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 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

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


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.