C An extension to productivity growth and productivity growth

The canonical Solow model includes exponential growth in population and productivity growth. That is, $A_{t+1}/A_t = 1+g$, where $g$ is a constant, and $L_{t+1}/L_{t} = 1 +n$ where $n$ is a constant. This changes the way we solve the model a little bit. We will assume that technology is “labor augmenting” – that is, having higher productivity is like having extra workers. We then define $A_t L_t$ as the number of “effective workers” in the economy. Steady state in the model is therefore characterized by a constant level of capital (and hence output) per unit of effective labor, rather than per-person.

It’s easiest to see this by assuming Cobb-Douglas production and assuming constant human capital per worker. Assume the aggregate production function is

\[ Y_t = K^\alpha (h A_t L_t)^{1-\alpha}\]

This is written a little differently than before, because we’re explicitly writing productivity as something that augments the number of workers in the economy. Notice that $h \times A_t L_t$ is the total amount of human capital per effective worker.

To get this in terms of output per effective worker, divide both sides by $A_t L_t$:

\[\frac{Y_t}{A_t L_t} = \frac{K^\alpha (h A_t L_t)^{1-\alpha}}{(A_t L_t)^{\alpha} (A_t L_t)^{1-\alpha}}\]

Then using lower-case variable names with tildes to denote things in per-effective-unit-of-labor units:

\[\widetilde{y_t} = \widetilde{k_t^\alpha} h \] We also need to re-write the law of motion for capital in terms of units of effective labor.

Here, we’re going to use a fact: Since $\widetilde{k_t} = K_t/(A_t L_t)$, $\ln(\widetilde{k}_{t}) = \ln(K_t) - \ln(A_t) - \ln(L_t)$

If we take a derivative of this expression with respect to $t$ (notice that each of these variables is a function of $t$), we get

\[\frac{\Delta k_t}{k_t} \approx \frac{\Delta K_t}{K_{t-1}} - \frac{\Delta A_t}{A_{t-1}} - \frac{\Delta L_t}{L_{t-1}}\]

(This becomes exact as $t $.)

From the law of motion for (aggregate) capital, $\Delta K_t/K_t$ = $\frac{I_{t-1}}{K_{t-1}} - \delta$. So then we can write

\[ \widetilde{k}_{t} -\widetilde{k}_{t-1} \approx \frac{I_{t-1}}{K_{t-1}} \cdot k_{t-1} -(\delta + n + g)\widetilde{k}_{t-1} \]

Multiplying the ($I/K$) term by $(1/AL)/(1/AL)$ we get $\widetilde{i}_{t-1}/{\tilde{k}_{t-1}}$. Simplfying we get

\[\widetilde{k_t} = \widetilde{k_{t-1}} + \gamma \widetilde{y_{t-1}} - (n + g + \delta)\widetilde{k_{t-1}}\] This looks very similar to the aggregate version of this expression, except now we have the $n+g$ term. The intuition is that the amount of units of capital per effective worker may decline for three reasons. One is that it may wear out ($\delta$). A second is that we have more actual human beings ($n$). That means a given amount of machines would have to be spread around more people. The third is each existing worker is more productive $(g)$. This means that a given amount of machines will be spread around more effective workers. A steady state will be when the amount of investment is enough to offset the wearing out of machines and the arrival of new (effective) workers.

Graphically, the story is the same as before except we now find steady state at the intersection of $\gamma \widetilde{k}^\alpha h$ and $(n+ g+ \delta)\widetilde{k}$. We no longer consider shifts in $A_t$ because it grows at a constant exogenous rate, but we can talk about what happens if that rate (or the rate of population growth) changes.

Finally, notice that if per effective worker variables are constant in steady state, that means that aggregates are growing at the rate $n + g$. That way, $\widetilde{x} = X /(AL)$ remains constant for any $\widetilde{x}$. This means that the Solow model predicts that aggregate GDP grows at the rate of $n+g$ in the long run. On a per capita basis (for similar reasons), the Solow model predicts that per worker variables grow at a rate $g$. Hence, the Solow model says that in steady state, the rate of productivity growth is what determines growth in GDP per capita. Hence, we still need productivity to change over time to explain sustained growth in GDP per capita over the long run.