What happens to effective consumption and capital in the Ramsey/Cass-Koopmans(RCK) model when technological growth decreases? 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.
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

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

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

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

in order to exclude infinite utility. 

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

$$\frac {\partial \hat c^*}{\partial g_x} &lt;0$$

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

See Barro &amp; 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.