Non-Walrasian Macro

The typical framework used by economists to make models renders generating monetary non-neutrality nigh impossible without the use of dubious assumptions like price adjustment costs or the existence of a 'Calvo fairy' that chooses at random when a firm can or cannot change its price.

Otherwise, it is posited, the firm would just set its price to whatever level is consistent with full employment at all times, rendering monetary policy useless at doing more than simply changing the inflation rate. Brushing aside possible empirical issues with this contention, this poses significant political problems for many economists; if monetary policy is impotent at all times, then either the economy is always at 'potential' thus rendering all intervention unnecessary, or the only viable means of economic stabilization is fiscal. As both of these options are highly undesirable to everyone who doesn't work at the St. Louis Federal Reserve of the University of Chicago, the problem must be solved.

This is when most of the profession turned to finding microfoundations for price stickiness (only to find that none of them are consistent with the actual behavior of individual prices, but never mind that) and thus New Keynesian economics was born. We now have the expectations augmented Phillips curve, in all its reduced form glory derived from a set of specific if highly unrealistic assumptions about the world. There is one point of interest here, however: if you remove both rational expectations and the dynamic aspect of the model, the basic three equation New Keynesian model reduces to IS-LM with sticky instead of completely stuck prices -- think Krugman's 1998 paper.

Evidently we spent 20 years trying to develop a model that has the exact same insight into the economy as we were able to get from a much simpler model from 1936 (Occam's razor, anyone), but at least this one can pretend to be quantitative (Smets-Wouters). Now I will not pretend to minimize the advantages of utility maximization when it comes to analyzing situations; sometimes it really helps to use utility maximization when you cannot simply draw an upward and downward sloping curve and call it a day (hooray for OLG models!), but when it comes to qualitative models of the business cycle, nothing beats IS-LM (say what you will about the consumption function, just have Y = a*G, for all I care).

Maybe it's time we took this all in a different direction, though. The main reason we went on the rabbit trail of New Keynesianism to get back to IS-LM was because of Walrasian ideas about supply and demand. That is, by some force of magic the market will determine the equilibrium price instantly (hence the rule that quantity supplied equals quantity demanded). But does this really make sense? even assuming individual firms knew the exact shape of their demand curve, is it reasonable to assume that they would be able to determine what (nominal) price to set so that their relative price is at the profit maximizing level? Or, more succinctly, how do firms magically know what price to set if there is no real Walrasian auctioneer?

But if we don't ignore tatonnement -- that gradual approach of a price to its equilibrium level -- then we naturally get Phillips curves everywhere, as firms slowly increase prices to fend of excess demand and lower wages to head off excess supply (of course wage bargaining is two sided but bear with me). In fact, replacing $D=S$ in models with $\dot{p} = \alpha(D-S)$ could circumvent the entire need for New Keynesian models; let the nominal rigidities run wild as freshwater economists cower in fear of non-Walrasian macro.

Not only would this alleviate the need for New Keynesian models, it would replace them with much simpler that are what I would like to call 'semi-microfounded.' That is models would have the typical upward and downward sloping curves -- who needs calculus -- and then have a specification for how tatonnement occurs. For instance, a labor market would have a labor demand equation: $L_d = -a W$ and a labor supply equation $L_s = b W$. Then tatonnement would be $\dot{W} = \alpha(L_d - L_s)$. Did the equilibrium wage move? Oh no, I guess we'll have to find it since uncle Walras can't just tell us what it is any more.

Freshwater economists, read and weep.

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.

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, $\varepsilon^i_t$:

New Keynesian Central Bankers Are Stupid

Imagine that there is a central bank that targets the inflation rate successfully every period because there are no real shocks in the economy. In this model, inflation looks like this:

$$ \pi_t = \pi^* $$

where $ \pi_t $ and $ \pi^* $ are the inflation rate and the inflation target, respectively. The nominal interest rate in this model will always be $ \pi^* $ higher than the constant (no real shocks) real interest rate, $ \rho $ and can be written as 

$$ i_t = \rho + \pi^* $$

Now suppose that the central bank is not omniscient and occasionally misses its target either on accident or because of some unforeseen shock. The inflation rate is now

$$ \pi_t = \pi^*+ \epsilon_t $$

where $ \epsilon_t $ is the central bank's error every period. Since the inflation rate is not serially correlated, the nominal interest rate remains equal to $ \pi^* \: \forall t $. Let's add some real shocks into this economy, so the real interest rate fluctuates over time, adjusts slowly, and is equal to $ r_t $.

$$ r_t = (1 - \rho^r)\rho  + \rho^r r_{t-1} + \nu_t $$

The nominal interest rate now moves around with the real shocks:

$$ i_t = r_t + \pi^* $$

For some unknown reason, the central bank decides to adopt a floating inflation target, $ \bar\pi_t $, and sets it so that it becomes a weighted average of $ \pi_{t - 1} $ and $ \pi^* $.

$$ \bar\pi_t = (1 - \rho^\pi)\pi^* + \rho^\pi \pi_{t-1} $$

The central bank still occasionally misses its target, so $ \pi_t $ is not always equal to $ \bar\pi_t $ and is instead

$$ \pi_t = (1-\rho^\pi)\pi^* + \rho^\pi \pi_{t-1} + \epsilon_t  $$

The nominal interest rate is related to the current inflation rate now because of the auto-regressive process that the rate of inflation fallows and can now be expressed as

$$ i_t = r_t + (1 - \rho^\pi)\pi^* + \rho^\pi \pi_t $$

By sheer assumption, let's say that $ \epsilon_t $ and $ \nu_t $ are negatively correlated. What is this model now? Well, it's New Keynesian, isn't it.

Think about it: There is a central banker that knows it could keep the real interest rate constant (or equal to its natural rate) by pegging the inflation rate, but instead he or she chooses to make it serially correlated by following a Taylor Rule. The real interest rate falls when inflation is above "target" and the nominal interest rate and the rate of inflation are positively correlated. Really, all you need to do to make the dynamics exactly like that of a New Keynesian model is to add a variable $ x_t $, call it the "output gap" and say that $ \dot x_t = r_t - \rho $.

Of course the real aspects of this "model" are really irrelevant (and pretty weak as assumptions go, replacing nominal rigidity with "shocks are negatively correlated" is pretty bad, the AR part of the real interest rate can make sense if the capital stock takes time to adjust, for example). What's really important is that NK central bankers are stupid. They know that they should be targeting a constant rate of inflation, but they abandon that for the sake of Taylor Rules and avoiding the money demand function.

I obviously don't think that central bankers can simply choose to their nominal target be achieved every period, but they should at least refrain from "endogenizing" the money supply in favor of a tool that can mean different things at different times depending on your assumptions.