Doing the Math: The Fiscal Multiplier Effect of the 2009 American Recovery and Reinvestment Act

Fiscal stimulus is everything I always dreamed it would be.

Any regular reader of this blog knows my opinion of fiscal stimulus and fiscal multipliers. I have shown evidence (here and here) that fiscal multipliers must be below 1 and are likely closer to zero or even negative. Pushing against this belief is the recent performance of the economy. GDP grew by a very healthy 3.5 percent in the third quarter, boosted by gains ranging from private consumption, to residential investment, to direct government expenditures. According to the Vice President, the increase in GDP is entirely attributable to the stimulus efforts by the administration.

I am inclined to agree.

I believe that in the absence of government stimulus the U.S. economy would have continued to contract in the third quarter. What’s more, I say this without changing my views on the multiplier. How is that possible? Let’s do some math.

The following table shows the GDP growth that would have occurred in the absence of fiscal stimulus under different assumptions for what counts as government stimulus using a multiplier of 1 (my maximum) and a multiplier of 3 (Romer’s base case). Because there is considerable uncertainty over the timing of the stimulus, I show the four-quarter change in GDP through the third quarter. Over this time period, GDP fell by 2.3 percent. The numbers in the table show the four-quarter change without stimulus and should all be viewed relative to the 2.3 percent fall.

The first row of the table assumes that the sum total of fiscal stimulus is the pay out from the ARRA. According to data from Recovery.gov, as of October 29, the government had actually spent $173 billion (this number includes tax relief and spending). This is the most conservative estimate of stimulus spent to date. (Romer would include both actual spending and money allocated ($310 billion). I agree with her but want to use the smallest number to start. My numbers get bigger fast anyway.) Assuming a multiplier of 1 ($173 billion spent adds $173 billion to GDP), counterfactual GDP growth is -3.6 percent. With a multiplier of 3, the counterfactual falls to -6.2 percent.

I find both of these numbers credible.

I actually believe GDP would have fallen more than 6 percent in the absence of the programs. And, if I believed that total stimulus was $173 billion, I would also have to join Romer in the multiplier is greater than 3 world. Fortunately for me (I am not the introspective type), the total amount of stimulus is much greater than $173 billion.

There are lots of ways of accounting for government stimulus, but the most time honored and method (used by the IMF, the OECD, and the Federal Reserve) is to simply examine the change in the government balance. Over the last four quarters, publically held debt rose by $1,732 billion. (Really, we should only count the increase in the growth rate; but over the previous four quarters, debt only increased by $280 billion.) This number is the correct measure of stimulus. It includes all Federal government spending and all implicit reductions in the tax burden. The number is still conservative because it does not include the increase in state borrowing, an additional $50 to $200 billion.

Using this number and a multiplier of 1, yields a counterfactual GDP growth of -15.4 percent. With a multiplier of 3, the number falls to -41.6 percent. These numbers are beyond the pale. GDP never fell by more than 40 percent in the Great Depression. Using this number and my belief of a counterfactual fall in output of 10 percent, the current multiplier is negative.

But, there is also the issue of the Fed’s balance sheet. The Fed has pumped almost $1,000 billion into the economy over the same period. This is measured by the expansion of the Fed’s balance sheet. There is no difference between receiving a tax break for $1 trillion dollars and receiving a $1 trillion dollars in cash from the Fed. We can argue about effectiveness but that is the exercise here.

Adding the Fed’s balance sheet expansion to the calculation, yields a counterfactual GDP growth of -22.4 percent. With a multiplier of 3, we have the absurd number of -62.5 percent. I believe that latter number would be a modern-era record. I suspect we would have to go back to the plague years in Europe to find an equivalent fall. I would love to see Summers or Romer stand up and make the case for this counterfactual.

So like many Banana Republics before us, we have managed to spend enough to turn GDP growth positive. But, the cost of achieving this number has been phenomenal. To achieve a paltry $212 billion increase in real GDP, we spent about $2 trillion dollars. This gives us a net return of 20 cents on the dollar.

Yes, GDP would have fallen without the spending. The probability that this recession would have scored as a depression in the absence of stimulus is high.

Was it worth it?

Housing Tax Credit: Did it boost the housing market?

The $8000 first time home buyers tax credit seems to have boosted the sales of housing. The following graphic gives my estimates of the total impact on the value of homes sold since the passage of the bill in January.

Including both new and existing homes and valuing the sales at their median price, total housing sales increased by about $16 billion through September. It is, of course, very difficult to measure how much of the increase is attributable to a natural increase in demand and how much is attributable to an increase in demand derived from the tax credit alone. I estimate two distinct effects: the direct effect of the extra money pouring into the housing market and the indirect effect from the induced change in prices.
In normal times according to the National Association of Realtors, about forty percent of sales go to first time home buyers. The tax credit likely increased this number. I don’t know how much but I assume 50 percent is a conservative number. Under this assumption, total outlays under the tax credit were $14.1 billion through the end of September, slightly more than the CBO’s scoring of the program ($11 billion) and slightly less than NAR’s estimate ($15.2 billion).

I assume that the $14.1 billion has a direct one-for-one impact on the demand for housing. This impact is shown in the figure as the difference between the solid black and the dashed black lines. The estimated impact has grown through the year. The sharp rise between April and June reflects ordinary seasonal fluctuations in demand. These data are not seasonally adjusted.

In addition to the direct effect, the subsidy has an indirect effect through induced price changes. Given an estimated price elasticity of demand, the subsidy pushed average house prices up by about 4 percent (My estimate is just below that of Goldman Sachs who estimated a little more than 5 percent. They must have estimated a slightly higher demand elasticity.) This price effect induces sales of existing homes. Because of the higher price, existing home owners have an incentive to sell their house and either rent (less likely) or buy a new home (more likely). This impact is small relative to the direct effect and is shown as the difference between the dashed red and dashed black lines.

In total, I estimate that the existence of the subsidy has boosted the value of housing sales between January and September by about $17.3 billion.

There is no question that this tax break helped the housing market. An extra $17 billion likely kept some home builders in business and it must have paid the bills of quite a few real estate agents. Whether or not the program was worthwhile depends on the public policy decision of whether or not we want to subsidy the housing market. There are no macro side effects of the program as designed.

Whether or not the tax credit is extended (and it looks like this is a done deal), the housing market is likely to decline in the coming months. Because the program was expected to expire this month, most households eligible for the program and able to buy a house have already taken advantage of the tax credit—see the 9 percent surge in existing home sales in September. The majority of these households would have bought a house sometime in the next year without the subsidy and the tax credit simply changed the timing of their decision. With the extension, I expect the value of sales to fall by about $9 billion over the next several months and to fall by the remainder as the credit is phased out.

Of course congress looking ahead to midterm elections hopes that underlying demand for housing will surge by the time the credit expires, disguising the tax credit induced slump. It is possible but we are going to have to see a more robust labor-market recovery before this can happen.

Estimated Taylor Rules: A Good Fit for Monetary Policy?

Several economists have recently used the Taylor rule to comment on the appropriateness of monetary policy (Rudebusch, Calculated Risk, Krugman, Altig). In some form or another, they regress the Fed Funds rate on a constant, the lagged policy rate, lagged inflation, and some measure of economic slack. They find that the fit of this equation over the last 20 years has been quite good, and that therefore, monetary policy has been appropriate.

In fact, the fit of these lines is so good that I have become a bit suspicious over the exercise. I decided to do two things: 1) I would extend the forecast back to the early 1950s and 2) I would drop any measure of economic slack.

The reason for the first is obvious: The Fed seems to have made systematic policy mistakes during the 1970s after performing admirably in the 1960s. Then, if the Taylor rule is to be a decent guide as to the appropriateness of policy, it had better deviate systematically in the 70s.

The reason for the second is both econometric and philosophical. If the lagged policy rate responds to economic slack and slack evolves on slowly, the system is over-identified: we are more or less estimating an identity. Philosophically, economic slack is impossible to measure. We have no idea how to measure it or how this measure would relate to inflation. (I know: there are a thousand ways to measure slack. I am just saying none of them are meaningful.)

The result of the exercise is shown below. I too was stunned.

The estimated Fed funds rate lies almost exactly on top of the effective funds rate, shown as a dashed line. Either this is a meaningless measure of policy or monetary policy has been equally appropriate from 1956 through 2009. I choose the former as the more reasonable explanation.

I am not saying monetary policy has been bad of late. I am saying that using a Taylor rule to evaluate policy is meaningless. (Not the theory behind the Taylor rule; just the practice of estimating such a rule.)

What gives?: The Taylor Rule is an Identity

The Taylor rule is really something of an economic identity rather than a model of economic decision making. In effect, estimating a Taylor rule is no different than estimating the national income identity and finding out that Y really is equal to the sum of its parts.

To see this, use the old quantity theory equation:

P = v(φ)*M/Y

Prices are a function of Money (M), the quantity of goods in the economy (Y), and perhaps some measure of the current state of the economy (v(φ)), where the variable φ simply stands in for the current state of everything not written explicitly.

This equation is old fashioned but is fundamentally an identity: more money chasing fewer goods leads to higher prices.

We can transform the identity by taking logs and then first difference the equation.

dln(P) = dln(v(φ)) + dln(M) – dln(Y)

Then rearrange the equation to put money on the left hand side:

dln(M) = dln(P) + dln(Y) – dln(v(φ))

This is the Taylor rule.

Wait! We started with an identity (of sorts) and we ended with a Taylor rule. Logic then insists that the Taylor rule is an identity.

Okay, I hear you. The Fed does not target the money supply; the Fed targets a nominal short-term interest rate and the Taylor rule uses a short-term interest rate not the supply of money.

To see the link, how does the Fed enforce its nominal target. The Fed buys or sells bonds until short bonds are trading at the desired price. What does it use to buy and sell bonds? Money, of course.

So, the money supply is determined by the desired interest rate. The Fed, in effect, adjusts the money supply such that the short-term bond market is in equilibrium at the desired policy rate. M = f(r) and r = G(M). Where G is the inverse function of f.

Then, we have finally derived the Taylor rule.

dln(f(r)) = dln(P) + dln(Y) – dln(v(φ))

Finally, it is a small step to transform the above equation to the more familiar:

r_t = αr_t-1 + ϐ₁Inf_t + ϐ₂Growth Rate of Output_t – ϐ₃Potential Something_t

The coefficients in the equation can be estimated to minimize the information loss in the last transformation. The final term simply reinterprets our state variable. Recall, v(φ) was just some term reflecting the economic environment. In my specification, I set this coefficient to zero. In a standard specification, the restriction sets this to some measure of potential output. We don’t have to argue over signs or magnitudes because we may freely estimate the coefficients.

The Taylor rule fits because it is an identity.

The Quantity Theory

For those of you who don’t believe in money or at least who don’t believe in the link between inflation and money. Take a look at the following picture. The graph shows the five-year change in M2 divided by output against the five year cumulative change in the CPI. If M/Y rises rapidly, so do prices.

NorthGG asked if the relationship is between core or headline. I would say: yes. The five-year changes should completely eliminate the influence of high-frequency changes in food or energy. Only prolonged increases in either food or energy prices will show through. And, prolonged increases are also known as inflation.

Is High Inflation Likely?

The Statement

In his blog, Krugman dismisses the possibility of inflation. He goes farther and calls the Fed irresponsible for even considering the possibility of inflation. Krugman’s analysis is completely misleading over the prospects of inflation. He is using a backwards looking indicator that ignores changes in current policy. I too believe that high inflation outcomes are likely avoidable. Krugman seems to believe high inflation will be avoided even if the Fed leaves rates at zero forever. I believe that positive action (tighter policy) on the part of the Fed will be necessary to avoid inflation.

The Method

To see the difference in our beliefs, let’s work within Krugman’s framework. Krugman proposes the Taylor rule as his model of monetary policy and uses the parameters estimated by Glen Rudebusch at the SF Fed to judge the current, appropriate policy stance. Krugman uses the following Taylor rule:

Target fed funds rate = 2.07 + 1.28 x inflation - 1.95 x excess unemployment

Rudebusch’s weight on the unemployment gap is much higher than most other estimates. Taylor called for a coefficient of 0.5 on excess employment relative to a coefficient of 1.0 on inflation, implying the Fed should care twice as much about inflation as unemployment. Krugman believes the Fed should only care about 2/3 as much about inflation as the unemployment gap. The change in weights is quite significant.

In Krugman’s specification, with current inflation at -.02 percent (core PCE, 4-quarter change), the first two terms imply a Fed funds rate of positive 2.1 percent. The unemployment gap is then driving the current negative policy rate. With current unemployment around 9.8 percent (it was only 9.3 percent in Q2) and using the CBO’s pre-recession estimates of the NAIRU, the implied policy rate is very, very negative; indeed, much more negative than Krugman reports, negative 7.7 percent. If we had used more standard weights and a constant of 2, the implied policy rate is just barely negative, -0.5 percent.

But, to be honest, I am with Krugman and don’t see any justification for Taylor’s weights. The Taylor rule is not a model of the policy rate but was rather designed as a descriptive rule for understanding policy setting. Given this view, we are free to estimate rates and if I estimate the Taylor rule I arrive at coefficients closer to Glen’s than to Taylor’s.

The Mistake

If you think the Taylor rule was a good guide to policy in the past, the Fed shouldn’t start to raise rates until the rule starts, you know, yielding a positive number.

The first part of the quote is wrong and the second part puzzling.

The Taylor rule was not a good guide to policy in the past: The Taylor rule had been a good description of past policy. The two statements are not equivalent.

The Taylor rule is essentially a linearized equation from a specific model of Fed policy and inflation and resource slack. The Taylor rule does not describe a fundamental relationship between policy, output and inflation. There are many models of output and inflation.

In particular, we can modify Krugman’s Taylor rule to make it forward looking. An optimally behaving Fed will set policy not based on the past behavior of variables but rather on their forward-looking expectations.

Money Matters

For some reason many Fed officials seem to view it as inherently unsound to stay at a zero rate for several years running — but I’m at a loss to understand what model, or even conceptual framework, leads them to that conclusion.

Maybe some of the Fed officials remember the adage ardently espoused by Friedman: Inflation is always and everywhere a monetary phenomena.

Krugman, along with the rest of the Fresh Water economists, seem to have completely forgotten about the link between money and inflation. Lucas, Freidman, Smith, Hume, and yes even Keynes believed in a link between the quantity of money and inflation. Lots of money: lots of inflation.

Every scholar who has ever seriously examined the relationship between inflation and money has found a positive relationship. Lucas found a positive relationship using both U.S. and international data. Friedman found the same in 1960s for data sets running from the mid 1880s to the early 1960s. Both Hume and Smith believed in a positive relationship between money and inflation, although there use of data was more anecdotal and conjectural than rigorous.

The most recent study of inflation and money, of which I am aware, was presented last Thursday at a Federal Reserve conference. “Money and Inflation,” by Bennet McCallum and Edward Nelson, finds a consistent positive relationship between money supply and inflation: money raises inflation about 1-for-1 with a two year lag.

The Fed has increased the money supply substantially over the last two years. According to the Federal Reserve’s H.6 release, M1 has increased 18.6 percent over the past twelve months, while the broader M2 has risen 7.8 percent. These are extremely high growth rates. According to the work of McCallum and Nelson, this growth rate will lead to inflation one to two years from now.

If we replace Krugman’s backward-looking PCE with a reasonable forward looking expectation of inflation driven by the increase in the money supply, we find that between one and two years from now the policy rate had better be above 2 percent. Further, if we believe the results of McCallum and Nelson, the policy rate probably needs to start increasing now. That is, the money supply has to be reduced now to avoid the high inflation in the future.

This is the model policy makers likely have in mind when they call for tighter policy.

I don’t think there is any particular rush to raise rates. I think the Fed can afford to be patient and watch the data.

Conclusion

Krugman and I agree: There is very little likelihood of high inflation. Krugman believes in the Taylor rule and so a passive Fed can achieve this outcome. I believe in money and so an active Fed will achieve this outcome.

Krugman’s own Taylor series framework implies high inflation within one to two years, if the Fed is passive. Fortunately, the Fed is not passive. The Fed has the power to control inflation. It simply has to withdraw the liquidity in a timely manner.

Gold Bugs, Exchange Rates, and Monetary Policy

Beware the dollar hawks says Krugman in a recent post to his blog. The dollar is depreciating quickly and many (Krugman’s many I have not fact checked) are calling for somebody to do something about it. Krugman believes there is a danger in this call. And for once, I am in unambiguous agreement with him.

Using monetary policy to control the value of the dollar would be a policy mistake.

The Fed is already trying to do too much with a single policy tool. Adding yet another criterion to their already long list is too likely to lead to policy mistakes. The Fed needs to keep its eyes on the balls already in the air and not add balls to impress like a foolish street juggler.

But, at the same time, the Fed should not disregard the exchange rate as a signal of overly loose policy. The exchange rate is perhaps the best, broadest, and most flexible dollar denominated price. A depreciation of the dollar is inflation—the dollar price of foreign goods goes up. It reflects the average relative price of dollar goods. Amongst their other price signals, the Fed should monitor the exchange rate. The value of this mechanism as a price signal ultimately lies behind the current dispute amongst various members of the FOMC, no different than the debate 4 years ago on the value of the Cleveland Fed’s median inflation rate.

The danger of the Fed ignoring these signals is exceptionally high at the moment. There is a group of economists (Roubini talks about this all the time.) that have been looking for a decline in the real value of the dollar for a long time. They view the real value of the dollar as an equilibrating mechanism to adjust the pattern of global demand. If the Fed shares these beliefs, they may confuse a nominal movement in the dollar with the long-looked-for real depreciation.

Remember, any price has two components, real and nominal. The real component reflects an equilibrium between supply and demand (of and for the good). The nominal value reflects and equilibrium between the good and money. The Fed controls the latter (to an extent) and never the former.

To tie it all together, the Fed should be careful not to confuse the real and nominal value of the dollar. From their perspective, unexplained movements in the dollar are probably nominal.

Watch the value of the dollar but don’t target it.