8 The AK model - Spillovers à la Romer(1986)
8.1 Introduction
We started from the standard Solow-Swan growth model. This model, as well as the Ramsey model, has neoclassical production function for the final good. With neoclassical production function these models cannot endogenously generate long run growth since the returns on capital decline with the accumulation of capital because of the decreasing returns assumption.
The model presented below assumes a neoclassical production function - at the “individual level.” In addition, it assumes that the labour augmenting technology is a function of average percapita capital stock. While doing so, it has in mind some “learning by doing” effects/spillovers (i.e., the workers learn/become more productive while working with desks, computers, etc.).
8.2 The Model
The main structure of the model is as follows:
The level of technology/ efficiency that augments the labour input in the production is a function of the average capital-labour ratio in the economy. The motivation for this is that investment of a firm brings productivity gains from its use from the labour. The firm builds up the knowledge (technical expertise) of how to efficiently use the capital by accumulating it, i.e., there is “learning-by-doing” (e.g., production lines). This learning-by-doing effect has an aggregate impact, when any individual firm’s technical efficiency is public knowledge, so that all firms can benefit from the technological advance for the use of capital in the production. This gives the link between ” \(A_{i}\) ” (i.e., some \(i\) th firm’s efficiency) and the average capital-labour ratio in the economy.
The level of technology is assumed to be \(A \equiv \bar{A} k\), where \(k\) is the average of per-capita capital stock and \(\bar{A}>0\) measures the efficiency of use of the capital
The production function takes the form: \(Y_{i}=F\left(K_{i}, A L_{i}\right)=L_{i} F\left(k_{i}, \bar{A} k\right) \equiv L_{i} \bar{A} k f\left(\frac{k_{i}}{\bar{A} k}\right)\). Thus, there are decreasing returns to capital at the firm-level since the firm is so small that it does not take into account its impact on \(A\). However, there are constant returns to capital in symmetric equilibrium since \(k_{i}=k\).
One-sector model of growth: \(\dot{K}=Y-C-\delta K\)
From the consumption-side, the representative household chooses its consumption and next period assets to maximize its lifetime utility \(U=\int_{0}^{\infty} e^{-(\rho-n) t} u(c) d t\), subject to its budget constraint: \(\dot{b}(t)=(r-n) b(t)+w-c\), where \(u(c)=\frac{c^{1-\theta}-1}{1-\theta}\).
Population grows at exogenous rate \(n\)
For any individual producer the level of efficiency \(A(\bar{A} k)\) is taken as given, along with the prices of inputs, when choosing capital and labour to maximize its per period profits \(\pi_{i}\), i.e.,
\[ \max _{K_{i}(t), L_{i}(t)} \pi_{i}=F\left(K_{i}(t), A(t) L_{i}(t)\right)-R(t) K_{i}(t)-w(t) L_{i}(t) \]
Therefore the optimal rules are
\[ \begin{aligned} \frac{\partial \pi_{i}}{\partial K_{i}} & =0 \\ \Leftrightarrow R & =F_{K_{i}}\left(K_{i}, A L_{i}\right) \\ & =\frac{\partial L_{i} \bar{A} k f\left(\frac{k_{i}}{\bar{A} k}\right)}{\partial K_{i}} \\ & =f^{\prime}\left(\frac{k_{i}}{\bar{A} k}\right) \end{aligned} \]
and
\[ \begin{aligned} \frac{\partial \pi_{i}}{\partial L_{i}} & =0 \\ \Leftrightarrow w & =F_{L_{i}}\left(K_{i}, A L_{i}\right)\\ & =A F_{A L_{i}}\left(K_{i}, A L_{i}\right) \\ & =\frac{\partial L_{i} \bar{A} k f\left(\frac{k_{i}}{\bar{A} k}\right)}{\partial L_{i}} \\ & =\bar{A} k f\left(\frac{k_{i}}{\bar{A} k}\right)-k_{i} f^{\prime}\left(\frac{k_{i}}{\bar{A} k}\right) \end{aligned} \]
Since all producers are identical, the equilibrium must be symmetric, i.e., \(k_{i}=k\) for \(\forall i\). Therefore, from the asset market equilibrium condition ( \(b(t)=k\) and \(r=R(t)-\delta\) ) it follows that the net rate of returns on assets in the economy is
\[ r=f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta \]
The standard optimal consumption path that comes from the intertemporal maximization problem of the households is
\[ \frac{\dot{c}(t)}{c(t)}=\frac{1}{\theta}(r-\rho) \]
As a result, the equilibrium is characterized by two dynamic equations (i.e., the law of motion of capital in per-capita terms and the optimal consumption path)
\[ \begin{aligned} \frac{\dot{c}(t)}{c(t)} & =\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] \\ \frac{\dot{k}(t)}{k(t)} & =\bar{A} f\left(\frac{1}{\bar{A}}\right)-\delta-n-\frac{c(t)}{k(t)} \end{aligned} \]
Meanwhile, the standard TVC applies, that the value of the assets (capital) is equal to zero in the end of the planning horizon
\[ \lim _{t \rightarrow \infty} k(t) \lambda(t) e^{-(\rho-n) t}=0 \]
8.3 Equilibrium and Balanced Growth Path
Given that in equilibrium the marginal product of capital is independent of the level of per capita capital, it is always constant, i.e.,
\[ r=f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta \] The constant returns to capital ensure that there can exist long-run growth driven by capital accumulation in this model. Consumption growth is constant in any equilibrium path, i.e.,
\[ \frac{\dot{c}(t)}{c(t)}=\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] . \] Increase of the per capita consumption over time requires that the externalities in the capital stock (as measured by \(\bar{A}\) ) be large enough, to increase the net marginal product of capital, \(f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta\), above the time preference rate, i.e.,
\[ f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta>\rho \] Given an initial level (choice) of consumption per capita, \(c(0)\), the economy is always along a BGP. In other words, there is no transition dynamics in this model and the economy immediately jumps to a balanced growth path. The proof is offered below. From the first dynamic equation (optimal path of consumption) follows that
\[ \begin{aligned} \int \frac{\dot{c}(t)}{c(t)} d t & =\int \frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] d t \Rightarrow \\ \int \frac{1}{c(t)} \frac{d c(t)}{d t} d t & =\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] t+\tilde{c}_{0} \Rightarrow \end{aligned} \]
where \(m_{0}\) is some constant,
\[ \begin{aligned} \int \frac{1}{c(t)} d c(t) & =\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] t+\tilde{c}_{0} \Rightarrow \\ \ln c(t) & =\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] t+\tilde{c}_{0} \Rightarrow \\ c(t) & =c(0) e^{\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] t} . \end{aligned} \]
with \(c(0)=e^{\tilde{c}_{0}}\). From the law of motion of per-capita capital it follows that
\[ \dot{k}(t)=\left(\bar{A} f\left(\frac{1}{\bar{A}}\right)-\delta-n\right) k(t)-c(0) e^{\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] t} \]
The solution to this differential equation is given by the (linear combination) sum of the general solution of the homogenous differential equation and a particular solution of the non-homogenous differential equation. In other words, solve first
\[ \dot{\hat{k}}(t)=\left(\bar{A} f\left(\frac{1}{\bar{A}}\right)-\delta-n\right) \hat{k}(t) \Rightarrow \hat{k}(t)=\hat{k}(0) e^{\left(\bar{A} f\left(\frac{1}{\bar{A}}\right)-\delta-n\right) t} \]
then find a solution to the general equation. A guess for such solution is
\[ \tilde{k}(t)=\tilde{k}_{1} c(0) e^{\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] t} \]
where \(\tilde{k}_{1}\) is found by plugging the \(\hat{k}\) to the law of motion of per-capita capital after differentiation w.r.t to time, i.e.,
\[ \begin{aligned} & \tilde{k}_{1} c(0) \frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] e^{\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] t}= \\ \Rightarrow & {\left[\bar{A} f\left(\frac{1}{\bar{A}}\right)-\delta-n\right] \tilde{k}_{1} c(0) e^{\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] t}-c(0) e^{\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] t} } \\ & \tilde{k}_{1}=1 /\left\{\left[\bar{A} f\left(\frac{1}{\bar{A}}\right)-\delta-n\right]-\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right]\right\} . \end{aligned} \]
Thus the solution of the differential equation is
\[ k(t)=\hat{k}(0) e^{\left[\bar{A} f\left(\frac{1}{\bar{A}}\right)-\delta-n\right] t}+\tilde{k}_{1} c(0) e^{\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] t} . \]
Given that \(f^{\prime \prime}<0\) the following inequalities hold
\[ \bar{A} f\left(\frac{1}{\bar{A}}\right)-\delta-n>f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho>0 \]
From household’s optimization problem, in turn, follows that the rate of return on capital accumulation in terms of utility is
\[ \begin{aligned} -\frac{\dot{\lambda}(t)}{\lambda (t)} & =r(t)-(\rho-n) \\ \lim _{t \rightarrow \infty} k(t) \lambda(t) e^{-(\rho-n) t} & =0 . \end{aligned} \]
Therefore, the transversality condition requires that \(\hat{k}(0)=0\) and
\[ k(t)=\tilde{k}_{1} c(0) e^{\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] t} \] which means that per-capita capital grows at the same (constant) rate as the consumption.
Another and more intuitive way to show that the growth rate of per-capita capital is always constant and equal to the growth rate of consumption.
Capital per head will grow at a constant rate only if the consumption-to-capital ratio remains constant over time. In order to examine the properties of the BGP, examine the behavior of the \(\frac{c}{k}\) ratio, i.e.,
\[ \frac{ \dot{\left(c(t) / k(t)\right)}}{c(t) / k(t)}=\frac{\dot{c}(t)}{c(t)}-\frac{\dot{k}(t)}{k(t)}=\frac{c(t)}{k(t)}+\frac{f^{\prime}\left(\frac{1}{\bar{A}}\right)-\theta \bar{A} f\left(\frac{1}{\bar{A}}\right)-(1-\theta) \delta-\theta n-\rho}{\theta} \]
Note that the above dynamic equation is unstable in \(\frac{c(t)}{k(t)}\). Moreover, unless \(\frac{\dot{c}(t)}{c(t)}=\frac{\dot{k}(t)}{k(t)}\), the growth rate of \(k(t)\) either should cease or should increase to infinity. Both cases would violate transversality condition. Therefore, it must be that the BGP is characterized by constant \((\frac{c}{k})^{*}=\) \(\frac{(1-\theta) \delta+\theta \bar{A} f\left(\frac{1}{\bar{A}}\right)-f^{\prime}\left(\frac{1}{\bar{A}}\right)+\rho-\theta n}{\theta}\). In the event of a structural change, consumption in this model makes a “discrete” shift to ensure that \(\frac{c(t)}{k(t)}=(\frac{c}{k})^*\) and the economy is set again on a BGP.
The steady-state growth rate of per capita consumption and capital is
\[ g=\frac{\dot{c}(t)}{c(t)}=\frac{\dot{k}(t)}{k(t)}=\frac{1}{\theta}\left[f^{\prime}\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] \]
Furthermore, as in the Solow model, the savings rate in this economy is constant but is an endogenous object since:
\[ s=\frac{Y(t)-C(t)}{Y(t)}=1-\frac{c^{*}}{k} \frac{1}{\bar{A} f\left(\frac{1}{\bar{A}}\right)}=\frac{f^{\prime}\left(\frac{1}{\bar{A}}\right)-(1-\theta) \delta-\rho}{\theta \bar{A} f\left(\frac{1}{\bar{A}}\right)} \]
8.4 Comparative Statics
- Increase in \(\bar{A}\) increases \(g\) and has ambiguous effect on savings rate \(s\)
- Higher \(\bar{A}\) implies higher growth rate \(g\) since it increases the effectiveness of capital per head in increasing the labor productivity
- Increase in \(\theta\) or \(\rho\) decrease both \(g\) and \(s\)
- The higher \(\theta\) and \(\rho\) imply lower saving rate \(s\) since the first one increases the consumption smoothing and the second one induces higher consumption at current period. In turn, lower savings rate implies lower growth \(g\) through lower capital accumulation and, thus, learning-by-doing spillovers.
8.5 Social Planner’s Problem
Note that the welfare theorems will not work in this model since knowledge externalities are assumed, which are inherent to the accumulated capital stock and are not internalized in competitive equilibrium. Due to these externalities the competitive equilibrium outcome may not be the first best (the socially optimal one).
The Social Planner (SP) internalizes these externalities and therefore he identifies that \(k_{i}(t)=k(t)\), when considering the marginal product of capital in the production. The SP selects the paths of quantities that deliver maximum utility (social welfare). The SP’s problem is
\[ \begin{aligned} & \max _{c} \int_{0}^{\infty} \frac{c(t)^{1-\theta}-1}{1-\theta} e^{-(\rho-n) t} d t \\ & \text { s.t. } \\ & \dot{k}(t)=\bar{A} k(t) f\left(\frac{1}{\bar{A}}\right)-c(t)-(n+\delta) k(t) \\ & k(0)>0\text { given } \end{aligned} \]
This problem in terms of current value Hamiltonian is
\[ \max _{c} H_{S P}=\frac{c(t)^{1-\theta}-1}{1-\theta}+\lambda(t)\left(\bar{A} k(t) f\left(\frac{1}{\bar{A}}\right)-c(t)-(n+\delta) k(t)\right) . \]
The first order conditions (optimal rules) are
\[ \begin{aligned} & { c(t)^{-\theta}=\lambda(t) \text {, }} \\ & { \quad \dot{\lambda}(t)=\lambda (t)(\rho-n)-\lambda(t) \left(\bar{A} f\left(\frac{1}{\bar{A}}\right)-(n+\delta)\right) \text {. }} \end{aligned} \]
Therefore,
\[ \frac{\dot{c}(t)}{c(t)}=\frac{1}{\theta}\left(\bar{A} f\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right) \]
The same way as in competitive equilibrium it can be argued that the capital per head grows at the same (constant) rate as the consumption. Thus,
\[ g^{S P} \equiv \frac{\dot{c}(t)}{c(t)}=\frac{\dot{k}(t)}{k(t)}=\frac{1}{\theta}\left[\bar{A} f\left(\frac{1}{\bar{A}}\right)-\delta-\rho\right] \]
and
\[ \left(\frac{c(t)}{k(t)}\right)^{S P} \equiv \frac{c(t)}{k(t)}=\frac{(1-\theta)\left(\delta-\bar{A} f\left(\frac{1}{\bar{A}}\right)\right)-\theta n+\rho}{\theta} \] - The SP achieves higher growth rate because \(\frac{f\left(\frac{1}{\bar{A}}\right) /(1 / \bar{A})}{f^{\prime}\left(\frac{1}{\bar{A}}\right)}>1\), which holds due to the concavity of the production function. - The social marginal product of capital exceeds the private one, because the latter does not account for the efficiency benefits delivered by the overall level of capital stock in the economy. As a result, the SP saves more since \(\left(\frac{c}{k}\right)^*=\left(\frac{c}{k}\right)^{S P}+\frac{\bar{A} f\left(\frac{1}{\bar{A}}\right)-f^{\prime}\left(\frac{1}{\bar{A}}\right)}{\theta}\), where the second term captures the gap between the social and private returns to savings. Thus, the SP achieves higher long-run growth.