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Is the conclusion that both decline correct? Both long and short run.

Let $g_x$ denote growth rate of $x$, $g_c$ the growth rate of $\hat c$. In the model there is no depreciation so,

$$ g_c = \frac 1{\theta}[r - \rho - \theta g_x] $$

$$\dot {\hat k} = f(\hat k) - \hat c - (n+g_x)\hat k$$

At steady state growth of $\hat k$ and $\hat c$ are both $0$. When that is the case:

$$\hat c^* = f(\hat k^*) - (n+g_x)\hat k^*$$

Also since $r = f'(k)$

$$f'(\hat k^*) = \rho + \theta g_x$$

Somehow based on these equations I reach the opposite conclusion, i.e. that $\hat c^*$ and $\hat k^*$ will increase rather than decrease if $g_x$ is reduced.

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1 Answer 1

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When the (exogenous) rate of technological change/efficiency is smaller, the corresponding steady-state levels of consumption and capital per unit of effective labor, increase.

For capital, we have

$$f'(\hat k^*) = \rho + \theta g_x \implies \frac {\partial}{\partial g_x}f'(\hat k^*) = \theta >0$$

So if $g_x \downarrow \implies f'(\hat k^*) \downarrow \implies \hat k^* \uparrow$ due to decreasing marginal product of capital.

For steady-state consumption per unit of effective labor we have

$$\hat c^* = f(\hat k^*) - (n+g_x)\hat k^*$$

$$\implies \frac {\partial \hat c^*}{\partial g_x}= f'(\hat k^*) \frac {\partial \hat k^*}{\partial g_x} - \hat k^* - (n+g_x) \frac {\partial \hat k^*}{\partial g_x}$$

$$=[f'(\hat k^*) - n - g_x]\cdot \frac {\partial \hat k^*}{\partial g_x} - \hat k^* $$

The term in brackets is assumed positive, i.e. we have already assumed

$$f'(\hat k^*) = \rho + \theta g_x > n + g_x \implies \rho > n + (1-\theta)g_x$$

in order to exclude infinite utility.

Moreover, evidently $\frac {\partial \hat k^*}{\partial g_x} <0$ so in all

$$\frac {\partial \hat c^*}{\partial g_x} <0$$

Therefore, if $g_x \downarrow \implies \hat c^* \uparrow $.

See Barro & Martin, ch. 2 page 102, where they discuss the $g_x \uparrow$ case (in p. 101 they discuss the constraint on the parameters).

Comment: This result may appear counter-intuitive, but a deeper examination of the model shows that if $g_x$ is lower, utility per capita is lower. Using "consumption per unit of effective labor" is a modeling tactic, what interests us is what happens per individual. So no, the model does not argue in favor of lower productivity/technology.

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Perfect! One more thing, the result of this in the phase diagram is that the $k^$ and $C^$ curves both move in the phase diagram (C/K coordinates). What is the intuition behind only one of the curves moving vs. both of them doing so? –  Dole Mar 9 at 2:07
@Dole I would suggest to go after Barro's book, it is legally freely downloadable and it presents the whole model in many details and extensions, as well as intuitive explanations. –  Alecos Papadopoulos Mar 9 at 2:32

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