Economics Stack Exchange is a question and answer site for professional and academic economists and analysts. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|improve this question
up vote 0 down vote accepted

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.

share|improve this answer
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 '15 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 '15 at 2:32

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.