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 \( P_t \).

Sunday, May 27, 2018

Foreign Policy is failing economics 101

From a recent Foreign Policy article on Turkish economic policy:

"Erdogan has repeatedly said he believes that high interest rates cause inflation, flying in the face of the economic rule that tightening the monetary supply reduces inflation."

Why do we think these things contradict each other? I have posted a model on the blog in which the impulse response to a permanent reduction in the money growth rate is a fall in both nominal interest rates and inflation simultaneously - there's no reason that high interest rates should be associated with a tightening of the monetary base. Besides, "tightening the monetary supply reduces inflation" is a rather ancient rule based on naive quantity theory ideas; it is not a good way to think about monetary policy in the 21st century. Central banks around the world have nominal interest rate targets, not monetary base growth targets.

"But economists caution that unrestrained growth can actually hurl a country into a recession"

Yes, some economists do. The word "unrestrained" is not a word that has an ex ante definition - it is an ex post characterization of periods of growth which are followed by big recessions, so the statement becomes a tautology once this definition is used. After there is a recession, some economists who think they are smart say things like "well, obviously the growth we were seeing was unsustainable", but the capacity of such economists to predict whether there will be a recession in the next 3 years is probably worse than a coin flip. Recessions are unpredictable, and by definition a recession is a period of falling output, so obviously a recession will always be preceded by a period of growth, which ex post will look "unrestrained" and "unsustainable".

"Every economy can grow at a certain pace, and if you grow faster than your capacity, you generate inflation,” says Atilla Yesilada, a Turkish economic consultant based in Istanbul. “‘Erdogan-omics’ denies that."

This sounds very much like Friedman's 1968 paper on the role of monetary policy, where he pointed out that if monetary stimulus was used to try to push the economy above the level of output which would prevail in the absence of monetary nonneutralities, the result would be a rise in inflation. However, actually measuring this level of output in practice is impossible. If we assume specific functional forms for the production function and such we can look at, say, the output elasticity of wages to get an idea of if there's a lot of wage stickiness, but such work is going to be rather unreliable. As such, all "Erdoganomics" needs to do is say that you have no evidence that the growth is coming from some kind of monetary nonneutrality, and there's no argument you can make which will save your position.

"The International Monetary Fund and other economists say that Turkey should be growing at no more than 4 percent."

How on earth are the IMF and "other economists" competent to make such a judgment? What does "should" mean here? If the history of the IMF's own growth forecasts for Turkey is any guide, I think they should not be taken very seriously. The IMF was forecasting %2.4 RGDP growth for Turkey in its April 2017 World Economic Outlook report. The actual growth figure came out at %7.4, which the IMF then attributed to "stimulus" measures and whatnot. You can also look at the IMF's 2013 growth forecasts and compare them to what actually happened. When it comes to Turkey, the IMF appears to have a visible downward bias in its RGDP growth estimates; the only IMF estimate which was above target was the growth estimate for 2016, which is the year of the coup attempt in Turkey.

Of course the sample in this case is too small to be statistically significant evidence that the IMF has a bias in its estimates, but if their forecasts are this bad, I wouldn't trust them to clairvoyantly and expertly determine the "correct" growth rate for Turkish RGDP.  (Of course you can argue that the only reason the IMF is unable to predict what's going to happen is because of uncertainty introduced by the Turkish government, not because of technocratic incompetence, but then you would have to point to some tangible harm that is done by the Turkish policies the IMF did not expect which produce growth rates above their forecasts.)

"Much of Turkey’s economic expansion over the last few years has been a result of a building explosion financed by easy credit that conveniently goes to construction and development giants that happen to be Erdogan’s political allies, by providing state loan guarantees and other tools to ease loans."

"Easy credit", whatever it is, is certainly not some kind of magic bullet which allows Turkish construction workers to produce more than they were before at zero opportunity cost. Of course the politicization of credit leads to misallocation of capital, but in that case it should be an even better sign about the fundamentals of the Turkish economy that its growth rate is high despite these bad policies. Yes, you can argue that the government is indirectly subsidizing construction projects whose public returns are questionable, but you can't argue that this is behind Turkish RGDP growth - if anything, this should be a drag on growth. It looks like FP writers took the economics class on Keynes, but missed the one on opportunity cost.

There's plenty of legitimate criticism which can be made of Turkish economic policy, but amazingly FP writers manage to miss the mark on so many points regardless.

Wednesday, May 23, 2018

Correlation and Jensen's inequality - failures of naive forecasting

A common situation we often find ourselves in when looking at market prices of assets is that there's insufficient data available to infer unbiased market forecasts of variables we would like to know more about. For instance, knowing unbiased market forecasts of random variables \( X \) and \( Y \), we may want to know an unbiased forecast for the ratio \( X/Y \), or the log difference \( \log(X) - \log(Y) \). Alas, the first moments of \( X \) and \( Y \) are insufficient data to recover this information. In practice, the ratio \( \mathbb E[X]/\mathbb E[Y] \) is then taken as a market forecast for \( X/Y \), ignoring both the nonlinearity of the map \( x \to 1/x \) and the possibility of correlation between \( X \) and \( 1/Y \). A substantial amount of empirical work in modern finance has been devoted to showing that these naive forecasts are almost never accurate, even to some reasonable tolerance for error.

What are examples of naive forecasts? Here are some that may be familiar to the reader:

  • Uncovered interest parity. This is a naive forecast of exchange rate movements. The exact expression for the exchange rate movement in a given period is the ratio of domestic and foreign discount factors for that period, \( Q_d / Q_f \). The yield of a zero-coupon bond over this period is \( 1/\mathbb E(Q) \), so by looking at bond prices in both countries we may infer the unbiased market forecasts for \( Q_d \) and \( Q_f \). Uncovered interest parity then forecasts \( \mathbb E[Q_d]/\mathbb E[Q_f] \) to be unbiased for the ratio of the end-of-period exchange rate to the start-of-period exchange rate. This forecast turns out to fail spectacularly in the data. Equivalently, we may phrase it as trying to forecast the movement in log exchange rates, in which case the correct forecast is \( \mathbb E[\log(Q_d)] - \mathbb E[\log(Q_f)] \) whereas the UIP forecast takes the expectation inside the logarithm. This forecast is also a failure - the concavity of the logarithm turns out to significantly affect the outcome.
  • Inflation compensation on indexed vs nominal bonds. This gives a naive forecast of inflation, very much of the same character as above. Nominal and real discount factors are linked by the relation \( Q \Pi= m \), where \( Q, \Pi, m \) are the nominal discount factor, inflation, and the real discount factor respectively. The inflation compensation gives the forecast \( \mathbb E[m]/\mathbb E[Q] \) for \( \Pi \), whereas the unbiased forecast is given by \( \mathbb E[m/Q] \). There's again a logarithmic version which should be obvious.
  • Futures prices. A futures contract indexed to a quantity \( X \) is just an asset whose payoff is equal to \( X \) itself (one can lever the contract up or down, but this doesn't affect the naive forecast when it is computed in a suitable way, it remains identical). The market forecast for \( X \) is then taken to be the price of the futures contract adjusted for the time value of money, i.e \( \mathbb E[QX]/\mathbb E[Q] \). As above, the correct unbiased forecast is \( \mathbb E[X] = \mathbb E[QX/Q] \).
Often, the difficulty of salvaging the naive forecast is the fact that the error term in it turns out to be related to the covariance between the discount factors \( Q, m \) and some other variables, and the Cauchy-Schwarz bound on the covariances (bounding in absolute value by the product of standard deviations) is too weak, since discount factors are highly volatile - the annual standard deviation of any discount factor pricing all US assets is estimated to be at least \( 0.5 \) (Hansen-Jagannathan bound), for instance. Taking the last bullet point above, we can only bound the error of the naive annual forecast (in the best case scenario where \( \sigma_Q \) is indeed around \( 0.5 \)) by around \( 0.5 \sigma_X / \mathbb E[Q] \approx 0.5 \sigma_X \) by empirical data on \( \mathbb E[Q] \) being fairly close to \( 1 \). When \( X \) itself is a highly volatile quantity, this error can be quite large. For example, real exchange rate growth across countries has an annual standard deviation of around \( 0.15 \), and international risks tend to be highly correlated as measured in the correlation of discount factors across countries, so our upper bound in this case is equal to \( 0.075 \) - that's a huge standard deviation, making forward rates as a forecast of where spot rates are going to go essentially useless.

The difficulty here is that there's no way of getting reliable forecasts out of markets as long as one can't ensure that the variable being forecasted has very small (in absolute value) correlation with the discount factor \( Q \). This is implied by the equivalence theorem between irrational and rational expectations - since any \( L^2 \)-continuous monotone linear functional can be written in the form \( X \to \mathbb E[QX] \) for some random variable \( Q \), the discount factor formulation can describe any pricing result whatsoever so long as basic arbitrage relations hold. Irrational probability estimates and rational estimates but different investor preferences are therefore indistinguishable from asset price data. Of course, if one had accurate knowledge about \( Q \) itself (even having an upper bound on its standard deviation would be nice, since then a lot of irrational probability estimates would be ruled out) one could say more, but so far \( Q \) remains a nebulous object beyond some basic facts we know about it, so market forecasts should always be taken with a grain of salt.

In some sense, the rational expectations view is depressing - the market contains all of this information and the information is indeed used efficiently for risk-sharing and resource allocation, but there's no way to ask the market for its forecast of almost any variable we care about. Ironically enough, the only variable we can easily obtain an unbiased forecast for is \( Q \) itself.

Sunday, May 20, 2018

Exchange rates

There's a lot of confusion about exchange rates in economic commentary in general. Comments such as "the exchange rate of a country measures how healthy its economy is" abound, and even though some basic economics is enough to refute this claim, the exact relationship between exchange rates and other macroeconomic variables like output, consumption and bond yields remains nebulous. In this post, I will mostly examine the asset pricing side of the exchange rate question using fairly standard methods in the field, and try to provide a basic framework for understanding exchange rates.

The most simple model of exchange rates was given in a perfect foresight environment. It is based on the following idea: if the risk-free bond yield paths of two countries over time (we will be working with discrete time in this post) \( R_d(t) \) and \( R_f(t) \) are known, then at time \( t \) one can short a bond of the domestic country \( d \) and long a bond of the foreign country \( f \), or do the arbitrage in reverse. The no-arbitrage condition then implies that the net return on this investment should be zero. Denoting the value of \( 1 \) unit of domestic currency measured in foreign currency by \( E(t) \), we get

\[ R_f(t) - \frac{E(t+1)}{E(t)} R_d(t) = 0 \]

This relationship is called uncovered interest parity. Writing it in terms of the log returns and exchange rate (lowercase letters denote logs), it takes the prettier form

\[ r_f(t) - r_d(t) =  e(t+1) - e(t) \]

This model makes a simple prediction: interest rate spreads between domestic and foreign bonds should forecast exchange rate movements. However, it makes a prediction which runs afoul of conventional wisdom: the currency of the domestic country should appreciate when its domestic bond yields are low. Here is what this looks like on a simple graph, assuming \( e(0) = 0 \) and a sinusoidal wave for the interest rate spread: (blue is the interest rate spread \( r_d - r_f \), purple is the exchange rate)

You can see that this is not the picture we want. The exchange rate is appreciating at the start when the bond yields of the domestic country are falling sharply.

What's the answer? Are we to conclude that the uncovered interest parity model is nonsense? Are exchange rates driven by "investor mood", "sentiment", or "animal spirits"? Or do we conclude that conventional wisdom is wrong?

The answer is, of course, none of the above. Our naive model has the strong assumption of perfect foresight. What if we allow for uncertainty, and weaken this assumption to one of rational expectations? In that case, the exchange rate will be a stochastic process, and asset pricing in the two countries will be driven by domestic and foreign discount factor processes \( \Lambda_d(t) \) and \( \Lambda_f(t) \). For now, assume that markets are complete, i.e all contingent claims are traded. This implies that the discount factors are unique, so it's safe to talk about the discount factor for both of our countries. (They are not equal because trade frictions prevent risks from being perfectly insured across countries.) In this case, we can check that both \( \Lambda_d(t+1)/\Lambda_d(t) \) and \( \Lambda_f(t+1)/\Lambda_f(t) E(t+1)/E(t) \) are discount factors pricing all assets in terms of the domestic currency by the law of one price, so they must be equal. Taking logarithms, this reduces to

\[ \Delta \lambda_d(t) - \Delta \lambda_f(t) = \Delta e(t) \]

Note that in the no uncertainty case we have \( \Delta \lambda_d(t) = -r_d \) and the same for the foreign country, so this relation generalizes uncovered interest parity. It is not a different model, it is a generalization of the same arbitrage idea to a stochastic environment.

Now, let's take the empirical regularity that exchange rates and bond yield spreads tend to move together and in the same direction. How can we produce this result? Well, the gross bond yield of a country \( c \) in its own currency is given by

\[ I_c(t) = \frac{1}{\mathbb E_t[\exp(\Delta \lambda_c(t))]} \]

whereas the exchange rate \( e(t) \) is equal to

\[ e(t) = e(0) + (\lambda_d(t) - \lambda_d(0)) - (\lambda_f(t) - \lambda_f(0)) \]

Phrased in terms of the discount factors, then, we want \( \mathbb E[\exp(\Delta \lambda_d(t))] \) to be low when \( \lambda_d(t) \) is high. We can already see that not every choice of \( \lambda_d(t) \) satisfies this requirement - our deterministic choice above is one such example. However, we can interpret this condition as a "mean reversion" condition: when the discount factor gets too high, there should be some force pulling it back down again, so that we expect it to fall more than it would otherwise. Indeed, some condition of this sort is necessary and sufficient to produce the patterns we see in the data. A simple, slow-moving stochastic process with this property is an AR(1): we define recursively

\[ \lambda_d(t+1) - s = \beta (\lambda_d(t) - s) + \varepsilon_d(t+1) \]
\[ \lambda_f(t+1) - s = \beta (\lambda_f(t) - s) + \varepsilon_f(t+1) \]

where \( s \) is the steady state value of the discount factor, the \( \varepsilon(t) \) are iid normal shocks with zero mean and \( 0 < \beta < 1 \). Here is the Wolfram Mathematica code for a simulation:

b = 0.99
v = 0.0004
s = 0.00011
s1 = RandomVariate[NormalDistribution[0, v], 300]
s2 = RandomVariate[NormalDistribution[0, v], 300]
m1 = FoldList[s + b*(#1 - s) + #2 &, s, s1]
m2 = FoldList[s + b*(#1 - s) + #2 &, s, s2]
ex = m1 - m2
i1 = 1/Exp[365*((b - 1)*(m1 - s) +  v/2)]
i2 = 1/Exp[365*((b - 1)*(m2 - s) + v/2)]
idf = (1 + i1)/(1 + i2) - 1
ListLinePlot[{ex, idf}]

and here is what the resulting picture looks like: (purple is the log bond yield spread, blue is the log exchange rate)

By assuming the time series structure of an AR(1) for the discount factors, we can produce the conventional wisdom result that bond yield spreads and exchange rates move together, and in the same direction. However, this has some implications: it's much more subtle than "higher interest rates prop up the currency". Indeed, our model is a generalization of uncovered interest parity. If we carried out a policy experiment of raising the mean bond yield of a country after some date, we would get the unconventional result that it would result in a currency depreciation. This is an example of the Lucas critique: just like Lucas' information-based Phillips curve said inflation only boosts output if it's unexpected, our model says that the conventional wisdom only works when movements in the discount factor are expected to be temporary - this is what the discounting factor \( 0 < \beta < 1 \) does in our model. Stochastic movements in the discount factor are not the same as an expected change in the distribution. So, for example, while our model is capable of fitting the !data and producing the conventional sign, it does not imply that a central bank can appreciate its currency by raising nominal interest rates (not unless its policy rule is seen by market participants to have the AR(1) structure for the nominal discount factor that we laid out above.)

So far, we have only done asset pricing. To relate this to macroeconomics, we need the usual form of the stochastic discount factor process relating it to the consumer's decision problem:

\[ \Lambda_{\textrm{nominal}}(t) = \frac{\beta^t U'(c(t))}{P(t)}, \, \Lambda_{\textrm{real}}(t) = \beta^t  U'(c(t)) \]

where \( U \) is any consumer's utility function, \( \beta \) is a time discounting factor (this is not the same \( \beta \) as in the above model for the discount factor) and \( P(t) \) is the price level at time \( t \). With this form, we can get more results that run afoul of conventional wisdom on exchange rates. Recalling the basic relation \( \Delta \lambda_d(t) - \Delta \lambda_f(t) = \Delta e(t) \), the law of diminishing marginal utility implies that when consumption is growing faster, \( \Delta \lambda_d(t) \) is smaller, so the exchange rate is depreciating faster! So much for real currency depreciation being a bad thing. This conclusion holds as long as the basic model linking the discount factor to the consumer's decision problem is accurate, and since this model is used in pretty much every modern macroeconomic model, it's fair to say that it has an equal amount of legitimacy.

If markets are incomplete, then the basic relation \( \Delta \lambda_d(t) - \Delta \lambda_f(t) = \Delta e(t) \) still holds for log discount factors having minimal variance, so a theory of exchange rates can be recovered. However, in that case the link to macroeconomic variables such as output and consumption is much harder to establish.

I may write a follow-up post to this one in the future, where I talk more about the international trade side of the coin.