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I have a discrete time Solow model and need to derive the golden rule for the economy by setting maximization problem. How can I do that, if I can’t take derivative?

Note: required investment can’t be approximated. enter image description here

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What is discrete is the time dimension, not the range of capital, the values that it can take. Also, at the steady-state we have $k_{t+1}=k_t = k^*(s)$. So proceed as in the continuous-time case.

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  • $\begingroup$ I need to find golden rule saving rate, i.e. maximize consumption. How can I do that without derivative? $\endgroup$
    – Ksenia
    Jan 21, 2023 at 15:02
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    $\begingroup$ @Ksenia But you can take derivatives. The other answer posted gives a detailed explanation. $\endgroup$ Jan 23, 2023 at 2:29
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As Alec Papadopulous said, Solow model in discrete time means that time cannot take all the values on the positive real line, but some discrete values only, so that the variables, thought of as function of time, take only values in correspondence of those discrete values of time. Continuous time model means that time varies continuously, that is it takes all the value of an interval (say the positive values) of the real line.

But that's all. The discrete values concern only time. Only time is discrete.

On the contrary, the other variables and functions are 'normal' variables and functions, defined on intervals of $\mathbb{R}$. So, functions as the intensive production function $f(k)$, the saving function $sf(k)$ and so on, are not functions defined on discrete sets, and they are usually assumed to be differentiable, considered as function of $k$ or other variables of the model.

When deriving the Golden Rule, you look for the value of $s$ that maximizes, in steady state, the per capita consumption. This is a hypothetical, conceptual experiment, time is not involved: you are comparing different steady states with different values of $s$.

Therefore, you mustn’t differentiate anything with respect to time. The variable considered, with respect to which you are maximizing consumption, is the saving rate $s$. You must take the derivative with respect to the saving rate $s$, and no discrete variable is involved.

So, there is no difference between discrete and continuous time approaches to find the Golden Rule rate of saving, you can use the derivative in the same way in both cases.

For an explanation of how to derive the Golden Rule level of saving, see the answer of Alecos Papadopoulos in

Deriving the golden-rule savings rate in a Solow Model

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  • $\begingroup$ In the task it was emphasized that I should derive the golden rule saving rate for discrete time, else “no points will be given”. Moreover, I don’t have any production function (the only function I have is the one in the picture). $\endgroup$
    – Ksenia
    Jan 21, 2023 at 16:55
  • $\begingroup$ You should refer to the version of Solow model you studied. The formula in your question is the difference equation that rules the system, I suppose, so you have to find the steady state $k^*$ setting $k_{t+1}=k_t$, then to find the steady state consumption, then maximizing it with respect to $s$. By the way, you have a production function, is $f(k)$ in your formula. I think you should review the Solow model. Without remembering the overall model it is difficult to solve your problem. It is not a matter of discrete or continuous time, but of remembering the overall functioning of the model. $\endgroup$ Jan 21, 2023 at 17:35

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