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2022 | OriginalPaper | Buchkapitel

6. Endogenous Growth Models

verfasst von : Alfonso Novales, Esther Fernández, Jesús Ruiz

Erschienen in: Economic Growth

Verlag: Springer Berlin Heidelberg

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Abstract

The continuous and discrete time versions of the AK model are examined in detail, showing the absence of transition and the existence of a balanced growth path with all per capita variables growing at the same constant rate. We show that transitory policy interventions or structural changes in endogenous growth models have permanent effects. We analyze dynamic Laffer curves, a possibility that is specific of endogenous growth models. We explain how to obtain numerical solutions to the stochastic, discrete time version of the AK model. After that, we consider Barro’s version of the AK model that includes government expenditures, and we introduce the Jones and Manuelli variant of the AK model, describing the transitional dynamics, characterizing the stability conditions, and explaining how to compute numerical solutions.

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Vorheriges Kapitel Numerical Solution Methods

Nächstes Kapitel Additional Endogenous Growth Models

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.

Without loss of generality, we will not use this convention with Hamiltonian or Lagrange multipliers.

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.

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.

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.

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.

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,

$$\displaystyle \begin{aligned} \lim_{T\rightarrow \infty }\beta ^{T}\tilde{k}_{T+1}\frac{\partial L}{ \partial \tilde{k}_{T+1}}=0, \end{aligned}$$

the partial derivative of the Lagrangian with respect to the last period’s stock of capital being equal to λ _T.

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.

The similarity between restriction (6.12) and the analogue constraint we found in the continuous time version of the model is evident.

When it is not needed, in what follows we skip the 1 + n factor from the transversality condition.

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

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left( \begin{array}{l} c_{t} \\ k_{t} \end{array} \right) & =&\displaystyle \left( \begin{array}{l@{\quad }l} \frac{\phi }{\sqrt{1+\phi ^{2}}} &\displaystyle 0 \\ \frac{1}{\sqrt{1+\phi ^{2}}} & 1 \end{array} \right) \left( \begin{array}{l@{\quad }l} 1 &\displaystyle 0 \\ 0 & \left[ \frac{A+1-\delta }{(1+n)(1+\gamma) }\right] ^{t} \end{array} \right) \left( \begin{array}{c@{\quad }c} \frac{1}{\phi }\sqrt{\left( 1+\phi ^{2}\right) } &\displaystyle 0 \\ -\frac{1}{\phi } & 1 \end{array} \right) \left( \begin{array}{l} c_{0} \\ k_{0} \end{array} \right) \\ & =&\displaystyle \left( \begin{array}{c} c_{0} \\ \frac{1}{\phi }c_{0}+\left( k_{0}-\frac{c_{0}}{\phi }\right) \left[ \frac{ A+1-\delta }{\left( 1+n\right) (1+\gamma)}\right] ^{t} \end{array} \right) , \end{array} \end{aligned} $$

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.

Remember that the left eigenvectors are obtained as the rows in the inverse of the matrix that has the right eigenvectors as columns.

To show this, first notice that the marginal product of capital changes directly with the tax rate. Furthermore,

$$\displaystyle \begin{aligned} \frac{\partial c_{{ ss}}\left( \tau \right) }{\partial \tau }=\frac{\partial k_{{ ss}}\left( \tau \right) }{\partial \tau }\left( \left( 1-\tau \right) f^{\prime }(k_{ss}\left( \tau \right) )-n-\delta \right) -f(k_{{ ss}}\left( \tau \right) )<0\;. \end{aligned}$$

The sign of this expression comes from $\frac {\partial k_{ss }\left ( \tau \right ) }{\partial \tau }<0$ and $\left ( 1-\tau \right ) f^{\prime }(k_{ss }\left ( \tau \right ) )-n-\delta >0$, 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.

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 $.

The reader can find in Novales and Ruiz [5] the analysis of dynamic Laffer effects in an endogenous growth model with human capital accumulation.

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.

To eliminate the possibility of non-zero Ponzi games.

Indeed, from the transversality condition, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \lim_{t\rightarrow \infty }\beta ^{t}\ \lambda _{t}\tilde{k}_{t+1} =0& \Leftrightarrow&\displaystyle \lim_{t\rightarrow \infty }\ \beta ^{t}\tilde{c} _{t}^{-\sigma }\tilde{k}_{t+1}=0\Leftrightarrow \lim_{t\rightarrow \infty }\ \left( \beta \left( 1+\gamma ^{1}\right) ^{1-\sigma }\right) ^{t}=0\\& \Leftrightarrow&\displaystyle \beta \left( 1+\gamma ^{1}\right) ^{1-\sigma } <1\Leftrightarrow 1+\gamma ^{1}<R^{1}/(1+n), \end{array} \end{aligned} $$

since $1+\gamma ^{1}=\left ( \frac {\beta R^{1}}{1+n}\right ) ^{1/\sigma }$.

As explained in Chap. 5, random Lagrange multipliers lead to a formulation of first order conditions involving conditional expectations.

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 $\ln \theta _{{ ss}}=0.$

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.

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,

$$\displaystyle \begin{aligned} 1+\gamma _{k_{t}}=\frac{1}{\left( 1+n\right) (1+\gamma )}\left[ A\theta _{t}+\left( 1-\delta \right) k_{t}-\frac{c_{t}}{k_{t}}\right] , \end{aligned}$$

to obtain the time series for capital itself, afterwards: $k_{t+1}=\frac {1+\gamma _{k_{t}}}{1+\gamma }k_{t}$.

Again, government expenditures become endogenous because of the structure of the financing policy.

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.

As a consequence of the fact that k ₀ is given and so is y ₀ which is a function of just k ₀.

Barro, R. J. (1990). Government spending in a simple model of endogenous growth. Journal of Political Economy, 98(5), S103–S126.CrossRef

Den Haan, W., & A. Marcet. (1990). Solving the stochastic growth model by parameterized expectations. Journal of Business and Economic Statistics, 8, 31–34.

Ireland, P. N. (1994). Supply-side economics and endogenous growth. Journal of Monetary Economics, 33, 559–572.CrossRef

Jones, L. E., & R. Manuelli. (1990). A convex model of economic growth. Journal of Political Economy, 98(5), 1008–1038.CrossRef

Novales, A., & J. Ruiz. (2002). Dynamic Laffer effects. Journal of Economic Dynamics and Control, 27, 181–206.CrossRef

Rebelo, S. (1991). Long-run policy analysis and long-run growth. Journal of Political Economy, 99(3), 500–521.CrossRef

Titel: Endogenous Growth Models
verfasst von: Alfonso Novales
Esther Fernández
Jesús Ruiz
Verlag: Springer Berlin Heidelberg
Buch: Economic Growth
Print ISBN: 978-3-662-63981-8

Electronic ISBN: 978-3-662-63982-5

Copyright-Jahr: 2022
DOI: https://doi.org/10.1007/978-3-662-63982-5_6

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