(E(t) = L(t)e^{\lambda t}\), with the production function \(Y = F(K, E)\).
Proceeding as before, defining \(\kappa = K / E\), one finds the equation of motion \begin{equation} \label{eq:solow_etc} \dot{\kappa} = sf(\kappa) - (\delta+n+\lambda)\kappa. \end{equation} The steady state is still defined by \(\dot{\kappa} = 0 \), where \begin{equation} \label{eq:capital_and_output_growth_rate} \dot{K}/K=\dot{Y}/Y=n+\lambda, \end{equation} but now \begin{equation} \label{eq:output_per_worker_growth} \frac{\partial_t (Y/L)}{Y/L} = \lambda. \end{equation} Now the output per person grows even on the steady state.
Therefore, the Solow model with exhogenous technological change \eqref{eq:solow_etc} accounts for all Kaldor's facts.
Fact 1: output per worker growth is \(\lambda\) given in \eqref{eq:output_per_worker_growth}. Fact 2, that is the constancy of capital-to-output ratio, is a direct implication of \eqref{eq:capital_and_output_growth_rate}. Fact 3 then follows from 1 and 2 combined. Fact 4 is explicitely stated in \eqref{eq:capital_and_output_growth_rate}. Fact 5 is built in the model rather than an implication, but part of the system nonetheless. Fact 6 can be addressed by explaining country heterogeneity as a function of different levels of productivity, a view which seems corroborated by empirical comparisons of total factor productivity (see e.g. [8]).

The neoclassical growth model

One of the main limitation of the Solow model is the assumed constancy of the saving rate, where no optimisation is involved. The problem with optimising agents was solved already back in the 1928 by British philosopher and mathematician Frank Ramsey [9], and independently rediscovered in the economic discipline only thirty years later by David Cass and Tjalling Koopmans [10,11]. The model, which often goes by the name of Ramsey-Cass-Koopmans model, is more widely known as the Neoclassical Growth Model (NGM).
Consider a closed economy with one representative household, whose members are both consumer and producers, infinitely lived, with population growth \(n\) as before[12], and seeking to maximise future discounted utility: \begin{equation} U(t) = \int_0^\infty u(c(t))e^{nt-\rho t}. \end{equation} Here it is assumed \(\rho>n\), that \(u'(c(t))>0\), \(u''(c(t)) < 0\), and that \(u\) satisfies the Inada conditions. The economy's dynamics now accounts for consumption \(C(t)\), but assumes no depreciation: \begin{equation} \dot{K}(t) = F(K(t), L(t))-C(t), \end{equation} which can be equivalently written in per capita units as \begin{equation} \dot{k}(t) = f(k(t))-nk(t)-c(t). \end{equation} Therefore, the model reads \begin{equation} \label{eq:ngm} \boxed{ \begin{split} \max_{c(t)}& \int_0^\infty u(c(t))e^{(n-\rho) t}\\ &\text{subject to}&\\ \dot{k}(t) &= f(k(t))-nk(t)-c(t) \end{split}} \quad . \end{equation} To solve \eqref{eq:ngm} one starts as usual by writing the present-value Hamiltonian \begin{equation} H = u(c(t))e^{(n-\rho) t} + \lambda(t)\left[f(k(t))-nk(t)-c(t)\right]. \end{equation} The co-state equation then reads \begin{equation} \label{eq:costateeq_ramsey} \dot{\lambda} = -\frac{\partial H}{\partial k} = -\lambda(f'(k)-n), \end{equation} while Pontryagin's maximum principle demands \begin{equation} \label{eq:pmp_ramsey} \begin{split} \frac{\partial H}{\partial c} &= 0\\ \lambda &= u'(c)e^{(n-\rho) t} \end{split}\quad , \end{equation} with transversality condition \begin{equation} \label{eq:transversality_condition_ramsey} \lim_{t\rightarrow\infty} \lambda(t)k(t) = 0. \end{equation} These are also referred to as first order conditions (FOC). Indeed, Pontryagin's principle provides only necessary conditions for optimality. Consider now the first order time-derivative of \eqref{eq:pmp_ramsey}: \begin{equation} \begin{split} \dot{\lambda} &= u''(c)\dot{c}e^{(n-\rho) t} + \lambda (n-\rho),\\ \frac{\dot{\lambda}}{\lambda} &= \frac{u''(c)}{u'(c)}\dot{c} + (n-\rho),\\ \frac{\dot{\lambda}}{\lambda} &= -\frac{1}{\sigma}\frac{\dot{c}}{c} + (n-\rho), \end{split} \end{equation} where the elasticity of intertemporal substitution in consumption[15] is defined by \begin{equation} \sigma = -\frac{1}{c(t)}\frac{u'(c(t))}{u''(c(t))} >0. \end{equation} Finally, using the co-state equation \eqref{eq:costateeq_ramsey} and rearranging, one obtains the Ramsey rule \begin{equation} \label{eq:ramsey_euler_equation} \frac{\dot{c}}{c} = \sigma\left[f'(k)-\rho\right], \end{equation} also referred to more generally as the Euler equation. The concept of Euler equation is key in dynamic equilibrium models as it embeds the optimal time preference for the control variables of the model (in this case consumption).

Let us know look at the steady state: the system has three solutions. First, the trivial solution at the origin is a repelling point. A second degenerate stable equilibrium is found at \(c^*=0\) and \(f(k^*)=nk^*\). Finally, an equilibrium exists for \begin{equation} \label{eq:ramsey_steady_state} \begin{split} f'(k^*) &= \rho,\\ c^* &= f(k^*)-nk^*. \end{split} \end{equation} This equilibrium point is a saddle point. The phase space of Ramsey's neoclassical growth model is depicted in Fig. 3: of all possible trajectories, only one is allowed, the so called saddle path solution (in red in Fig. 3), which is also the only trajectory intersecting the steady state defined by \eqref{eq:ramsey_steady_state}. All other trajectories are ruled out by the transversality condition \eqref{eq:transversality_condition_ramsey} .

Notice that in the steady state both per capita capital and consumption are constant at \(k^*\) and \(c^*\), implying that per capita output remains constant at \(y^*=f(k^*)\). Therefore, as in the Solow model, the NGM shows no per capita growth in the steady state if not via exhogenous innovations in the technology level.

The NGM represents the first example of microfounded dynamic general equilibrium framework for modelling the macroeconomy, providing, in its variants, a workhorse model for much of modern macroeconomic analysis.

Endogenous growth

Despite the progress made, both Solow's model and the NGM are still unable to endogenously explain long-term growth. Growth in per capita terms occurs only in the transition to the balanced growth path, or because of exhogenous innovation in the technology level. One intrinsic reason comes from the diminishing return to the use of capital which underpins the neoclassical production function defined above.
Are there mechanisms other than pure technological innovation that can generate long-term growth? In order to explain long-term growth there seem to be two options. The first, is to somehow adjust the production function so as to allow for constant or increasing returns to scale. The second option, is to model technological change endogenously.
As briefly mentioned for the Solow model, one possible way to extend the production function is to account for increased efficiency in the labour force, what can be referred to as human capital. This effectively allows the production function to escape marginal returns to scale. The following model presents instead what is arguably the simplest instance of microfounded dynamical framework allowing for endogenous growth: the AK model.

The AK model

Following the same line of reasoning as for the NGM, consider an economy where households own the capital and use it to produce according to the production function \(Y = A k(t)\), with \(A>0\) [16], so that the equation of motion for capital (resource constraint of the economy) is \begin{equation} \label{eq:ak_eom} \dot{k}(t) = Ak(t) - c(t). \end{equation} The utility from consumption which the household seeks to maximise is given by \(\int_0^\infty u(c(t)) e^{-\rho t}dt\) again with the rate of time preference \(\rho>0\).
From \eqref{eq:ramsey_euler_equation} we know the Euler equation is \begin{equation} \label{eq:ak_euler_equation} \frac{\dot{c}}{c} = \sigma(A-\rho). \end{equation} The main difference from the NGM is in the nature of the technology, particularly the AK model is characterised by constant returns to capital [17], which here implies long term growth in the steady state: \begin{equation} \label{eq:ak_constant_return} \frac{\dot{k}}{k} = \frac{\dot{y}}{y} = \frac{\dot{c}}{c} = \sigma(A-\rho). \end{equation} Moreover, it is also clear [18] that consumption is proportional to capital \begin{equation} c(t) = ((1-\sigma)A+\sigma\rho)k(t). \end{equation} Finally, given an initial state \(k(0)=k_0\) for the capital at time zero, it is trivial to find the following explicit solution for the AK model: \begin{equation} \begin{pmatrix} k(t) \\ y(t) \\ c(t) \end{pmatrix} = \begin{pmatrix} 1 \\ A \\ (1-\sigma)A + \sigma\rho \end{pmatrix} k_0 e^{\sigma(A-\rho)t}. \end{equation} One interesting feature emerging is therefore the permanency of the effects of shocks, even for transitory ones. Indeed, notice that changes in \(\sigma\), \(\rho\), or \(A\) affect the whole long term trajectory of the system. This is in contrast to the results of the Solow model and NGM where after a temporary shocks the system would transition back to the same equilibrium. Interestingly, this dynamics seems to explain empirical evidence of the permanency of shocks' effects, as suggested by Fig. 4.

References and notes

[1] "A Model of Economic Growth", Nicholas Kaldor, The Economic Journal, Vol. 67, No. 268 (Dec., 1957), pp. 591-624
[2] "Eight centuries of global real rates, R-G, and the 'suprasecular' decline", Paul Schmelzing, 1311-2018 - BoE SWP 845, 2020.
[3] "The Global Decline of the Labor Share", Loukas Karabarbounis, Brent Neiman, The Quarterly Journal of Economics, 2014.
[4] "The New Kaldor Facts: Ideas, Institutions, Population, and Human Capital", Charles I. Jones and Paul M. Romer, 2010.
[5] "A Contribution to the Theory of Economic Growth", Robert M. Solow, The Quarterly Journal of Economics, 1956.
[6] Theorem 3.2.1 in Zhang (2005).
[7] Since the rate of change of \(f(k)\) is decreasing (\(f''(k) < 0\)), the tangent line (with slope \(f'(k)\)) will always have a steeper slope than the line connecting the origin to the point on the curve (with slope \(f(k)/k\)). This is because as \(k\) increases, the curve flattens out, making the tangent line approach the \(k\)-axis faster than the line from the origin.
[8] "Barriers to Riches", Stephen L. Parente and Edward C. Prescott, MIT press, 2002.
[9] "A Mathematical Theory of Saving", F. P. Ramsey, The Economic Journal, Vol. 38, No. 152 (1928).
[10] "Optimum Growth in an Aggregative Model of Capital Accumulation", David Cass, The Review of Economic Studies, Volume 32, Issue 3, 1965.
[11] "On the Concept of Optimal Economic Growth", Tjalling Koopmans, No 163, Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University, 1963.
[12] While some of these assumptions may seem extreme extreme and unrealistic, they are actually well justified. The one household assumption is acceptable to the extent that one is not interested in the heterogeneity of households reactions to the model's parameters, and to distributional dynamics. For example, people respond rather similarly to changes in interest rates - in this sense the one household approximation can be seen as a reasonable assumption. A similar argument holds for having one single utility function. That's reasonable to the extent that people respond ina similar way to the same tradeoffs. Interestingly, it can be shown that under certain conditions, many heterogeneities in skills, taste, etc. can still be incorporated into a single representative household[13]. Infinite life also becomes reasonable in light of intergenerational transfers, where the older generation may include in its utility an expectation of the future utility of their children and grandchildren [14].
[13] "A Representative Consumer Theory of Distribution", Francesco Caselli and Jaume Ventura, AMERICAN ECONOMIC REVIEW, VOL. 90, NO. 4, 2000, (pp. 909-926).
[14] "Are Government Bonds Net Wealth?", Robert J. Barro, Journal of Political Economy, Vol. 82, No. 6 (1974).
[15] Its inverse \(1/\sigma\) is instead referred to as elasticity of marginal utility with respect to consumption.
[16] Notice this is just a limit case of the Cobb-Douglas production function \eqref{eq:cobb_douglas} with \(\alpha=1\).
[17] This follows in particular from the Euler equation \eqref{eq:ak_euler_equation} and from the equation of motion \eqref{eq:ak_eom}, which implies \(\frac{\dot{k}}{k} = A-\frac{c}{k}\).
[18] To see that is sufficient to write the equation of motion \eqref{eq:ak_eom} as \( c = Ak - \dot(k) \), then to use \eqref{eq:ak_constant_return}.
[19] "Growth based on increasing returns due to specialization", Romer, P. M., The American Economic Review, 77(2), 56–62, (1987).
[21] "Geography, Demography, and Economic Growth in Africa", D.E. Bloom, J.D. Sachs, P. Collier, C. Udry, Brookings Papers on Economic Activity, Vol. 1998, No. 2 (1998).