## 30 November 2015

### Using Demographics to Estimate Potential Output in Japan

I don't stray into empirical matters very frequently because I'm really not all that adept at them, but I thought it would be interesting to feed Japan's working age population growth into a basic Solow growth model.

Skip the following if you already understand the Solow model:
In the Solow model, it is assumed that output is produced using three inputs: capital, labor, and productivity. The production function is Cobb-Douglas for capital and labor (with constant returns to scale), but is multiplied by what's called the Total Factor of Productivity, or TFP, which represents technological progress. Defining output as $Y_t$, capital as $K_t$, labor as $N_t$, and TFP as $A_t$, the production function can be written as
$$(1)\: Y_t = A_t K_t^\alpha N_t^{1-\alpha}$$
where $\alpha$ is capital's share in production and $1-\alpha$ is labor's share in production. Workers devote a constant share $c$ of production to consumption ($C_t$), so $C_t = c Y_t$. Non-consumed income is used to increase the capital stock, which exogenously depreciates at $\delta$. Defining $s$, the share of income put toward investment in each period as $1-c$ allows us to write the capital accumulation equation as such:
$$(2)\: K_{t+1} = (1 - \delta) K_t + s Y_t$$
The labor force is assumed to grow at constant rate $n$, so next period's labor forced is defined as
$$(3)\: L_{t+1} = L_t (1 + n)$$
TFP is assumed to grow at constant rate $g$, so TFP evolves according to
$$(4)\: A_{t+1} = A_t (1 + g)$$
Given $L_0$, $A_0$, and $K_0$, the economy will eventually converge to a balanced growth path in which all variables grow at the same rate as productivity: $g$. If one of the parameters ($\alpha$, $\delta$, $s$, $n$, or $g$) changes, then the economy will take time to adjust to new equilibrium levels.

 Figure 1
As you can see in figure one, Japan began to see a secular decline in it's working age population growth rate in about 1990. This decline coincides roughly with Japan's lost decade -- the period between the mid 90s and the early 2000s characterized by low growth and high unemployment. Given the low working age population growth, it may be possible to explain some of this lack of economic activity with the Solow model. Assuming constant technological growth of 1%, a capital depreciation rate of 2.5%, a capital share of 33%, and a savings rate of 10% (I have no clue how close to accurate this calibration is, if someone wanted to find the average values of each variable over the last 20 years or so in Japan, I'll update them, but right now I can't be bothered to find the information myself), I was able to come up with an estimate of 'potential' output in Japan -- i.e. what Japanese output would be absent any shocks to productivity, government spending, or monetary policy (or natural disasters, which explain the 2011 output contraction).

Here are a couple of graphs relating actual output to demographically-adjusted potential output:
 Figure 2
 Figure 3
Figure 2 plots my estimate of potential output against actual output, assuming potential output was 2% above actual output in 1995 and figure 1 plots the output gap, or the percentage gap between actual and potential output. An interesting note here is that potential output, absent any demographic or technological changes, is predicted converge to a decay rate of roughly 0.5% per year and potential output is currently growth at about zero percent per year, meaning that, not only is potential growth for the next couple of years zero, the economy should be expected to shrink without being in a recession in the future. That is, unless the workforce stops decaying so quickly.

Another interesting observation is that Japan's lost decade seems to closely resemble the experience that the United States has had since the Great Recession. This is entirely unsurprising given that both periods are characterized by monetary policy ineffectiveness (the zero lower bound), but the post 2007 experience in Japan could possibly be used to predict the outcome of another large recession in the US absent monetary policy normalization. Perhaps more on this later.