Now that kk is defined as the amount of capital per effective worker, increases in the effective number of workers because of technological progress decrease kk:

gg

0.02
00.15

The introduction of technological progress also modifies the criterion for the Golden Rule, which is now defined as the steady state that maximises consumption per effective worker. Following the same argument as before, steady state consumption per effective worker is maximised if:

MPK=δ+n+gMPK = \delta + n + g

Because in practice economies experience both population growth and technological progress, we must use this criterion to determine if the economy is in the Golden Rule steady state.

We will now explore how key variables behave in the steady state with technological progress. As we have seen, capital per effective worker kk is constant in the steady state. Because y=f(k)y = f(k) output per effective worker is also constant in the steady state.

We can now infer what is happening with the other variables that are not expressed at the per effective worker level. We consider output per actual worker Y/L=y×EY/L = y \times E. In the steady state, yy is constant, whilst EE grows at rate gg, hence Y/LY/L is also growing at rate gg in the steady state.

Similarly, we consider how total output Y=y×(E×L)Y = y \times (E \times L) evolves in the steady state. We know that yy is constant in the steady state, whilst EE grows at rate gg and LL grows at rate nn. Hence total output YY grows at rate n+gn + g in the steady state.

We can summarise the growth rates of key variables in the steady state as follows:

VariableSymbolSteady-State Growth Rate
Capital per effective workerk=K/(E×L)k = K/(E \times L)0
Output per effective workery=Y/(E×L)y = Y/(E \times L)0
Output per workerY/L=y×EY/L = y \times Egg
Total OutputY=y×(E×L)Y = y \times (E \times L)g+ng + n

With the introduction of technological progress, we can finally explain the long run increases in per capita income that we observe. We have shown that technological progress leads to a steady state with growth in output per worker. Once the economy is in the steady state, the rate of growth in output per worker is the rate of technological progress. We can conclude that according to the Solow model, only technological progress can lead to sustained growth in income per capita and hence to persistantly rising living standards.