## 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.

## 25 November 2015

### Demystifying Neo Fisherism

Misunderstanding of monetary economics abounds in the econoblogosphere. Since I'd like to think I know a decent bit about this issue, I think I might try and clarify some things with a pretty simple model.

There exists a household with the utility function $U = E_0 \sum^\infty_{t=0} \beta^t \left(u(c^1_t) + u(c^2_t)\right)$ where $0 < \beta < 1$ is the household's discount factor, $E_t$ is the rational expectations operator given information known in period $t$, $c^1_t$ is a consumption good that can be purchased using cash only, and $c^2_t$ is a good that can be purchased using cash or credit. The household uses government bonds and money carried from the last period as well as a constant endowment to purchase government bonds, money, and both consumption goods and to pay a lump sum tax levied by the government. The household's budget constraint is
$$(1.1)\: M_{t-1} + B_{t-1} + P_t y = M_t + Q_t B_t + P_t \tau_t + P_t (c^1_t + c^2_t)$$
where $M_t$ is the money supply that will be carried into the next period, $B_t$ is the stock of government bonds that will be carried into the next period, $Q_t$ is the price of government bonds maturing in period $t+1$, $P_t$ is the price of both consumption goods, $y$ is the endowment, and $\tau_t$ is the real lump sum tax. $c^1_t$ must be paid for in cash, so the household faces a cash in advance constraint where it must hold at least enough money to cover $P_t c^1_t$.
$$(1.2)\: M_t \geq P_t c^1_t$$

I assume that the government sets $B_t = 0\: \forall t$, so the government's budget constraint, given zero government bonds, is
$$(1.3)\: M_t + P_t \tau_t = M_{t-1}$$
The government sets the lump sum tax so that $M_t = \mu_t M_{t-1}$ where $\mu_t$ is an exogenous policy parameter set by the central bank.

The household maximizes $U$ subject to $1.1$ and $1.2$ which gives the following maximization problem
$$(2)\: \mathcal{L} = U + \lambda_t \left(M_{t-1} + B_{t-1} + P_t y - M_t - Q_t B_t - P_t \tau_t - P_t (c^1_t + c^2_t)\right) + \gamma_t \left(M_t - P_t c^1_t\right)$$
which  yields
$$(2.1)\:\frac{\partial \mathcal{L}}{\partial c^1_t} = \beta^t u'(c^1_t) - \lambda_t P_t - \gamma_t P_t = 0$$
$$(2.2)\: \frac{\partial \mathcal{L}}{\partial c^2_t}= \beta^t u'(c^2_t) - \lambda_t P_t = 0$$
$$(2.3)\: \frac{\partial \mathcal{L}}{\partial B_t} = -\lambda_t Q_t + E_t \lambda_{t+1} = 0$$
$$(2.4)\:\frac{\partial \mathcal{L}}{\partial M_t}=-\lambda_t + E_t \lambda_{t+1} +\gamma_t=0$$

$2.1-4$ and $1.3$ can be combined to form an equilibrium for $P_t$, $c^1_t$, $c^2_t$, $M_t$, and $Q_t$:
$$(3.1)\: u'(c^1_t) = u'(c^2_t) (2 - Q_t)$$
$$(3.2)\: M_t = P_t c^1_t$$
$$(3.3)\: y = c^1_t + c^2_t$$
$$(3.4)\: u'(c^2_t) = \beta u'(c^2_t) \frac{1}{Q_t}E_t\frac{P_t}{P_{t+1}}$$
$$(3.5)\: M_t = \mu_t M_{t-1}$$

With the equilibrium, it is possible to get a bit of an answer to the questions that Neo-Fisherians raise. Firstly, the long run inflation rate is equal to the growth rate of the money supply and the euler equation shows that, in the long run, the inflation rate is a constant different from the nominal interest rate. This means that, were the central bank to choose a low path for $\mu_t$, both inflation and the nominal interest rate would be lower. Of course, that's completely standard, it's just a lot more sensible to have a model where it's clear that this is a long run tightening of monetary policy. (Point Neo Fisherians)

This means that a disinflation, i.e. a reduction in the path of $\mu_t$, is consistent with a low nominal interest rate in the long run. In the short run, a higher value of $\mu_t$ can either take the form of higher inflation or lower interest rates. This is because a higher value of $Q_t$ (the inverse of the nominal interest rate) induces the household to shift demand from the credit good to the cash good because the nominal interest rate represents a cost to holding cash (and therefore buying the cash good) which can almost be considered a "shadow price" for the cash good. When the "shadow price" falls, as happens when the nominal interest rate falls, $c^1_t$ goes up which, given equation $3.2$, puts downward pressure on the price level. Because of this effect, increases in $\mu_t$ in the short run result in lower interest rates. The effect is exacerbated if the money supply is assumed to be auto-regressive. (Point everyone else)

The real problem with Neo Fisherism, as John Taylor points out in the post that Cochrane links to, is that the money supply is not modeled. High interest rates mean that the future price level is high relative to the current price level, but does that mean that the current price level has fallen to produce this, or that the future price level has increased? Adding the money supply solves this entirely. Interest rates can be high because the future money supply has been raised relative to today or because the current money supply has been reduced; only now the central bank has complete control over it.

The addition of the cash and credit goods to the basic cash in advance framework helps to illustrate that some (pseudo) non-neutrality of money can cause low interest rates and high expected inflation to coincide, something that doesn't happen in New Keynesian models unless the Taylor Rule has extremely persistent shocks. Also key here is that the interest rates are indicative of expected inflation, not current inflation and any apparent relationship with current inflation is either coincidence -- because the money supply auto-regresses, e.g. -- or a result of temporary money non-neutrality.

Also, the idea that forcing interest rate to be low actually causes high inflation is completely wrong; it's all about the money supply, and high inflation only happens if the money supply is growing quickly. Deliberately setting a low nominal interest rate must eventually result in low money growth (unless you are in a liquidity trap. See here), so it's pointless to suggest such a policy in the hopes of deliberately causing higher inflation. The endgame is to stop thinking about monetary policy in terms of interest rates at all and switch to thinking about movements in the money supply.

## 07 November 2015

### Using Fiscal Policy to Escape a Liquidity Trap

Read the last post before you read this one; this post builds off of the analysis from that one.

In my last post, I explained my reasoning for monetary policy ineffectiveness at the zero lower bound on nominal interest rates. There, I explained that, at the zero lower bound, there is no equilibrium path of the price level (i.e., the model does not pin down a specific price level in all current and future periods). In this case, the central bank is powerless to escape the zero lower bound and must hope that the household randomly selects and equilibrium in which the cash advance constraint will bind in the future so that it can engage in expansionary monetary policy (in the future) in order to escape the zero lower bound.

In my analysis, I did not model fiscal policy because I was specifically writing about monetary policy ineffectiveness. Nevertheless, fiscal policy could be used to pin down an equilibrium price level when monetary policy can't. To begin with, let's take the budget constraint from the last post:

$$(1a)\: M_{t-1} + (1+i_{t-1})B_{t-1} + P_t y = P_t c_t + B_t + M_t + T_t$$

Since the household sets $c_t$ equal to $y$, it is possible to rewrite the household's budget constraint as the government's budget constraint:

$$(1b)\: M_{t-1} + (1+i_{t-1})B_{t-1} = B_t + M_t + T_t$$

We can also plug in the consumption Euler equation (equation $5$ in the last post) to express the budget constraint without the nominal interest rate

$$(1c)\: M_{t-1} + \left(\frac{1}{\beta} \frac{P_t}{P_{t-1}}\right)B_{t-1} = B_t + M_t + T_t$$

Assuming that the zero lower bound is binding, the budget constraint can be further reduced to

$$(1d)\: M_{t-1} + B_{t-1} = B_t + M_t + T_t$$

With this budget constraint, it is possible to determine an equilibrium price level from the specification of fiscal policy and monetary policy. Essentially, the government can monopolize on four sources of revenue: issuing government bonds, increasing the supply of money, levying taxes, and inflation. To understand why inflation can be used for revenue, it is useful to divide $1d$ by the price level and get all the variables in real quantities (lower case letters indicated real variables, except for $T_t$ which is changed to $\tau_t$):

$$(1e)\: m_{t-1}\left(\frac{P_{t-1}}{P_t}\right) + b_{t-1}\left(\frac{P_{t-1}}{P_t}\right) = b_t + m_t + \tau_t$$

In order to make $1e$ true, the government can obviously either increase $b_t$, $m_t$, or $\tau_t$. Alternatively, it can increase $P_t$, which will reduce the value of the entire left side of the budget constraint. Because of this, the fiscal authority can essentially choose to be irresponsible and the monetary authority will be forced to comply. Normally this means that the central bank must increase the growth rate of the money supply, but because of the zero lower bound, revenue from the central bank can either be more money or more inflation. Assuming the fiscal authority promises to not pay its debts (technically, this is called non-ricardian fiscal policy), only a money supply growth rule is needed to determine the price level. Given the money growth rule, all fiscal policy has to do is be just irresponsible enough to push inflation onto target.

From equations $4a$ and $4b$ in the last post, we know that $P_t/P_{t-1} = M_t/M_{t-1}$ if $P_t/P_{t-1} > \beta$ and that $P_t/P_{t-1} \leq M_t/M_{t-1}$ if $P_t/P_{t-1} = \beta$. This means that the growth rate of the money supply always represents an upper bound for the rate of inflation. Because of this, it makes sense for the central bank to grow the money supply at exactly the desired rate of inflation throughout the liquidity trap. This way, when the fiscal authority switches to non-recardian policy, the inflation rate has an upper bound and when the inflation rate goes up and the cash-in-advance constraint binds again monetary policy doesn't have to change.

## 03 November 2015

### Monetary Policy Effectiveness In Liquidity Traps

As I've argued here, conventional money demand models suggest that the price level becomes indeterminate at the zero lower bound and monetary expansion can not do anything to change inflation. In a recent conversation with Scott Sumner, Scott pointed to Paul Krugman's 1998 paper about this issue. Krugman suggests in his paper that only current monetary expansions are useless, but commitments to larger money supplies in the future (or, as Scott would probably like me to say, commitments that the current monetary expansion will be permanent) can both alleviate the liquidity trap and raise the current price level.

So, in line with Krugman's model, let's assume that there is a representative household that maximizes the utility function

$$(1)\: U = \sum^\infty_{t=0}\beta^t\left(u(c_t)\right)$$

where $\beta$ is the household's discount factor and $u(c_t)$ is the utility that the household gains from its consumption, $c_t$, in period $t$. The household is endowed without output $y$ every period and participates in an asset market where it trades one period government bonds and government money. The household's budget constraint is

$$(2)\: M_{t-1} + (1 + i_{t-1}) B_{t-1} + P_t y = P_t c_t + B_t + M_t + T_t$$

where $M_t$ is the money supply, $B_t$ is the household's holding of government bonds, $i_t$ is the nominal interest rate that government bonds pay, $P_t$ is the price level, and $T_t$ is the lump sum tax from the government. The household also faces a cash-in-advance constraint; it must finance its consumption with government cash. This constraint takes the form

$$(3)\: M_t \geq P_t c_t$$

Notice the fact that this is an inequality constraint. The household can hold as much money as it wants, but must at minimum have enough cash on hand to pay for its consumption. The household maximizes $1$ subject to $2$ and $3$ which yields the following first order conditions:

$$(4a)\: M_t = P_t y\: \mbox{if}\: i_t > 0$$
$$(4b)\: M_t \geq P_t y\: \mbox{if}\: i_t = 0$$
$$(5)\: 1 + i_t = \frac{1}{\beta}\frac{P_{t+1}}{P_t}$$

If, like Krugman did, we assume hat next period's price level is constant, we can draw a nice diagram with $4a$, $4b$, and $5$:
The solid blue line is the curve from $5$, the dotted blue line marks the zero lower bound, and the black lines represent the money supply. Normally, the central bank is in complete control of the price level and can move it around by moving the money supply around. But, because the cash-in-advance constraint does not bind at the zero lower bound, increases in the money supply at the zero lower bound will not be immediately spent by the household. This means that, given a constant future price level, the central bank can only push the price level up until it hits the zero lower bound. After that, no amount of current monetary expansion can increase the current price level.

Of course, all that was exactly in line with Krugman. Here's where it gets interesting, though. Krugman assumes in his paper that the central bank has control of the future price level the entire time and can easily increase the future money supply to end the liquidity trap. If we drop the assumption that the cash-in-advance constraint must bind in the next period, can the monetary expansion, regardless of permanence be effective? In order to escape the liquidity trap, the central bank needs to make the household expect that the price level next period will be higher than the price level this period (this would shift the solid blue curve in the graph to the right).

I'm having a lot of trouble wrapping my head around it, but I think that everything hinges on expectations. The cash-in-advance constraint will only bind in the next period if the price level two periods ahead is expected to be higher than the price level next period and so on, ad infinitum. This means that the central bank can only exit the liquidity trap if the household subjectively expects inflation to be greater than the rate of time preference (the inverse of the discount factor subtracted by one) in the future. This is independent of the path of the money supply; not only is there no equilibrium for the price level in the static analysis at the zero lower bound, there is no equilibrium for the entire path of the price level once the zero lower bound has been reached.

The alternative is to reduce the current money supply until the cash-in-advance constraint binds once again; basically to cause a bunch of deflation now instead of in the future. The problem with this is that prices are sticky and a massive monetary contraction would cause a recession.

## 01 November 2015

### It's not time to blow up the New Keynesian model

Scott Sumner wrote a blog post recently in which he questioned the validity of New Keynesian models. He listed five of it's predictions that he finds troublesome:

1. The NK model implies that higher taxes on wages can be expansionary.
2. The NK model implies that higher capital gains taxes can be expansionary.
3. The NK model implies that raising the aggregate wage level by government fiat can be expansionary.
4. The NK model implies that an increase in the fed funds target can be expansionary.
5. The NK model implies that an fiscal austerity can be expansionary, if done by slowing the growth in government spending
His first three claims are hardly criticisms applicable to New Keynesian theory in general, but they do accurately point out that New Keynesian models turn a bit wonky at the zero lower bound. It is the last two claims that are particularly nefarious.

The fourth claim is ignorant of the fact that Neo-Fisherian results are entirely dependent on the existence of multiple equilibrium which arise from ambiguous fiscal policy and the absence of monetary policy rules. Inflation should really be considered indeterminate during an interest rate peg (as it was until the Neo-Fisherians decided to implicitly assume active fiscal policy). Alternatively, a theory of money demand could be added to New Keynesian theory that could solve the problem than discretionary interest rate control policy creates.

Sumner refers to Nick Rowe's recent (and quite good) post on New Keynesian fiscal policy as evidence for his fifth attack on New Keynesian models. What he doesn't realize, though, is that what Nick Rowe describes in his post is not consistent with higher output. Actually, the government is causing potential output to change by moving government spending around. Lower government spending is consistent with lower potential output. To understand why, it is necessary to look at Real Business Cycle theory. In RBC models, permanent changes in government spending have real (supply side) effects that change output. These supply side effects are present in New Keynesian models, and in this case they move potential output around which, in turn moves the natural rate of interest around. Expected austerity does raise the natural rate of interest, but it does not actually raise output.

Contrasting his own views with those he associates with New Keynesian models, Sumner provides four characteristics of the "musical chairs model":
1. In the short run, employment fluctuations are driven by variations in the NGDP/Wage ratio.
2. Monetary policy drives NGDP, [sic] by influencing the supply and demand for base money.
3. Nominal wages are stick in the short run, and hence NGDP shocks cause variations in employment in the same direction.
4. In the long run, wages are flexible and adjust to changes in NGDP. Unemployment returns to the natural rate (currently about 5% in the US.)
As I noted in my comment, this set of four characteristics is not a model. It can be made into a model perhaps, but it falls short of actually being a model. If someone were to write down the "musical chairs model" as Sumner describes it, it would likely closely resemble a New Keynesian model where the primary friction is changed from sticky prices to rigid nominal wages and the central bank uses the monetary base, rather than the nominal interest rate, as the instrument for monetary policy.

Naturally, this leaves Sumner with the task of coming up with a money demand function that is both empirically accurate and gets around the problems that I wrote about here and here so that he doesn't have to drop his assertion that monetary expansion, even at the zero lower bound, is always expansionary, rather than useless as most plausible models of money demand (and the empirical evidence) seem to suggest. I suggest Sumner add his money demand function to this model by Stephanie Schmitt-Grohe and Martín Uribe and see if it performs as well as New Keynesian models when put to the data.