Abstract
The AK model, introduced by Rebelo [6], is characterized by a constant returns to scale technology, linear in physical capital
with A representing the constant average and marginal productivity of capital, and K t the aggregate stock of capital. As we saw in Chap. 2, aggregate constant returns to scale in the cumulative inputs is a necessary condition for endogenous growth. This assumption is a violation of the Inada condition \(\lim _{K_{t}\rightarrow \infty }F^{{\prime}}\left (K_{t}\right ) = 0,\) which is assumed to hold in neoclassical growth models under decreasing returns to scale.
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- 1.
In presence of technological growth, per-capita variables grow in steady-state in exogenous growth model, but there is no growth when variables are considered in units of efficient labor. Introducing technological growth in the AK model, variables in units of efficient labor would still display non-zero growth in steady-state.
- 2.
Without loss of generality, we will not use this convention with Hamiltonian or Lagrange multipliers.
- 3.
In this simple version of the AK economy policy interventions do not directly affect the rate of growth, which depends on the values of A, n, δ, θ, σ. Later on, we will see that policy choices may also affect growth.
- 4.
A quite natural condition, that requires that the rate of growth of the consumption argument in the single period utility function be lower than the rate of time discount, θ.
- 5.
But this is exactly the same condition (2) we obtained before to guarantee that the transversality condition will hold, although the latter also requires the linear relationship between consumption and capital we characterized in the previous section.
- 6.
Since there are no taxes, money or any public expenditures in this simple version of the AK economy, the planner’s problem is the same as that of the representative agent.
- 7.
As usual, the transversality condition comes from taking derivatives in the finite horizon version of the Lagrangian with respect to \(\tilde{k}_{T+1}\), and imposing the condition,
$$\displaystyle{ \lim _{T\rightarrow \infty }\beta ^{T}\tilde{k}_{ T+1} \frac{\partial L} {\partial \tilde{k}_{T+1}} = 0, }$$the partial derivative of the Lagrangian with respect to the last period’s stock of capital being equal to λ T .
- 8.
This would be physical depreciation as well as the loss of resources due to providing the newly born with the same stock of capital as owned by existing workers.
- 9.
The similarity between restriction (6.12) and the analogue constraint we found in the continuous time version of the model is evident.
- 10.
When it is not needed, in what follows we skip the 1 + n factor from the transversality condition.
- 11.
There is nothing specific of the normalization we use. In fact, if we normalized the eigenvectors to have unit norm, these would be \(\left (\begin{array}{l} \frac{\phi } {\sqrt{ 1+\phi ^{2}}} \\ \frac{1} {\sqrt{1+\phi ^{2}}} \end{array} \right )\) and \(\left (\begin{array}{l} 0\\ 1 \end{array} \right ),\) and the system could be written
$$\displaystyle\begin{array}{rcl} \left (\begin{array}{l} c_{t} \\ k_{t} \end{array} \right )& =& \left (\begin{array}{l@{\quad }l} \frac{\phi } {\sqrt{ 1+\phi ^{2}}} \quad &0 \\ \frac{1} {\sqrt{1+\phi ^{2}}} \quad &1 \end{array} \right )\left (\begin{array}{l@{\quad }l} 1\quad &0 \\ 0\quad &\left [ \frac{A+1-\delta } {(1+n)(1+\gamma )}\right ]^{t} \end{array} \right )\left (\begin{array}{c@{\quad }c} \frac{1} {\phi } \sqrt{\left (1 +\phi ^{2 } \right )}\quad &0 \\ -\frac{1} {\phi } \quad &1 \end{array} \right )\left (\begin{array}{l} c_{0} \\ k_{0} \end{array} \right ) {}\\ & =& \left (\begin{array}{c} c_{0} \\ \frac{1} {\phi } c_{0} + \left (k_{0} -\frac{c_{0}} {\phi } \right )\left [ \frac{A+1-\delta } {\left (1+n\right )(1+\gamma )}\right ]^{t} \end{array} \right ), {}\\ \end{array}$$the same representation we obtained before, so the same argument could be made to characterize the single stable trajectory. Normalizing the eigenvectors to have their second component equal to one would again give raise to the same characterization of stability.
- 12.
Remember that the left eigenvectors are obtained as the rows in the inverse of the matrix that has the right eigenvectors as columns.
- 13.
To show this, first notice that the marginal product of capital changes directly with the tax rate. Furthermore,
$$\displaystyle{ \frac{\partial c_{\mathit{ss}}\left (\tau \right )} {\partial \tau } = \frac{\partial k_{\mathit{ss}}\left (\tau \right )} {\partial \tau } \left (\left (1-\tau \right )f^{{\prime}}(k_{ss}\left (\tau \right ))-\delta \right ) - f(k_{\mathit{ss}}\left (\tau \right )) < 0\;. }$$The sign of this expression comes from \(\frac{\partial k_{ss}\left (\tau \right )} {\partial \tau } < 0\) and \(\left (1-\tau \right )f^{{\prime}}(k_{ss}\left (\tau \right ))-\delta > 0\), since the latter is the marginal product of capital net of taxes and depreciation, which will coincide in equilibrium with the real interest rate in the economy, which must be positive.
- 14.
For any ε > 0, there is always a time period t ε such that \(t > t_{\epsilon } \Rightarrow \mid \left (k_{t}^{0}/k_{t}^{1}\right ) - 1\mid <\epsilon\).
- 15.
The reader can find in Novales and Ruiz [5] the analysis of dynamic Laffer effects in an endogenous growth model with human capital accumulation.
- 16.
The presence of the product of real interest rates in the denominator of this fraction is due to solving backwards for the Lagrange multiplier that usually appears in the transversality condition.
- 17.
To eliminate the possibility of non-zero Ponzi games.
- 18.
Indeed, from the transversality condition, we have
$$\displaystyle\begin{array}{rcl} \lim _{t\rightarrow \infty }\beta ^{t}\ \lambda _{t}\tilde{k}_{ t+1} = 0& \Leftrightarrow &\lim _{t\rightarrow \infty }\ \beta ^{t}\tilde{c}_{t}^{-\sigma }\tilde{k}_{ t+1} = 0 \Leftrightarrow \lim _{t\rightarrow \infty }\ \left (\beta \left (1 +\gamma ^{1}\right )^{1-\sigma }\right )^{t} = 0 {}\\ & \Leftrightarrow &\beta \left (1 +\gamma ^{1}\right )^{1-\sigma } < 1 \Leftrightarrow 1 +\gamma ^{1} < R^{1}/(1 + n), {}\\ \end{array}$$since \(1 +\gamma ^{1} = \left ( \frac{\beta R^{1}} {1+n}\right )^{1/\sigma }\).
- 19.
As explained in Chap. 5, random Lagrange multipliers lead to a formulation of first order conditions involving conditional expectations.
- 20.
Taking again into account the fact that \(\frac{\partial F} {\partial \ln \theta _{t+1}} = \frac{\partial F} {\partial \theta _{t+1}} \theta _{t+1}.\) Additionally, θ ss = 1, so that lnθ ss = 0.
- 21.
Those with absolute size above \(\frac{1} {\sqrt{\beta }}\).
- 22.
Note that the ratio \(\frac{c_{t}} {k_{t}}\) can be written with the growth trend or without it \(\frac{\tilde{c}_{t}} {\tilde{k}_{t}}\) since the growth rates of both variables are the same.
- 23.
Again, either the version with growth or the one without growth of the global constraint of resources, could be used to obtain the stock of capital. Alternatively, that constraint could be used to obtain time series for the rate of growth of capital,
$$\displaystyle{ 1 +\gamma _{k_{t}} = \frac{1} {\left (1 + n\right )(1+\gamma )}\left [A\theta _{t} + \left (1-\delta \right )k_{t} - \frac{c_{t}} {k_{t}}\right ], }$$to obtain the time series for capital itself, afterwards: \(k_{t+1} = \frac{1+\gamma _{k_{t}}} {1+\gamma } k_{t}\).
- 24.
Again, government expenditures become endogenous because of the structure of the financing policy.
- 25.
The remaining eigenvalues, if any will be smaller than one in absolute value. In the two models considered, there is a control variable and a single state variable, so an eigenvalue is equal to one and the other one is greater than one in absolute value.
- 26.
As a consequence of the fact that k 0 is given and so is y 0 which is a function of just k 0.
References
Barro, R.J. 1990. Government spending in a simple model of endogenous growth. Journal of Political Economy 98(5): S103–S126.
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Novales, A., and J. Ruiz. 2002. Dynamic Laffer effects. Journal of Economic Dynamics and Control 27: 181–206.
Rebelo, S. 1991. Long-run policy analysis and long-run growth. Journal of Political Economy 99(3): 500–521.
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Novales, A., Fernández, E., Ruiz, J. (2014). Endogenous Growth Models. In: Economic Growth. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54950-2_6
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