Abstract
We present the continuous time Cass–Koopmans model, characterizing the rate of capital accumulation that maximizes some social welfare criterion. Thus, we no longer consider a constant savings rate. We show the existence and stability of a unique optimal path. We discuss some numerical exercises on the long-run effects of changes in structural parameters, paying attention to the relevance of the different structural characteristics of the economy in characterizing the transition path between steady-states. The chapter closes with an economy with government. We examine the potential inefficiency of its competitive equilibrium with government, showing that the Ricardian doctrine, on the irrelevance of the financing tools used by the government, may not hold under some types of distortionary taxation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The steady-state stock of capital is \(k_{{\textit {ss}}}=\left ( \frac {\alpha }{n+\delta +\theta }\right ) ^{\frac {1}{1-\alpha }}\). Under the assumed technology, \( y_{t}=k_{t}^{\alpha }\), steady-state output and consumption satisfy: \( y_{{\textit {ss}}}=f\left ( k_{{\textit {ss}}}\right ) =k_{{\textit {ss}}}^{\alpha }\), \(c_{{\textit {ss}}}=y_{{\textit {ss}}}-\left ( n+\delta \right ) k_{{\textit {ss}}}\).
So that the investment to output ratio is:
$$\displaystyle \begin{aligned} 1-\frac{c_{{\textit{ss}}}}{y_{{\textit{ss}}}}=\alpha \frac{n+\delta }{n+\delta +\theta }=\frac{ \alpha }{1+\frac{\theta }{n+\delta }}, \end{aligned}$$smaller than the output share of capital, α. It will approach α only if the rate of time discount is small, relative to total depreciation.
- 2.
Alternatively, it can be said that the utility of future generations receives too much weight. Along the Golden Rule, individuals from successive generations all receive the same weigh in the utility function. However the size of generations grows at a rate n, thereby future generations receiving a higher weight in the planner’s objective function.
- 3.
Or any other fraction of that distance, of course.
- 4.
To integrate by parts, we define: \(\ u=e^{-\int _{0}^{t}r_{s}{\textit {ds}}}\) and: υ = K t, so that: \(d\upsilon =\dot {K}_{t}{\textit {dt}}\), and applying Leibniz’s rule: \(du=e^{-\int _{0}^{t}r_{s}{\textit {ds}}}r_{t}{\textit {dt}}\).
- 5.
Turnovsky [4, p. 228].
- 6.
In the simpler situations, the government is supposed to act passively, just taking care of expenditures and revenues. Alternatively, the government may be considered to conduct an optimal policy exercise, thereby designing policy optimally, so a to maximize consumers’ welfare. This is the so-called Ramsey Problem, usually subject to technical difficulties.
- 7.
Note that the planner chooses not only private but also public consumption. On the other hand, at a difference of a government, the planner does not have anything to do with taxes or debt, but only with allocating physical resources in the economy.
- 8.
The production function has now the form: Y t = F(K t, L t l t) where L t l t is the total number of hours worked. Homogeneity of the production function allows us to normalize,
$$\displaystyle \begin{aligned}\frac{Y_{t}}{N_{t}}=F\left(\frac{K_{t}}{N_{t}},\frac{L_{t}}{N_{t}}l_{t}\right).\end{aligned}$$In equilibrium, N t = L t, and the production function can be written in per capita terms as, y t = F(k t, l t) , where k t denotes, as usual, the capital–labor ratio.
- 9.
By discounting depreciation from output, we are considering depreciation allowances in the tax base. The alternative formulation would be,
$$\displaystyle \begin{aligned} c_{t}+\dot{k}_{t}=\left( 1-\tau ^{y}\right) f\left( k_{t}\right) -(n+\delta )k_{t}+g_{t}. \end{aligned}$$
References
Blanchard, O., & Fischer, S. (1989). Lectures on macroeconomics. Cambridge: MIT.
Cass, D. (1965). Optimum growth in an aggregative model of capital accumulation. Review of Economic Studies, 32, 233–240.
Koopmans, T. C. (1965). On the concept of optimal economic growth. In The economic approach to development planning. Amsterdam: North-Holland.
Turnovsky, S. (2000). Methods of macroeconomic dynamics. Cambridge: MIT.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer-Verlag GmbH Germany, part of Springer Nature
About this chapter
Cite this chapter
Novales, A., Fernández, E., Ruiz, J. (2022). Optimal Growth: Continuous Time Analysis. In: Economic Growth. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63982-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-662-63982-5_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-63981-8
Online ISBN: 978-3-662-63982-5
eBook Packages: Economics and FinanceEconomics and Finance (R0)