# Question on Ramsey model with shock on technological level

I have a question on Ramsey Model.

Consider the following one-sector, closed, representative household economy.

I have the following Cobb Douglas production function in intensive form

$$f(k)=Ak(t)^{\alpha}$$

A is the constant technology level: There is no technological progress: Let denote n the population growth rate. This production function displays constant returns to scale in both Capital and labor ; hence each factor is paid its marginal product.

$$\delta$$ is depreciation rate for capital. And $$p$$ is time preference rate.

I found the characteristics of the equilibrium in this model as follows

Euler equation is

$$\frac{\dot{c}}{c}=A\alpha k^{\alpha -1}- \delta -p$$

Resource constraint is

$$\dot{k} = Ak^{\alpha} - (\delta +n) k - c$$

suppose at some t = 0, a permanent shock raises technological level A to $$A>A^*$$. Show the effect of this shock on the phase diagram of the model. In your answer, be sure to compare the equilibrium paths for capital stock and consumption in the economies with and without technological shock.

My answer The economy is initially at $$E_0$$.

If $$A$$ increases, then the return to saving $$A\alpha k^{\alpha -1}- \delta$$ increases, so saving increases, this implies that capital accumulates. So the $$\dot{c}=0$$ locus shifts to right. As capital accumulates, consumption jumps up and so, consumption rises to point B.

If $$A$$ increases, then the Output $$f(k)$$ is much higher than the break even investment $$(\delta +n)k$$, so c will be also increasing such that $$\dot{k}=0$$ holds. Then the locus $$\dot{c}=0$$ shifts up.

At the end, the economy reaches the new equilibrium point $$E_N$$.

The graph is as follows.

However, my instructor gives such a feedback about my answer. And my answer is not enough for her.

you need to show the old and new stable arms and also show that at the time of the shock the c jumps up and the new path remains above the old path.

How can I correct my answer according to this feedback?

Now suppose at some t = 0, a permanent shock raises technological level A to $$A^* > A$$. Show the effects of this shock on the phase diagram of the model. Please compare the equilibrium paths for capital stock and consumption in the economies with and without the technological shock on the phase diagram.
• With no technological progress and $α<1$, how does your economy achieve the initial steady state? Commented Jul 21, 2022 at 17:24