# What happens to aggregate C, K, and Y when TFP increases permanently?

I was wondering what would happen to aggregate capital, consumption and output (i.e. K, C, Y) in the Solow model with constant population growth (i.e. n > 0) and no technological growth (i.e. a = 0) if TFP suddenly increases permanently at time $t_{0}$ (i.e. $A_{0}$ --> $A_{1}$). I am assuming that all three K, C, and Y will increase and converge to the new steady state level. But I wanted to know how it would adjust(i.e. would it be an instant increase or gradual increase from previous to new steady state level).

Take a look at the dynamics of the capital: $k_{t+1}=sA_ty_t+(1-\delta-n)k_t$. A sudden positive shock to TFP in period $t$ increases the capital stock of the next period $k_{t+1}$. So, there is no contemporaneous effect on $k$, convergence to the new steady state will be gradual.

The other variables have contemporaneous relationship with TFP.

EDIT: A google search for a good explanation lead me to lecture notes from MIT which you can find here. Start reading slide # 62+. You will see the graphical explanation as well.

• Thank you very much london! I understand how k_{t+1} converges to its new state. Here, k_{t+1} is per worker term, and I was wondering if the dynamics of K, the aggregate capital would be similar to k_{t+1}. Also, could you explain the term "contemporaneous relationship"? Is it to describe the instant change? – David Kim Sep 16 '17 at 3:41
• @DavidKim If you suddenly become more productive today, what would happen to your wage, output and consumption today? Also, the answer to all your questions can be seen one single graph ($k_t$ in x-axis, and $f(k_t)$ in y-axis). I suggest you do attempt to get this graph. – luchonacho Sep 16 '17 at 7:50
• @DavidKim, see the updatd post above. Contemporaneous, in this context, means 'instant'. – london Sep 16 '17 at 20:59
• @DavidKim, yes, the answer is the same for aggregate values. – london Sep 17 '17 at 0:07