## 22 December 2015

### What's the Significance of Low Real Interest Rates?

[Note: I started writing this post a while ago, so it ostensibly has no connection with these two posts that Nick Rowe and Scott Sumner wrote recently. I just realized that this is somewhat relevant, so I decided to finish it]

For the last twenty years or so, real interest rates on government bonds have continued to fall from their high of about 9%. Determining the cause of such a fall is by no means an easy task; after all economic theory generally suggests that real interest rates on safe assets -- like government bonds -- should be relatively constant in the long run and reflect the rate at which consumers discount future spending relative to current spending. Economic theory tells us that low real interest rates mean that current consumption is high and future consumption is low relative to what it otherwise would have been. This certainly is a possibility; perhaps falling real interest rates are indicative of a shift in consumer spending patterns away from saving and into borrowing, although the causality seems to be backwards if that is truly the case, which leaves the question of what has caused this decline in real interest rates open once again.

Perhaps the basic models in which the government has no power over the real interest rate in the long run are incorrect; given the sharp increase in the real interest rate on government bonds during the 1980s, this certainly seems plausible. In this case, it may be useful to switch to looking at this problem through the lens of an OLG model instead of a basic representative agent RBC/Neo Classical one. Every period, a new young agent is born with the endowment $y$ which can be used to buy either consumption ($c^y_t$) or government bonds ($b_t$), or to invest in capital ($k_t$). The young agent faces the budget constraint
$$(1)\: y = c^y_t + b_t + k_t$$
In the next period, the young become old and use income from interest on government bonds, $R_t b_t$, and from income generated from capital, $f(k_t)$ to finance their consumption and the taxes levied by the government. Old agents face the budget constraint
$$(2)\: c^o_{t+1} = R_t b_t + f(k_t) - \tau_t$$
Agents are born wanting to maximize their consumption in both periods of their life, with consumption when old discounted at rate $\rho$. The agents' discount factor is $\beta = \frac{1}{1 + \rho}$. Utility it derived from the log of current young consumption and the log of future old consumption:
$$U = \log c^y_t + \beta \log c^o_{t+1}$$
Agents maximize their utility function subject to both of their budget constraints. Young agents choose their consumption so that
$$(3)\: \frac{1}{c^y_t} = \beta \frac{1}{c^o_{t+1}} R_t$$
That is, young agents take as given the interest rate the they can receive by saving now and consuming later or that they would pay if they consumed now and saved later and decide to save more if the interest rate is high -- since their lifetime income can be increased by their saving -- and save less if the interest rate is low. The government sets the number of bonds that it issues by discretion each period which, given the young agent's consumption decision, determines the level of capital investment.

Another first order condition of the model is the the real interest rate on government bonds is equal to the marginal productivity of capital. That is,
$$(4)\: R_t = f'(k_t)$$
Since the level of government bonds determines capital investment, it also determines the real interest rate on government bonds. More government debt means less capital which, per $4$, means a higher real interest rate (assuming that $f(k)=k^\alpha$ where $\alpha < 1$). This works because agents must be indifferent between holding more government bonds or more capital in equilibrium; otherwise they would end up demanding more or less capital than they wanted.

In this model, low real interest rates are a result of high capital expenditure and low government debt. The prescription for low interest rates, then, is to engage in a large fiscal expansion that would increase the amount of government bonds in the economy. Less capital demand would have to be justified by a higher real interest rate. Of course, this seems empirically slightly dubious. After all, the amount of government debt skyrocketed in 2008 and interest rates failed to rise. To understand why this wouldn't necessarily be consistent with higher real interest rates, it's important to think along the lines of a demand for government bonds.

Agents in this model are willing to demand more government bonds at higher interest rates, so if the government sets the supply of government bonds higher, then the demand must correspondingly rise through an increase in the real interest rate. The reason that massive increases in government debt in 2008 and 2009 are not consistent with higher real interest rates is that demand for government debt increased; perhaps even by more than the increase in supply. This was likely caused by the sudden illiquidity associated with other assets that were previously considered safe - e.g. mortgage backed securities or Greek government bonds. The resulting surge in demand for government bonds is known almost colloquially as a 'flight to quality.'

The ideal fiscal response to this is to satiate demand for government debt by running large deficits (note that this is the exact opposite of the policy actions taken by the majority of governments since 2008). In a way, this is a non-Keynesian reason for pursuing fiscal stimulus; more government debt would be useful for raising the real interest rate. Not only would this make the economy closer to a competitive equilibrium (one without government intervention), it would likely make monetary policy more effective. Narayana Kocherlakota, president of the Minneapolis Fed, made this point in a speech in July. The basic argument he presents is that the government can raise the long-run neutral real interest rate by increasing the amount of government debt. The higher neutral rate of interest (i.e. the real interest rate in this model, since there is no money) will make it so that the Fed will be less likely to hit the zero lower bound when trying to ensure that target is hit.

Effectively, fiscal policy should be used to remedy situations in which the demand for money is indeterminate and the central bank cannot adequately influence the real interest rate (see, e.g., here).

## 16 December 2015

### A Novel New Keynesian View of Fiscal Policy

Usually when I read New Keynesian economists on fiscal policy, they tend to focus more on fiscal multipliers or the effectiveness of tax cuts at the zero lower bound. But what about fiscal policy in general? I have come across some literature on this, but it usually limits itself to comparing the relative roles of monetary and fiscal policy - e.g. what is the optimal coefficient for the output gap in the fiscal policy rule? Here, I'd like to present a somewhat novel approach to New Keynesian fiscal policy (at least I've never seen or read this anywhere else).

Consider first the basic Consumption Euler equation that determines how household's allocate consumption between the present and the future given an interest rate.

$$(1)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} \left(\frac{1 + i_t}{1 + E_t\pi_{t+1}}\right)$$

In simple New Keynesian models, real GDP is composed of just government spending and consumption since there is no capital accumulation, so $1$ can be rewritten as a function of output, $y_t$, and government spending, $g_t$.

$$(2)\: (y_t - g_t)^{-\sigma} = \beta E_t (y_{t+1} - g_{t+1})^{-\sigma} \left(\frac{1 + i_t}{1 + E_t\pi_{t+1}}\right)$$

It is useful to linearize $2$ to make it a bit easier to work with, but first, to make the math a little easier, it is helpful to notice that $y_t - g_t$ is the same as $y_t(1 - \frac{g_t}{y_t})$. Given this, defining $\theta$ as $\frac{1}{\sigma}$, and defining $\beta$ as the inverse of the gross time preference rate, $\rho$, it is possible to write $2$ in log-linear form - i.e. all equations are written as percentage gaps from their long run level.

$$(3)\: \hat{y}_t = E_t\hat{y}_{t+1} - E_t \Delta \hat{g}_{t+1} - \theta (i_t - E_t\pi_{t+1} - \rho)$$

Keep note that, in this case, $\hat{g}_t$ is the gap of the government spending to GDP ratio from trend rather than simply government spending from trend. For my purposes, this is basically irrelevant.

If the goal of fiscal policy is to ensure that the output gap is zero at all times - not too weak of an assumption in my opinion - then it's pretty simple to solve for optimal policy given $3$:

$$(4)\: E_t \Delta \hat{g}_{t+1} = -\theta(i_t - E_t\pi_{t+1} - \rho)$$

In English, equation $4$ tells us that the role of fiscal policy is simply to offset any failure of the monetary authority to set the right real interest rate ($i_t - E_t \pi_{t+1}$). If, for example, the monetary authority has set a real interest rate that is too high, then government spending should be expected to shrink relative to trend in the next period. This can be accomplished either through stimulus - raising current government spending now and reducing it in the future - or through causing expected temporary austerity - decreasing next period's government spending then allowing government spending to return to trend.

The first option is preferable for a couple of reasons. For one thing, government spending also has real effects (see my previous blog post), so austerity might have unintended supply side consequences. Also, the austerity must be reversed at some point for the policy to work - since $\hat{g}_{t+1}$ would fall to zero if the austerity were permanent - so future fiscal policy may be impaired if the central bank continues to set an interest rate that's too high.

Another way of approaching this is to rewrite $3$ to incorporate a 'natural real rate of interest.' In this case, $3$ can be rewritten as

$$(5)\: \hat{y}_t = E_t\hat{y}_{t+1} - \theta (i_t - E_t\pi_{t+1} - \rho + \frac{E_t\Delta\hat{g}_{t+1}}{\theta})$$

Defining the natural real rate of interest at which the output gap is zero, it is clear that $\rho - \frac{E_t\Delta\hat{g}_{t+1}}{\theta}$ is equal to the natural rate, $r^n_t$.

Assuming the central bank tries to set the real interest rate equal to the natural rate unless the zero lower bound is binding, i.e. $i_t = \max\left(0,\: E_t \pi_{t+1} + r^n_t\right)$, the job of the government can be seen as preventing the zero lower bound from ever binding, or, in other words, setting $E_t\pi_{t+1} + r^n_t > 0\: \forall t$.

$$(6)\: 0 < E_t\pi_{t+1} + \rho - \frac{E_t\Delta\hat{g}_{t+1}}{\theta}$$

or

$$(7)\: E_t\Delta\hat{g}_{t+1} < \theta(E_t\pi_{t+1} + \rho)$$

From $7$, it is clear that expected growth in government spending relative to trend should always be less than a function of the expected inflation rate. That is, the lower the expected inflation rate, the bigger the stimulus that should be undertaken. Effectively the goal of fiscal policy is to offset failures in monetary policy and to make sure that the zero lower bound never binds in the first place.

## 15 December 2015

### Shut Up About Ricardian Equivalence

Economists that are both opposed to and in favor of fiscal stimulus frequently cite Ricardian equivalence as a reason that, in models with perfect credit markets, it doesn't matter whether stimulus is funded through increased taxes or through deficits. The problem with this analysis is that it assumes lump sum taxation. That is, taxes are not collected from things like consumption expenditures, which are effectively no different than deficits because 1) they don't discourage people from working, consuming, investing, etc and 2) they are expected to rise at some point in the future to retire the current debt, so the present value of taxes goes up with government spending. In reality, the argument is about distortionary taxes vs. deficits (= lump sum taxes, as per Ricardian Equivalence).

Distortionary taxes are different than their lump sum counterparts since they directly act to disincentivize working (in the specific case of income taxes, which I will limit my analysis to from now on) and can thus either partially or fully negate the effects of a fiscal stimulus. So, when John Cochrane says something like "'Ricardian Equivalence,' which is the theorem that stimulus does not work in a well-functioning economy," [1] he's clearly confusing the two types of taxation as well as ignoring the fact that neoclassical economics predicts a positive multiplier on government spending [2]. To illustrate this, I wrote down a standard Real Business Cycle model and ran two simulations: one in which a temporary fiscal expansion was financed entirely with an income tax and another in which the same fiscal stimulus was financed partially by deficits (see the appendix for a derivation of the model).
 Figure 1: Impulse Response Function of Output to the Stimulus
 Figure 2: Impulse Response Function of the Income Tax Rate to the Stimulus
 Figure 3: Government Spending in both simulations; Government Debt in the second simulation

The Ricardian Equivalence argument would be irrelevant if 1) the stimulus had a positive effect on output and 2) the tax funded stimulus was initially less effective than the partially deficit funded one. As you can see in figure 1, both of these are true; the stimulus positively impacted output in each simulation and the stimulus was initially more effective when taxes were not increased to fully finance the stimulus on impact. The effectiveness of the stimulus is slightly less sound of a result, though. The fiscal multiplier in neoclassical models is highly dependent on calibration (see, e.g., [2]) and can range anywhere from zero to one, without distortionary taxation, depending on the specific calibration used. Regardless, the most important part of this argument is sound; the Ricardian Equivalence argument against deficit funded stimulus is wrong and should be ignored completely as it applies to a form of taxation that doesn't actually exist.

References:

[1] John Cochrane, 2011. "Krugman on Stimulus" The Grumpy Economist.

[2] Woodford, Michael. 2011. "Simple Analytics of the Government Expenditure Multiplier." American Economic Journal: Macroeconomics, 3(1): 1-35.

Appendix:

The following is a derivation of model that I used to generate the impulse response functions in figures 1, 2, and 3.

Household:

There is a representative household who maximizes the utility function $U = E_0 \sum^\infty_{t=0} \beta^t\left(\frac{c_t^{1-\sigma}}{1-\sigma} - \frac{n_t^{1+\phi}}{1 + \phi}\right)$ where $E_t$ is the rational expectations operator given information available in period $t$, $c_t$ is the household's consumption, $n_t$ is the labor supply, and $\beta$ is the household's discount factor - the rate at which future utility is discounted relative to current utility. The household can use net-of-taxes income from labor ($(1-\tau^w_t)w_t n_t$, where $w_t$ is the real wage), government bonds carried from last period ($R_{t-1} B_{t-1}$, where $R_t$ is the interest rate that bonds maturing in period $t$ - $B_t$ - pay), and net-of-depreciation income capital, $(1 + r_{t-1} - \delta)k_{t-1}$ to purchase consumption, new government bonds, or new capital. The budget constraint can be written as

$$(1.1)\: (1-\tau^w_t)w_t n_t + R_{t-1} B_{t-1} + (1 + r_{t-1} - \delta)k_{t-1} = c_t + B_t + k_t$$

The household maximizes $U$ subject to $1.1$  in order to determine its behavior:

$$(1.2)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} (1 + r_t - \delta)$$
$$(1.3)\: R_t = 1 + r_t - \delta$$
$$(1.4)\: (1-\tau^w_t)w_t = c_t^\sigma n_t^\phi$$

Additionally, it is useful to define investment, $i_t$ as the instrument of capital accumulation:

$$(1.5)\: k_t = (1-\delta)k_{t-1} + i_t$$

Firm:

The firm bundles capital carried from last period and labor using a Cobb-Douglas production function to form output, $y_t$

$$(2.1)\: y_t = k_{t-1}^\alpha n_t^{1-\alpha}$$

The firm maximizes profits, $y_t - w_t n_t - r_{t-1}k_{t-1}$ subject to $2.1$ in order to determine labor and capital demand

$$(2.2)\: w_t = (1 - \alpha)\frac{y_t}{n_t}$$
$$(2.3)\: r_t = \alpha E_t\frac{y_{t+1}}{k_t}$$

Government:

The government issues new government bonds and collects tax revenue to pay for both government spending and interest on government bonds carried from last period. The government budget constraint can be written as

$$(3.1)\: B_t + \tau^w_t w_t n_t = g_t + R_{t-1} B_{t-1}$$

In the first simulation, it is assumed that the government ensures $B_t = 0\: \forall t$, so government spending is simply financed by taxes

$$(3.2)\: \tau^w_t w_t n_t = g_t$$

In the second simulation, the government sets the tax rate as a function of the tax rate consistent with the long run level of government spending, $\tau^w_{SS}$ and the level of government debt issued in the previous period, $B_t$. The rule for the tax rate in the second simulation is

$$(3.3)\: \tau^w_t = \tau^w_{SS} + \phi_b B_{t-1}$$

In both simulations, government spending follows an autoregressive process and returns to its long run trend trend at decay factor $\rho$. Government spending follows

$$(3.4)\: g_t = (1 - \rho)g_{SS} + \rho g_{t-1} + \eta_t$$

Where $\eta_t$ also follows an autoregressive process with the same decay factor an is hit with with the shock $\epsilon^g_t$

$$(3.5)\: \eta_t = \rho \eta_{t-1} + \epsilon^g_t$$

Equilibrium:

Combining $1.1$, $1.5$, and $3.1$ yields the resource constraint for the economy

$$(1)\: y_t = c_t + i_t + g_t$$

Equations $1.2$-$3.5$ can be used to determine the equilibrium for the rest of the endogenous variables:

$$(2)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} (1 + r_t - \delta)$$
$$(3)\: R_t = 1 + r_t - \delta$$
$$(4)\: (1-\tau^w_t)w_t = c_t^\sigma n_t^\phi$$
$$(5)\: k_t = (1-\delta)k_{t-1} + i_t$$
$$(6)\: y_t = k_{t-1}^\alpha n_t^{1-\alpha}$$
$$(7)\: w_t = (1 - \alpha)\frac{y_t}{n_t}$$
$$(8)\: r_t = \alpha E_t\frac{y_{t+1}}{k_t}$$
$$(9)\: B_t + \tau^w_t w_t n_t = g_t + R_{t-1} B_{t-1}$$
$$(10a)\: \tau^w_t w_t n_t = g_t\: \mbox{in simulation 1}$$
$$(10b)\: \tau^w_t = \tau^w_{SS} + \phi_b B_{t-1}\: \mbox{in simulation 2}$$
$$(11)\: g_t = (1 - \rho)g_{SS} + \rho g_{t-1} + \eta_t$$
$$(12)\: \eta_t = \rho \eta_{t-1} + \epsilon^g_t$$

## 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.

## 31 October 2015

### Economic Philosophy on the Left and Right

The usual explanation for different approaches to economic policy from the political left and right leaves much to be desired. To suggest that economic policy debates are really, at their deepest, just disagreements about ways to achieve the same goal of maximum prosperity is ignorant of the severe differences between proposals about fiscal policy on the left and right. These debates do not stem from disagreements about how to set up the welfare system or how big the welfare system should be; they arise out of a disconnect on what kinds of behavior and outcomes ought to be prioritized by the government.

The discourse on the right is dominated by an idolization of employment and for Pareto efficiency. Proposals seeking to substantially reduce taxes on labor and investment reflect these priorities. Lower and flatter income taxes are consistent with incentivizing people to work longer hours and with removing some of the alleged Pareto suboptimality that arises when higher incomes are deliberately taxed more than lower incomes. Lower taxes on investment serve this purpose as well; they reduce the problematic progressive taxation that, rather than helping to bring the poor up and the rich down, reduce everyone’s income. Meanwhile, anti-welfare policies are supported because of their effectiveness at bringing conservatives closer to their two-fold economic purpose. Programs like unemployment insurance and food stamps make it easier for people to not work and increase the income of the poor at the expense of everyone and must therefore be annihilated.

In contrast, left-wing discussion is primarily concerned with minimizing involuntary employment and raising the average level of welfare in the population. These two ideals inspire proposals that would provide healthcare for all and make sure that anyone in the country has the ability to survive regardless of circumstances. A completely private health system may very well be maximally efficient, but it is heavily tilted against those on the lower income scale. Nationalization makes sense; some efficiency will be lost because of the monopoly, but the health of the average constituent can allegedly be raised and so must be raised. More broad welfare programs are also consistent with these goals. Unemployment insurance, pensions, and food stamps serve as subsidies for those who find themselves unable or unwilling to work at any point in time and serve the purpose of raising the average income, especially the income for those at the low end of the income and wealth distribution.

The clash between left and right should not be seen simply as a clash over the appropriate size of government or the optimal method of achieving the same goals. It is more fundamental, more substantive than that. Conservatives emphasize the inefficiency of government provision and the distortion of progressive taxation while liberals emphasize the broad welfare gains that can be had in spite of the distortion and inefficiency that government necessitates. The most probable reality is one in which both sides are correct and the costs and benefits of each government program must be carefully weighed before implementation.

## 30 October 2015

### The Fed Should Cut IOR

It's difficult to understand why people and financial institutions would willingly carry an asset that earns less interest than other types of asset. A typical way to get around this problem is by assuming that some or all of the goods in an economy must be paid for in cash. Because agents must pay for goods in cash, they choose to hold only as much as they need to pay for those goods. If they were to hold any extra cash, then they would be missing out on valuable interest that they would earn from other assets. The cost that agents face by holding money as opposed to other assets (e.g. government bonds) is called the opportunity cost of holding money.
But what happens when the opportunity cost of holding money is zero (i.e., the interest rate on government bonds is equal to the interest rate on money)? In this situation, it makes no difference to agents what kind of asset they hold, so the distribution of government bonds and money is indeterminate. This indeterminacy breaks the link between inflation and the money supply. Typically, the money supply is linked directly with the nominal value of spending on cash-goods in this economy, so a higher money supply would necessitate higher nominal spending and, assuming flexible prices and wages, higher prices. Now, because there is no opportunity cost of holding money, agents will freely hold any money that the central bank gives them without needing to spend it.

This is the situation that the Federal Reserve is currently in. There is no incentive for financial institutions to do anything will all the money that the Fed has injected into the system since 2009 because the interest rate that money pays (interest on reserves or IOR) is equal (after adjusting for risk/liquidity) to the interest rate that other assets pay. Financial institutions are happy to sit on interest bearing and highly liquid (easy to buy and sell) cash. This is why the vast majority of the increase in the monetary base since 2009 has taken the form of "excess reserves".

In order to reverse this, it is necessary to make reserves less attractive to financial institutions (or, in the case of cash-in advance models, make cash less attractive to agents). This means that there must be an opportunity cost of holding money; that money needs to pay less interest than other assets. Of course, this can be accomplished one of two ways: the Fed can either reduce the interest it pays on reserves increase the interest rate that other assets pay. Since economists widely agree that raising interest rates would have a negative effect on the rate of inflation, it seems clear that the way to increase inflation in the US is to cut IOR.

## 05 October 2015

### Effect of Austerity on US GDP

I decided I would make a little chart comparing the US output gap [1] to a counterfactual of the output gap without austerity [2]. The graph shows the actual output gap and counterfactual output gaps for government expenditure multipliers of 0.25, 0.5, 0.75, and 1.

[1] The output gap measure I used assumes that potential GDP grows as roughly 2% a year and that the output gap was zero in Q1 2005.

[2] By "without austerity" I mean "if the government spending to potential GDP ratio stayed at 20% from Q1 2005 to Q2 2015."

(The GDP measure I used is the 'GDPC96' series from fred and the government spending measure I used is the 'GCEC96' series)

## 03 October 2015

### Inflation ≠ Expected Inflation

Neo-Fisherian arguments seem to rest on the idea that expected inflation is somehow related to current inflation - almost to a point of equivalence. A typical argument would be: look at the fisher relation $i_t = r + E_t \pi_{t+1}$. Notice that the nominal interest rate, $i_t$, and the expected inflation rate, $E_t \pi_{t+1}$, are related. Increasing the nominal interest rate must therefore cause the rate of inflation to increase. Before you accuse me of debating a straw man, read this from John Cochrane's recent post:

If you parachute down from Mars and all you remember from economics is the Fisher equation, this looks utterly sensible. Expected inflation = nominal interest rate - real interest rate. So, if you peg the nominal interest rate, inflation shocks will slowly melt away. Most inflation shocks are individual prices that go up or down, and then it takes some time for the overall price level to work itself out.

The problem with this argument is that the current rate of inflation is never modeled; the central bank can choose expected inflation, but there is no reason that the actual rate of inflation must change in response to higher expected inflation. There are a couple of ways around this problem. In the interest of keeping the model as simple as possible, you could assume that the central bank sets the nominal interest rate in response to the current inflation (i.e. a Taylor Rule) or, in the interest of coming of with a more structural model, you could try and come up with a variable that actually does cause current inflation (e.g. the money supply).

The Taylor Rule approach is the way that most economists have gone in the last twenty years or so. Positive deviations of the nominal interest rate from the level implied by the Taylor Rule result in lower rates of inflation. This is itself enough to prove that, as long as a central bank follows a Taylor Rule, a Neo-Fisherian analysis is wrong. There are still some valid contentions that a Neo-Fisherian might make though: a.) central banks set interest rates by discretion, not by adherence to a Taylor Rule b.) Taylor Rules don't actually produce a unique equilibrium value for the initial rate of inflation or the initial price level. In order to deal with contention a, it is clear that a more structural model of inflation is necessary since interest rates clearly do not cause inflation. Contention b is a bit more complicated. In order to make sense of it, it is helpful to look at the coefficient on inflation in the Taylor Rule. If that coefficient is less than one, then any initial rate of inflation will converge to the central bank's inflation target; there are multiple equilibria. Alternatively, the coefficient can be greater than one which will cause the rate of inflation in the future to explode unless the initial rate of inflation is equal to the target rate. The only reason that this calibration works is because economists have chosen to rule out explosive solutions which may make sense for real variables, but does not make sense for nominal variables like inflation.

Contentions a and b leave two options for revision to the conventional approach: come up with a more structural model of inflation or come up with a model that determines a unique equilibrium for "passive" Taylor Rules (i.e. Taylor Rules where the coefficient on inflation is less than one). For some reason, the price determination literature failed to go down the first route and instead chose to come up with 'the fiscal theory of the price level'. Basically, the fiscal authority can threaten to disobey its budget constraint unless the initial inflation rate does not jump to the correct level. I don't really understand how this is all that much better than the trick with active Taylor Rules though. After all, both involve threats to either cause a hyperinflation or not pay off debt at some point that force the initial inflation rate to be on target.

Because of this, it seems obvious to me that the structural path should be taken. There should be a way to determine a unique equilibrium rate of inflation without requiring that fiscal or monetary policy be intentionally unstable. Of course, a cursory analysis using something like the money supply is easy. Current inflation is caused by current money growth and expected inflation is caused by expected money growth, so interest rates and inflation will go up in response to an increase in the growth rate of the money supply that is expected to be persistent. The debate should end there.

P.S. I don't necessarily mean to say that applies to the current situation; the zero lower bound is special both in theory and in practice.

P.P.S. For the more visually oriented, here's a graph that illustrates the explosive behavior of inflation under a Taylor Rule:

## 01 October 2015

### 'Liberal' and 'Conservative' Theories

Recently I found myself unfortunately coerced into reading "Jeb Bush Keeps Repeating A Phrase That's Central To A Liberal Economic Theory" (link). The article brought to my attention that the public sees theory in a much different light than I - and hopefully most members of academia - do.

The author asserts the Keynesian economics is a "liberal theory" as if scientific theories (social or otherwise) simply exist to advocate different political positions. This view represents a complete failure to understand the way that science is done. Rather than assuming a conclusion and attempting to prove that conclusion by any means possible, a good scientist will not assume that any position is correct and instead see what the data suggests and come of with a theory that attempts to match that data. Keynesian theory is no more 'liberal' than monetarism is 'conservative'. Yes, both economic ideologies are characterized by differing assumptions, but those assumptions do not come from inherent political bias.

The view of theory presented in the article can be especially dangerous because it allows for politicians to write off any bit of theory on the grounds that it is a 'liberal' theory or a 'conservative' theory. This is how the Cameron government in the UK can ignore the fact that austerity reduces GDP in even the most friction-less models (e.g. here) by arguing that any theory that suggests austerity is contractionary has a liberal bias.

Behind this whole issue seems to be the idea that every side in any political argument is at least somewhat right regardless of whether their opinions are backed up by academia. Essentially, politicians can redefine the political center whenever they choose and the media won't try and stop them. No matter how insane a policy proposal is, it can not be one hundred percent wrong in the eyes of the public and the media.

For this reason, the media ought to shift its primary concern from representing each position fairly to determining the facts or the consensus theories and criticizing those who fail to understand or accept them. Policy proposals that are in direct contempt of the scientific consensus should not be tolerated.

## 28 September 2015

### Yet Another Way That QE is Deflationary

Suppose the cash-credit model (here and here) of money demand is roughly correct, so when short term interest rates on safe assets (e.g. government bonds) are equal to the interest rate central banks pay on reserves, money demand is indeterminate. In this, case, increasing the money supply does nothing to the price level; monetary expansion just increases the real money supply.

Part of government revenue is seigniorage which can take the two forms: a.) inflation b.) real money growth. Since QE causes real money growth and not inflation, a lot of the seigniorage that would otherwise come in the form of extra inflation is already taken care of by extra real money.

Therefore QE is deflationary in a fiscal-theoretic way (this isn't even really FTPL, as actual fiscal policy doesn't matter). Q.E.D.

## 26 September 2015

### A Detailed Derivation of My Favorite Monetary Model

WARNING: This post contains an excessive amount of math. If you find math unbearable and/or difficult to understand, do not attempt to read this.

A little bit ago, I decided to combine a New Keynesian model with Rotemberg style pricing and a Cash-Credit goods model. Here is a derivation of that model:

Households

Households maximize

$$U = E_0 \sum^\infty_{t=0} \beta^t \left(\theta \log c^1_t + (1 - \theta) \log c^2_t - \gamma \log n_t \right)$$

subject to

$$M_{t-1} + R_{t-1} B_{t-1} + W_t n_t = P_t C_t + B_t + M_t + P_t \tau_t$$
$$M_t \geq P_t c^1_t$$
$$C_t = c^1_t + c^2_t$$

Where $c^1_t$ is the part of the consumption good that the household buys in the cash market, $c^2_t$ is the part of the consumption good that the household buys in the credit market, $C_t$ is total spending on the consumption good, $W_t$ is the nominal wage rate, $n_t$ is hours worked by the household, $M_t$ is the nominal money supply, $B_t$ is the supply of government bonds, $P_t$ is the price of the consumption good, and $\tau_t$ is lumps sum taxes/transfers from the government.

The households maximization problem can be written as

$$\mathcal{L} = U + \lambda^0_t \left(M_{t-1} + R_{t-1} B_{t-1} + W_t n_t - P_t C_t - B_t - M_t - P_t \tau_t \right) + \lambda^1_t \left(M_t - P_t c^1_t \right) + \lambda^2_t\left(C_t - c^1_t - c^2_t \right)$$

Solving the Lagrangian gives the following First Order Conditions:

$$(1) \: \frac{1}{c^2_t} = \beta E_t \frac{1}{c^2_{t+1}} R_t E_t \frac{P_t}{P_{t+1}}$$
$$(2) \: \frac{W_t}{P_t} = \frac{\gamma}{1 - \theta} \frac{c^2_t}{n_t}$$
$$(3) \: \frac{\theta}{c^1_t} = \frac{1-\theta}{c^2_t} \left(2 - \frac{1}{R_t}\right)$$
$$(4) \: M_t = P_t c^1_t$$

Retail Firms

Retail firms maximize profits, $P_t Y_t - \int^1_0 P_t(i) y_t(i) di$ subject to the production technology $Y_t = \left[\int^1_0 y_t(i)^\frac{\epsilon-1}{\epsilon}di\right]^\frac{\epsilon}{\epsilon-1}$.

Substituting the production technology into the profit function yields

$$P_t \left[\int^1_0 y_t(i)^\frac{\epsilon-1}{\epsilon}di\right]^\frac{\epsilon}{\epsilon-1} - \int^1_0 P_t(i) y_t(i) di$$

Taking the derivative of this with respect to $y_t(i)$ gives the retail firm's first order condition:

$$y_t(i) = Y_t \left(\frac{P_t(i)}{P_t}\right)^{-\epsilon}$$

Since the retail firm is perfectly competitive, its profits are equal to zero. We can therefore set profit equal to zero and plug in the first order condition to get the definition of the price level

$$P_t^{1-\epsilon} = \int^1_0 P_t(i)^{1-\epsilon}$$

Wholesale Firms

There is a continuum of monopolistically competitive wholesale firm who are subject to the quadratic price adjustment cost

$$\frac{\varphi}{2}\left(\frac{P_t(i)}{P_{t-1}(i)} - 1 \right)^2 Y_t$$

First, each wholesale firm minimizes total costs, $\frac{W_t}{P_t} n_t(i)$ subject to the production function $y_t(i) = a_t n_t(i)$. This problem can be set up as

$$\mathcal{L} = -\frac{W_t}{P_t} n_t + mc_t \left( a_t n_t(i) - y_t(i)\right)$$

which yields

$$(5) \: \frac{W_t}{P_t} = mc_t n_t(i)$$

The Lagrangian multiplier in this problem is the marginal cost of production (hence the name $mc_t$).

Each retail firm now maximizes the expected sum of all future profits which is discounted by the 'stochastic discount factor' with the real interest rate replacing the time preference rate and is subject to the retail firm's demand function, $y_t(i) = Y_t \left(\frac{P_t(i)}{P_t(i)}\right)^{-\epsilon}$. Since the maximization problem for this is so obscenely long, I won't write it down, I'll just skip to the first order condition.

$$0 = (1-\epsilon)\frac{Y_t}{P_t} + \epsilon mc_t \frac{Y_t}{P_t(i)} - \varphi \left(\frac{P_t(i)}{P_{t-1}(i)} - 1 \right)\frac{Y_t}{P_{t-1}(i)} + \beta E_t \left( \frac {c^2_{t+1}}{c^2_t} \right)^{-1}\varphi \left(\frac{P_{t+1}(i)}{P_t(i)} - 1 \right)\frac{P_{t+1}(i) Y_t}{P_t(i)^2}$$

Consider the fact that, since each firm has the same level of technology, the same demand curve, and the price adjustment costs, every firm chooses the same  price. Given this as well as the fact that the rate of inflation, $\pi_t$ is equal to $\frac{P_t}{P_{t-1}}$, the 'New Keynesian Phillips Curve' above can be written as

$$(6) \: 0 = (1-\epsilon) + \epsilon mc_t - \varphi \pi_t (1 + \pi_t) + \beta E_t \left( \frac {c^2_{t+1}}{c^2_t} \right)^{-1}\varphi\pi_{t+1}(1+\pi_{t+1})\frac{Y_{t+1}}{Y_t}$$

Equilibrium

Equations 1-6 can be combined with a description of government policy to complete this model. The money supply and the wage have been rewritten in real terms.

$$(1) \: \frac{1}{c^2_t} = \beta E_t \frac{1}{c^2_{t+1}} \frac{R_t}{1 + \pi_{t+1}}$$
$$(2) \: w_t = \frac{\gamma}{1 - \theta} \frac{c^2_t}{n_t}$$
$$(3) \: \frac{\theta}{c^1_t} = \frac{1-\theta}{c^2_t} \left(2 - \frac{1}{R_t}\right)$$
$$(4) \: m_t = c^1_t$$
$$(5) \: w_t = mc_t n_t$$
$$(6) \: 0 = (1-\epsilon) + \epsilon mc_t - \varphi \pi_t (1 + \pi_t) + \beta E_t \left( \frac {c^2_{t+1}}{c^2_t} \right)^{-1}\varphi\pi_{t+1}(1+\pi_{t+1})\frac{Y_{t+1}}{Y_t}$$
$$(7) \: \log R_t = \frac{\beta - 1}{\beta} + \phi_\pi \pi_t + \upsilon_t$$
$$(8) \: \log a_t = \rho \log a_{t-1} + \varepsilon^a_t$$
$$(9) \: Y_t = C_t + \varphi \pi_t^2 Y_t$$
$$(10) \: C_t = c^1_t + c^2_t$$
$$(11) \: \upsilon_t = \rho \upsilon_{t-1} + \varepsilon^i_t;$$

Impulse Response Functions

Here is the impulse response function (in log deviations from steady state) for the technology shock, $\varepsilon^a_t$ where $V$ is the velocity of money:
And here is the impulse response function for the monetart policy shock, $\varepsilon^i_t$:

## 21 September 2015

### The Trouble With The Zero Lower Bound

Most of the time, it seems that the monetarist view of inflation is pretty much correct. Inflation roughly tracks the monetary base and velocity is pretty stable and almost directly follows short term interest rates. Unfortunately, there is this thing called the zero lower bound that seems to throw monetarism off.

The US has been at the zero lower bound twice in the last 150 years, and both times monetary expansion has seemed to have an irrelevant - even a negative - impact on inflation.

Here's 1934-1945:

And here's 2009-2015:
Most monetarists seem to have trouble coping with the irrelevance of the monetary base at the zero lower bound, even though it does seem to be part of a lot of basic monetary models. Take the most simple of money demand functions - cash-in-advance. It is easy to figure out that as long as there is a cost to the household incurred by holding money, the cash-in-advance constraint will bind, but whenever there isn't a cost, the constraint ceases to bind. This effectively means that, rather than being stuck at unity, the velocity of money is indeterminate; increases in the money supply will no longer have any effect on the price level.

Money-in-the-utility-function models have similar properties in the sense that velocity also becomes indeterminate. MIUF models are slightly strange though because money demand itself actually goes to infinity when the zero lower bound binds. But, MIUF is a pretty bad assumption anyway, so it's fine to ignore this.

An easy modification to CIA models that, when calibrated properly, might be able to make them match the data pretty well is the addition of a non-cash good to the economy. The income-velocity of money will now fluctuate with the nominal interest rate while the effects above will still be present.

I digress, the key idea of this kind of rambling post is that the zero lower bound seems to do strange things to monetary policy which precludes central banks from being omnipotent as some would suggest...

## 11 September 2015

### Two Papers Every Republican Candidate (And Everyone) Should Read

Here's a list of a few papers that all the Republican (this applies less to Democrats, at least at the moment) candidates for US president in 2016 should make themselves familiar with:

1. "How Far Are We From The Slippery Slope? The Laffer Curve Revisited" - Mathias Trabandt and Harald Uhlig:

I recently found this paper while browsing ideas for estimates of the US Laffer curve based off of a neoclassical growth model. This paper (download here) has a couple of key points for the Republican candidates. Namely 1. The US is currently on the left side of its Laffer curve and 2. Because of this, tax cuts will not be self-financing (take that Jeb!).

2. "Simple Analytics of the Government Expenditure Multiplier" - Michael Woodford:

This is actually one of the papers that has most influenced my understanding of fiscal policy. Woodford's model doesn't have all the bells and whistles of typical DSGE's, so the analysis is extremely clear and identifies the effects of fiscal policy given different monetary policy choices. Three main insights that I get from it are: 1. If the central bank pegs the real interest rate, the multiplier equals one 2. If the central bank pegs the inflation rate, the multiplier equals the flexible price multiplier 3. If money is neutral, the multiplier is greater than zero 4. At the zero lower bound, the multiplier can exceed one (download here).

## 09 September 2015

### Wikipedia Books

I recently discovered the amazing fact that you can create books with Wikipedia pages. Naturally, I decided to try my hand at a ~150 page book on macroeconomics. Here it is in all its awesomeness: https://dl.dropboxusercontent.com/u/92766758/Macroeconomics.pdf

(And here's the per-existing book on evolution that I found on Wikipedia for anyone whose interested: https://dl.dropboxusercontent.com/u/92766758/Evolution.pdf)

PS: Sorry about the short post/the lack of posts in general lately, I'm in the process of writing a couple that should come out soon.

## 31 August 2015

### I Don't understand Market Monetarist Logic

So the typical market monetarist view on business cycles is that low NGDP causes low RGDP. Let $p$ indicate whether or not NGDP is lower than normal and $q$ indicate whether or not RGDP is lower than normal. The market monetarist contention can be represented as such:

$$p \rightarrow q$$

If a central bank successfully targets inflation, then NGDP should track RGDP (because NGDP growth is always equal to RGDP growth plus the inflation target). This looks like

$$q \rightarrow p$$

These two statements don't seem to make sense when paired with each other... According to market monetarists, low NGDP caused the great recession, but, because of the inflation targeting regime in the US, low RGDP causes low NGDP... Do market monetarists think that the great recession caused itself? Their logic seems to imply either that recessions come from something like multiple equilibria when central banks target inflation or that they just can't happen because in inflation targeting regimes, NGDP doesn't fall unless RGDP does and RGDP doesn't fall unless NGDP does, so neither ever fall. If they think that inflation targeting produces multiple equilibria, then why don't they say so? If the multiple equilibria logic is correct, then they shouldn't be strictly advocating and NGDP target; they should be telling everyone to switch to any target that doesn't make NGDP depend of RGDP...

This is all extremely confusing.

## 30 August 2015

### Notes on Taylor Rules, Inflation, and Neo-Fisherism

I've been working on writing a paper outlining my views on three topics in monetary economics (the three things in the title). Click here for the pdf of what I have written so far. Here is the text if you don't want to download the pdf:

UPDATE: The pdf link should update automatically to changes, but I won't change the text in the blog post. Just download a copy of the pdf every time you want the most up to date version.

Introduction:

There has been quite a bit of discussion about the relationship between the nominal interest rate and the rate of inflation recently among economists. To my knowledge, the problem began when Cochrane (2007) challenged the idea that the inflation rate could be determined with simply a Taylor Rule and a Fisher relation in combination with a commitment to active monetary policy and implicitly passive fiscal policy (see Leeper (1991) for example). Cochrane's key insight was that, in these models, the central bank is essentially committing to cause inflation to explode by increasing the nominal interest rate (effectively the expected inflation rate) more than one for one with current inflation. Because economists had ruled out explosive solutions, the only other equilibrium – one in which the inflation rate jumps immediately at period zero to the central bank's target – was considered. As Cochrane noted, there is not necessarily any reason to rule out explosions in nominal variables as they have no impact on the real economy in the models in question.
\par In the years since then, the failure of zero interest rate policies to generate inflation became of interest. Pretty soon, a similar yet entirely different debate came into existence. A few of economists (to my knowledge, Williamson and Cochrane) had the novel idea that the nominal interest rate had a causal relationship with the rate of inflation. This notion had long existed in the literature and is even a property of just about every macroeconomic model; the problem, in fact, was not the notion that high inflation and high interest rates happened at the same time. Rather, it was the idea that central banks could deliberately cause inflation by setting the nominal interest rate at a higher level. The consensus that active monetary policy was required for inflation stabilization and that positive deviations from the target interest rate implied by a Taylor Rule would result in lower inflation was in direct opposition to these "Neo-Fisherian" claims, so a debate that pulled in a slew of other economists ensued.

The difficulty in this case is that both sides are right in their own way. The consensus was correct that, so long as the central bank uses a Taylor Rule to target inflation, positive deviations from that target would result in a lower inflation rate. The Neo-Fisherian view is correct in the sense that if the central bank does not follow a rule and deliberately loosens monetary policy, the inflation rate and the nominal interest rate will increase. The issue with both views is that the underlying assumptions are either not understood or not made clear by there proponents. Economists putting forward the conventional wisdom don't make it clear that the Taylor Rule is the sole cause of inflation dynamics in their model and Neo-Fisherians fail to put forward that the result that they purport is highly dependent on how the money supply (or in some cases fiscal policy) acts when the nominal interest rate is increased. Each model relies heavily on a set of implausible assumptions about the way central banks behave. It is clear that central banks don't behave in the way implied by the consensus models and it is equally clear the the Neo-Fisherian result only occurs when monetary policy has taken a permanently more accommodative stance; even though this assumption is not put forward by its proponents.

If, as I suggest, the "Neo-Fisherian problem" and the "Taylor Rule problem" are all about assumptions, then their respective solutions are simple: just add some microfoundations. When it comes to arguments about monetary policy, the necessary microfoundation is painfully obvious. These models all need money in order for their implications to be understood. Interest elastic money demand functions solve Cochrane (2007)'s critique as they prohibit real explosions of the money supply – something that would happen if the nominal interest rate expanded or collapsed infinitely and money demand functions in general can determine when high interest rates mean tight monetary policy and when high interest rates mean loose monetary policy without appealing to dynamics implied by implicit monetary policy rules and without simply assuming that all high interest rates are do to loose money.

Model:

We will begin by adding a simple ad-hoc money demand function to a two equation frictionless New Keynesian model and looking into the implications of the simple addition for monetary modeling. As usual, there is a Fisher equation relating the nominal interest rate to expected inflation and a Taylor Rule relating current inflation to the nominal interest rate.

$$i_t = \rho + E_t \pi_{t+1}$$

$$i_t = \rho + \phi \pi_t$$

$i_t$ is the nominal interest rate, $\pi_t$ is the inflation rate, $E_t$ is the period $t$ rational expectations operator, $\phi$ is the "inflation reaction parameter" on the Taylor Rule, and $\rho$ is the constant real interest rate. The sole addition that we will add to this basic model is a simple money demand function which sets real money demand equal to

$$m_t - p_t = y - \eta i_t$$

where $m_t$ is the nominal money supple, $p_t$ is the log price level ($\pi_t = \Delta p_t$), $y$ is the (constant) level of output, and $\eta$ is the interest-elasticity of the money supply.
\par With the addition of the money demand function, so long as $\left|\eta\right| > 0$, Cochrane's problem with ruling out nominally explosive equilibria disappears. Now, a real variable depends on the nominal interest rate and prevents hyper inflations that are not caused by excessive money growth. In fact, adding money demand changes nothing about the dynamics of the model; following a Taylor Rule still gives the conventional wisdom about monetary policy without having to deal with the difficult problem of ruling out nominal explosions.
\par The interesting thing about this model is that it can replicate the Neo-Fisherian result easily. Consider a deterministic economy where the central bank permanently increases the growth rate of the money supply, $m^g_t = \Delta m_t$, from $m^g_0$ to $m^g_1 > m^g_0$.

If you have any feedback or suggestions before I continue to write, feel free to comment. I intend to continue by expanding my analysis to a more full fledged New Keynesian model with different types of money demand ranging from Money-In-The-Utility-Function to Cash-In-Advance and explain the mixed signal problems of using interest rates as an indicator of the stance of monetary policy. (I also plan to refer to more of the relevant literature than just "Determinacy and Identification with Taylor Rules")