Tuesday, August 14, 2018

Simple labor search model (a la Pissarides)

In this post I will outline a search model of the labor market, similar to some found in the literature on labor economics. I think macroeconomists have focused too much on sticky prices and wages as a mechanism for explaining fluctuations in unemployment, and not nearly enough on mechanisms of economic disruption and search frictions in the labor market. You can view this simple model as a module that may be put into a larger model incorporating various kinds of shocks.

The model has infinite periods. There are a continuum of firms with measure \( 1 \) and workers of measure \( N \).

For simplicity, I assume that workers have a fixed reservation wage \( w \), above which they are willing to supply labor to any firm, and firms make take-it-or-leave-it offers to workers in bilateral meetings. Labor is indivisible, i.e each worker can choose between supplying 0 or 1 unit of labor to one firm. This assumption simplifies the analysis without changing the resulting dynamics very much.

Firms produce goods using a decreasing returns to scale production function \( f(L) \) where \( L \) is the amount of labor used in production, and they post vacancies \( V \) at a flow cost (means it must be paid in every period the vacancy remains posted) of \( c \) per vacancy which have a chance \( Q \) of being filled that firms take for granted. We assume there's no firm-level uncertainty in the filling of vacancies in the sense that a firm which posts \( V \) vacancies will fill \( QV \) of them surely. They maximize the present value of future profits, discounted at a constant gross interest rate \( R \), i.e they maximize

\[ V = \sum_{k=0}^{\infty} R^{-k} (f(L_k) - w L_k - c V_k) \]

subject to the constraint \( L_{k+1} = (1 - \delta) L_k + Q_k V_k \) and a nonnegativity constraint \( V_k \geq 0 \).

The firm chooses \( V_k \), the number of vacancies it will post in every period. The first order condition for the firm's optimization problem (assuming the nonnegativity constraint does not bind) is

\[ c - c \frac{(1 - \delta) Q_k}{Q_{k+1}}= Q_k R^{-1} f'(L_{k+1}) - Q_k R^{-1} w \]

Now, we need to pin down \( Q_k \). We do this by specifying a constant returns to scale matching function \( g(V, U) \) giving the number of vacancies filled in the next period if in the current period there are \( V \) vacancies and \( U \) unemployed workers. This means we have

\[ Q_k = \frac{g(V_k, N - L_k)}{V_k} \]

These two conditions pin down the time evolution of the labor market, and by linearizing them around the steady state one may find the time evolution of \( V \) and \( L \). I will restrict myself to doing quasistatic analysis here, i.e look at the stationary equilibrium. This is pinned down by the relations

\[ \delta R V c = g(V, N - L) (f'(L) - w) \, \, (1)\]
\[ g(V, N - L) = \delta L \, \, (2) \]

In \( L-V \) space, the first equation pins down a downward sloping curve, whereas the second pins down an upward sloping curve. A slightly more detailed analysis of the behavior of these curves proves that there is a unique intersection point with positive \( V \) and positive \( L \), so the model has a unique steady state. You can graph these curves and do some quasistatic analysis of your own. For instance, an increase in vacancy costs \( c \) shifts the curve (1) to the left in \( L-V \) space, resulting in a lower number of vacancies and higher unemployment. A constant factor increase in the production function, say something like an increase in \( A \) if \( f(L) = A h(L) \) for some other function \( h \) and a constant \( A \), has the opposite effect: it increases the number of vacancies and decreases unemployment. In particular, this model produces unemployment naturally in the event of a negative productivity shock and produces a Beveridge curve - a negative correlation between unemployment and posted vacancies. The reason is that a negative productivity shock decreases the returns on posting new vacancies, so given the fixed cost \( c \) of doing so the firm chooses to post less of them.

In this model, labor \( L \) adjusts slowly to shocks, while vacancies adjust quickly. In other words, the immediate response in the event of a negative productivity shock is a fall in posted vacancies, which gradually increases unemployment as convergence to the stationary equilibrium takes place. The basic model fits quite well with data on the 2008 recession in the US, for instance.

The model may easily be augmented by introducing a more complex utility function for the worker, which would add a third relation to this set of two equations relating the wage \( w \) to labor supply \( L \) through some kind of bargaining solution between the firm and the worker. I won't do this here, but it's a straightforward extension.

Monday, August 6, 2018

Minimum wages and inelastic product demand

The Wikipedia page on the minimum wage has a paragraph claiming that a possible explanation for why a minimum wage increase would not reduce employment is that companies producing goods for which the demand is inelastic could raise prices to pay for the higher wage instead of laying off workers. Apparently Paul Krugman thought this is an unreasonable explanation, so let me help him out with a simple model.

Say that a firm has profits \( p f(L, X) - wL - r \cdot X \) where \( L \) is labor, \( w \) is the wage paid per worker, \( X \) denotes other factors of production with rent vector \( r \) (we assume these are fixed) and \( p \) is the price of the output good. \( f \) is a constant returns to scale production function. In this case, in a competitive labor market we have the usual relation \( p f_1(L, X) = w \) relating wages to the marginal product of labor. Say the goods are sold in a competitive market where the overall demand for the product as a function of the price is \( q(p) \). Then, the equations \( q(p) = f(L, X) \) and \( p f_1(L, X) = w \) determine prices and employment given wages. Now we can do some comparative statics to get

\[ q'(p) \frac{dp}{dw} = f_1(L, X) \frac{dL}{dw} \]
\[ \frac{dp}{dw} f_1(L, X) + p f_{11}(L, X) \frac{dL}{dw} = 1 \]

which gives

\[ \frac{dL}{dw} = \frac{1}{f_1(L, X)^2/q'(p) + p f_{11}(L, X)} \]

Since \( q'(p) < 0 \) and \( f_{11}(L, X) < 0 \), this is always negative, i.e a rise in the labor market wage, whether it is caused by a wage floor or by increasing productivity in other parts of the economy, will reduce employment at this specific firm. However, this increase will be less if \( q'(p) \) is small (with all other variables held constant, same thing as lower elasticity) relative to \( f_1(L, X)^2 = (w/p)^2 \), since then the denominator is large and the entire expression is closer to \( 0 \). In other words, yes, inelastic demand can reduce the sensitivity of employment to wage floors.

Friday, August 3, 2018

Are expectations of tariffs on Chinese exports affecting Chinese stock prices?

The answer seems clear - the Shanghai composite index fell by nearly %20 in the past five months, coinciding with an increase in the amount of noise being made related to a possible "trade war" between the US and China. Based on this, some people have concluded that "obviously the fall has been caused by changes in the expectations of market participants about future tariffs or trade barriers". This came up in a recent conversation I was having, so I decided to test this claim by a simple regression model of the form \( R = \beta X + \epsilon \) where \( R \) is the return on the Shanghai composite index in a given time period, \( X \) is an indicator of shocks to market expectations of increases in trade barriers, and \( \epsilon \) is a homoskedastic mean zero shock. I use weekly data on stock market returns, and I use data from Google Trends to proxy for market expectations.

I test two different identifications - one with \( X \) equal to the weekly Google Trends score \( S \) of the phrase "trade war", and one with \( X_t = \max \{S_t - S_{t-1}, 0 \} \) - an intermediate model could model \( S \) as a more general AR(1) of shocks to expectations, but I take the two extreme cases in which the decay rate in the AR(1) is either equal to \( 0 \) or \( 1 \). You can see the spreadsheet, along with the computed statistics, here. Neither gives a statistically significant value of \( \beta \), and the sample correlations between \( X \) and \( R \) in the two identifications are -0.17 and -0.28, respectively - if squared to get the proportion of explained variance, we get %2.8 and %7.6, so the second identification gives a significantly better fit than the first. The conclusion is that there does not seem to be a significant effect of the talk about tariffs on stock prices - market prices adjust quickly to new information, but the timing of the price movements does not match with news flow, even at a weekly resolution.

The answer to the question in the title is probably "yes", from general economic principles, but such principles say nothing about the magnitude of the effect. It's entirely possible that the costs imposed on the Chinese economy are small due to possible shifts in global trade flows, for instance - China sells the goods it now sells to the US to European countries, the exporters who were previously selling goods to European countries shift to selling the same goods to the US, and the costs imposed on the Chinese economy are second order costs coming from the need to accommodate such a shift in the direction of trade. The Chinese economy is also well-diversified, so it's possible that factors of production now producing goods being exported to the US could be diverted and used in other sectors, once again at some smaller cost than the static effects of the tariff. In the data, there's no evidence of any effect whatsoever, but that's also partly because of the high volatility of Chinese stocks making it difficult to identify the effect of the tariffs. In other words, the true magnitude of the effect is small enough to be obscured by usual volatility in the Chinese stock market in our small sample size.

Saturday, June 23, 2018

A simple model of tariffs

Here are the assumptions of the model:

There are two periods and a continuum of identical households of measure \( 1 \). They consume two goods \( A, B \) and have utility

\[ \frac{A_1^{1 - \gamma} - 1 + B_1^{1 - \gamma} - 1 + \beta (A_2^{1 - \gamma} - 1 + B_2^{1 - \gamma} - 1)}{1 - \gamma} \]

with \( \gamma > 0, 0 < \beta < 1 \). The goods are perishable (can't be stored), and in each period the household receives an endowment \( (A_e, B_e) \) of the two goods. There are open international trade and credit markets with no frictions, with a constant interest rate \( r \) and constant international prices \( p_A, p_B \) with \( p_A > p_B \) for the two goods. The household's intertemporal budget constraint is therefore (assuming that the tariff \( \tau \) on good \( B \) is small enough that in equilibrium it is always imported):

\[ \left(1 + \frac{1}{1+r} \right) (p_A A_e + p_B B_e) + T_1 + \frac{T_2}{1+r} = p_A A_1 + p_B B_1 + \tau p_B (B_1 - B_e) + \frac{p_A A_2 + p_B B_2 + \tau p_B (B_2 - B_e)}{1+r} \]

where the \( T_i \) are lump sum transfers to households by the government. The first order conditions of this optimization problem are

\[ \left( \frac{A_i}{B_i} \right)^{-\gamma} = \frac{p_A}{(1 + \tau) p_B} \]
\[ \beta \left( \frac{A_2}{A_1} \right)^{-\gamma} =  \frac{1}{1+r} \]

and the government runs a balanced budget, so that \( T_i = \tau p_B (B_i - B_e) \) in both periods. I set \( \gamma = 1 \) (natural log), \( \beta = 0.99 \), \( r = 0.02 \), \( p_A/p_B = 2 \) and \( A_e = B_e = 1 \) to calibrate the model, and solve for \( A_i, B_i \) for different values of \( \tau \).
  • With \( \tau = 0 \), we have \( A_1 = 0.746379, A_2 = 0.753693, B_1 = 1.49276, B_2 = 1.50739 \). The trade balance in units of good \( B \) in period \( 1 \) is \( 0.01448 \) - a trade surplus. We can understand why this happens - the impatience \( 1 - \beta \) is small relative to the interest rate \( r \), so households prefer to lend their goods in the first period rather than consume them. Utility is equal to \( 0.234 \).
  • With \( \tau = 0.5 \), we have \( A_1 = 0.895655, A_2 = 0.904432, B_1 = 1.19421, B_2 = 1.20591 \). The trade balance in units of good \( B \) in period \( 1 \) is \( 0.01448 \) (!). Utility is equal to \( 0.153 \).
In other words, our model gives the conclusion that a tariff
  • reduces the volume of trade,
  • decreases welfare,
  • has no effect on the trade balance.
The first two are fairly standard results. However, where does the third result come from? Shouldn't a tariff reduce the trade balance in absolute value by decreasing openness?

The answer is that in this model, the first order conditions fix the ratio of consumption in period \( 1 \) to consumption in period \( 2 \) in terms of the parameters \( r, \gamma \). Given this assumption, the intertemporal budget constraint then fixes the market value of consumption in both periods, which given endowments determines the trade balance independently of \( \tau \). This is a property of the specific utility function we use - constant relative risk aversion utility displays this behavior, regardless of the value of \( \gamma \).

You can play around with the simple model with different utility functions to get an idea of the effect of a tariff on the trade balance in a more general setting - I'm too lazy to type out the computations here.

Wednesday, June 6, 2018

Can the Fed buy up all of Planet Earth and not produce any inflation?

This is a reply to a Scott Sumner post on "liquidity traps". Short answer: yes.

As I have said many times in the past, in the balance sheet/fiscal theory of the price level, what matters is the central bank's willingness to eventually swap the money in the system with assets on the other side of its balance sheet. This fundamental commitment is what gives money its value in a frictionless environment. When the central bank transacts at market prices, it does not reduce the amount of backing of the currency on the other side of the balance sheet, so it doesn't need to have any effect at all. If the Fed bought up all assets on Planet Earth, another intermediary could simply buy up all of the dollars on Planet Earth and undo the asset transformation done by the Fed.

Of course Sumner knows this (even though he believes in monetary frictions more than I do), which is why his focus is on "permanent" monetary expansion. However, the problem here is that the "permanent" can be pushed arbitrarily far into the future. As long as the money in circulation will eventually be redeemed by the central bank's assets in an appropriate sense, no amount of conventional or unconventional monetary policy done by a central bank will have any effect. The central bank can still create as much inflation as it wants - either by raising the nominal interest rate it pays on reserves, or by making transfer payments to agents in the economy, i.e transacting at below market prices. To create inflation, you need to find some mechanism by which you can increase the total dollar value of the liabilities of the central bank without increasing the real value of its assets proportionally. IOR has this effect, and so do transfer payments.

There is, of course, a third way, which is to abandon the implicit commitment that all of the central bank's assets are backing the currency. In that case, talking and setting policy targets can have quite dramatic effects on the price level without any open market operations at all. (Of course, there is always a commitment to do some such operations in the future, but this commitment can be pushed arbitrarily further into the future.) The simplest example of this is the dollar devaluation Sumner mentions in the article: in this case, the amount of backing the dollar had was an explicit commitment enshrined in the convertibility of the dollar to gold, and by simply devaluing the currency (i.e reducing the total amount of real assets available to be redeemed for the currency) the US succeeded in creating inflation. The ability to do this is of course why Sumner wants an institutional structure like a "NGDP futures standard", so that the central bank can easily communicate its NGDP growth target and be supported by an institutional structure which means the amount of backing given to the currency will be dynamically adjusted as new information arrives to offset any changes which would push NGDP growth off track.

So, liquidity traps are real in the sense that open market operations have no effect - they already have no effect at any positive nominal interest rate! (This relies crucially on the absence, or relative unimportance, of monetary frictions.) The way to have an effect is to reduce the amount of real backing each dollar has on the central bank's balance sheet. The central bank can do this by a higher IOR target (which is a distribution-neutral way of increasing the growth rate of the monetary base), transfer payments, or engendering expectations that not all of the central bank's assets will be swapped for dollars in the future (reducing the backing of the currency). By using these tools, the central bank can indeed control any one nominal quantity as it pleases: a nominal exchange rate, nominal GDP, inflation, nominal interbank lending rates...

Monday, June 4, 2018

Curing neo-Fisherian denial

In this post, I will be trying to address some common criticisms of the neo-Fisherian proposition. First, let's start with some definitions:

Strong Neo-Fisherism is the proposition that raising the central bank's policy rate target permanently raises inflation permanently, both in the short run and in the long run. It does not imply that an expansionary monetary policy (one that depreciates nominal exchange rates) is always accompanied by a rise in short term nominal interest rates, nor does it say anything about the impact of other kinds of interest rate targeting policy.

Weak/Mild Neo-Fisherism is the proposition that nominal interest rate pegs are stable, which follows directly from long-run monetary neutrality, along with the long-run Fisher effect. This proposition says nothing about the short run response of inflation to a change in the interest rate target, so it's weaker than the strong version of neo-Fisherism. (This is my view.) Another way to phrase it is that a monetary policy which ensures permanently low short term nominal interest rates is a contractionary monetary policy, and one which ensures permanently high short term nominal interest rates an expansionary monetary policy.

A nominal interest rate peg is characterized by the central bank doing whatever it takes in terms of monetary policy to hold its policy rate within a small target range permanently. Raising policy rates for two years and then lowering them to a low level again is not a permanent rise in the target policy rate unless market participants believed that the initial rise was going to be permanent, so episodes like the Volcker disinflation are not evidence against either kind of neo-Fisherism.

Let's start:

  • Empirical work suggests that the Fisher effect is only one-for-one in long, not short, term interest rates: This is true, but also irrelevant for several reasons. First, unless there is a nominal interest rate peg in place, no empirical work is going to disprove the neo-Fisherian proposition. Second, this literature either assumes constant real interest rates, or uses inflation compensation on indexed vs nominal bonds as a proxy for inflation expectations. Given that the same logic gives abysmal results in foreign exchange markets (forward rates don't forecast where spot rates are going to go), I don't think ignoring the risk premia involved is going to give reliable results. Third, most of this research was either done before 2008 or the datasets used come from before 2008, so they are inadequate to analyze monetary policy under a large balance sheet regime with IOR and no scarcity of bank reserves. Fourth, since the empirical data supports the long-run Fisher effect, there's no contradiction between these findings and the weak neo-Fisherian proposition. Finally, the empirical estimates of the short-run Fisher effect in the existing literature fail in the sense of producing coefficients on inflation which are below one, but for the strong neo-Fisherian proposition to fail definitively it is necessary to show that the coefficient on inflation is in fact negative, so the data does not even disprove the strong neo-Fisherian view in this case.
  • Neo-Fisherism is reasoning from a price change: This criticism is unfounded, as neo-Fisherism is about the effects of a specific monetary policy regime. It is interesting to ask whether two distinct paths for monetary policy can support the same time series for the nominal interest rate, but it is also irrelevant - if both kinds of monetary policy give a nominal interest rate peg as the result, they should both have the neo-Fisher effect in them. This possibility is not a problem for the neo-Fisherian proposition. Moreover, neo-Fisherism does not say that an easy/tight money policy always leads to high/low short term nominal interest rates respectively, as pointed out in the definitions.
  • Deflation spirals: These are actually not a result of any macroeconomic model that I know. Even old-school IS-LM models, when solved properly to rule out real output explosions, do not give any deflation spirals as an equilibrium. The empirical data also contradicts this view, with 20 years of zero interest rate policy in Japan not producing any deflation spiral to speak of. 
  • Neo-Fisherians confuse correlation with causation: Not at all. The neo-Fisherian view comes with many causal mechanisms going from high nominal interest rate targets to high inflation. One way to think of it is that a high nominal interest rate means that nominal quantities in the economy have to be growing over time, so if real quantities are unaffected by monetary policy, then the result is a rise in the nominal price of real goods and services. Then, as long as you acknowledge long-run monetary neutrality, this leads you to weak neo-Fisherism. The strong version makes another claim about the magnitude of the short run liquidity effect, so it's not as robust, but certainly there's no confusion about the direction of causality anywhere in it. For instance, under an IOR regime of monetary policy, raising the IOR is the same thing as raising the growth rate of central bank liabilities outstanding, so it should be fairly obvious why that would lead to higher inflation. (The nice thing about this is that it works even in models where expanding the balance sheet through open market operations or large-scale asset purchases is ineffective at creating inflation, so it's in fact a very robust result.)
  • Short and long rates are decoupled: This is not an argument I've seen people make explicitly, but it seems to me that they have this idea in the back of their head. Let's make this explicit: in the absence of frictions, if a central bank commits to pegging overnight short rates at some level for 10 years, then the 10 year bond yield must be equal to the central bank's short rate target annualized. In other words, a central bank can fix the entire yield curve by only controlling overnight rates if it can precommit to a specific path of rates in the future. In this scenario, it's impossible for long-term bond yields to rise if investors don't anticipate the central bank raising its short-term policy rate in the future. In practice, central banks don't precommit to things like this, which means there are risk premia on bonds that would not exist in the scenario I described here. However, the basic idea that you get long rates by chaining together short rates seems to prove elusive for many people who think of short rates as being controlled by the central bank while long rates are pinned down by long-term inflation expectations that somehow can't be controlled by a short term interest rate peg, i.e they treat long and short rates as objects that are somehow decoupled from one another.
If you know of other arguments made against neo-Fisherism, let me know so I can add them to this post along with my responses.

Wednesday, May 30, 2018

The balance sheet theory of the price level (Backing theory of money)

The fiscal theory of the price level is sometimes phrased in terms of taxes, which I think may be confusing for some people who want to think of a monetary arrangement similar to the Eurozone, where government debt is not nominally risk-free and there is an independent central bank. (It can also lead to confusion - the FTPL is not a chartalist theory of the value of money, even though it is sometimes incorrectly phrased that way.) The way to do this is to correctly interpret the fiscal theory of the price level as saying something very simple: the value of the government's assets (including its claims to future taxes/transfers, but other assets as well) has to equal the value of the government's liabilities. When phrased this way, we may replace "government" with "the central bank" and not lose any content - the vital part about this theory is that we deal with an entity which can issue money, so that these equations are valuation equations rather than intertemporal budget constraints. The reason that we use the government instead of the central bank in the fiscal theory is that if government debt is nominally risk-free, then the central bank is going to have to issue the money the government needs to pay back its debts anyway, so a negative fiscal shock necessarily shows up on the central bank balance sheet as well.

A simple way to think of this is that if the markets for all goods and assets except for money clear in the economy, then the market for money must clear as well. For instance, we can imagine an economy inhabited by agents \( \{ 1, 2, \ldots, n \} \) along with a central bank, and in each period we may write agent \( i \)'s budget constraint (under the assumption that money does not relax some kind of constraint or enter into the utility function, i.e we're in a frictionless setting) as

\[ \frac{M^i_t + T^{c, i}_t}{P_t} + w^i_t + Y^i_t = c^i_t  + \sum_{j \neq i} T^{i, j}_t + \mathbb E_t \left( m_{t, t+1} \left(\frac{M^i_{t+1}}{P_{t+1}} + w^i_{t+1} \right) \right) + I^i_t \]

where the \( T^{i, j} \) are transfer payments between the agents, \( m_{t, t+1} \) is the real discount factor for pricing time \(t+1 \) payoffs at time \( t \), \(M^i_t \) is money balances for agent \( i \), \( P_t \) is the price level, \( w^i \) is the net asset holdings of agent \( i \), \( I^i_t \) is physical investment by agent \( i \) and \( Y^i \) is the income of agent \( i \). If we sum up these budget constraints for all agents, we find that

\[ \frac{M_t + T^c_t}{P_t} - w^c_t + Y = c_t + \mathbb E_t \left( m_{t, t+1} \left(\frac{M_{t+1}}{P_{t+1}} - w^c_{t+1}\right) \right) + I_t \]

(we assume that the central bank has no income other than the returns on its assets) where \( w^c_t \) is the central bank's net asset holdings at time \( t \), since the net value of all financial assets in the economy is zero (every asset is a liability). The accounting identity for national income then gives some cancellation, and solving the remaining equation forward under a no-real-explosion condition gives the simple result

\[ \frac{M_t}{P_t} = w^c_t - \sum_{k=t}^{\infty} \mathbb E_t \left( m_{t, k} \frac{T^c_k}{P_k} \right) \]

i.e the value of the central bank's liabilities (outstanding real money balances) equals the value of its assets (its private sector asset holdings) minus the present value of real transfers \( T_c/P \) it makes to the agents in the economy, by issuing money or otherwise. In other words, there's nothing special about the central bank compared to any other private intermediary beyond the fact that we use its liabilities as the numeraire (unit of account), as we would expect.  All of the results which are true under the fiscal theory of the price level then carry over immediately to this new setting. Because of this reason, I think it may actually be more appropriate to call this theory "the balance sheet theory of the price level" - all it says is that in the absence of any frictions, the real value of assets = the real value of liabilities. We can already see in this simple model the ineffectiveness of large scale asset purchases, for instance - an increase in \( M_t \) which is backed by a corresponding increase in \( w^c_t \) will have no effect on the price level \( P_t \).