Excess reserves and inflation risk: A model

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Note: The following is an edited version of my original post. Thanks to Nick Edmonds for pointing out an inconsistency in my earlier analysis. Nick's comment forced me to think through the properties of my model more carefully. In light of his observation, I have modified the original model to include capital investment. My earlier conclusions remain unchanged. 
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I should have known better than to reason from accounting identities. But that's basically what I did in my last post and Nick Rowe called me out on it here. So I decided to go back and think through the exercise I had in mind using a simple model economy.

Consider a simple OLG model, with 2-period-lived agents. The young are endowed with output, y. Let N denote the number of young agents (normalize N=1). The young care only about consumption when they are old (hence, they save all their income y when young). Agents are risk-averse, with expected utility function E[u(c)]. There is a storage technology. If a young agent saves k units of output when young, he gets x*f(k) units of output in the next period, where x is a productivity parameter and f(.) is an increasing and strictly concave function (there are diminishing returns to capital accumulation). Assume that capital depreciates fully after it is used in production.

If x*f'(y) > 1, the economy is dynamically efficient. If x*f'(y) < 1, the economy is dynamically inefficient (and there is a welfare-enhancing role for government debt).

Now, imagine that there are two such economies, each in a separate location. Moreover, suppose that a known fraction 0 < s < 1 of young agents from each location migrate to the "foreign" location. The identity of who migrates is not known beforehand, so there is idiosyncratic risk, but no aggregate risk.

Next, assume that there are two other assets, money and bonds, both issued by the government supply (and endowed to the initial old). Let M be the supply of money, and let B denote the supply of bonds. Let D denote the total supply of nominal government debt:

[1] D = M + B

Money is a perpetuity that pays zero nominal interest. Bonds are one-period risk-free claims to money. (Once the bonds pay off, the government just re-issues a new bond offering B to suck cash back out of the system.) Assume that the government keeps D constant maintains a fixed bond/money ratio z = B/M, so that [1] can be written as:

[2] D = (1+z)*M

In what follows, I will keep D constant throughout and consider the effect of changing z (once and for all). Note, I am comparing steady-states here. Also, since D and M remain constant over time, and since there is no real growth in this economy, I anticipate that the steady state inflation rate will be equal to zero.

Let R denote the gross nominal interest rate (also the real interest rate, since inflation is zero). Assume that the government finances the carrying cost of its interest-bearing debt with a lump-sum tax,

[3] T = (R-1)*B

The difference between money and bonds is that bonds (or intermediated claims to bonds) cannot be transported across locations. Only money is transportable. The effect of this assumption is to impose a cash-in-advance constraint (CIA) on the young agents who move across locations. (Hence, we can interpret the relocation shock as an idiosyncratic liquidity shock).

Young agents are confronted with a portfolio allocation problem. Let P denote the price level. Since the young do not consume, they save their entire nominal income, P*y. Savings can be allocated to money, bonds, or capital,

[4] P*y = M + B + P*k

There is a trade off here: money is more liquid, but bonds and capital (generally) pay a higher return. The portfolio choice must be made before the young realize their liquidity shock.

Because there is idiosyncratic liquidity risk, the young can be made better off by pooling arrangement that we can interpret as a bank. The bank issues interest-bearing liabilities, redeemable for cash on demand. It uses these liabilities to finance its assets, M+B+P*k. Interest is  only paid on bank liabilities that are left to mature into the next period. (The demandable nature of the debt can be motivated by assuming that the idiosyncratic shock is private information. It is straightforward to show that truth-telling here in incentive-compatible.)

Let me describe how things work here. Consider one of the locations. It will consist of two types of old agents: domestics and foreigners. The old foreigners use cash to buy output from the domestic young agents. The old domestics use banknotes to purchase output from the young domestics (the portion of the banknotes that turn into cash as the bond matures). The remaining banknotes can be redeemed for a share of the output produced by the maturing capital project. The old domestic agents must also pay a lump-sum tax.

As for the young in a given location, they accumulate cash equal to the sales of output to the old. After paying their taxes, the old collectively have cash balances equal to D. The young deposit this cash in their bank. The bank holds some cash back as reserves M and uses the rest to purchase newly-issued bonds B. The bank also uses some of its banknotes to purchase output P*k from the young workers, which the bank invests. At the end of this operation, the bank has assets M+B+P*k and a corresponding set of (demandable) liabilities. The broad money supply in this model is equal to M1 = M+B+P*k. The nominal GDP is given by NGDP = P*y + P*x*f(k).

Formally, I model the bank as a coalition of young agents. The coalition maximizes the expected utility of a representative member:  (1-s)*u(c1) + s*u(c2), where c1 is consumption in the domestic location and c2 is consumption in the foreign location. The maximization above is constrained by condition [4] which, expressed in real terms, can be stated as:

[5] y = m + b + k

where m = M/P and b = B/P (real money and bond holdings, respectively).

In addition, there is a budget constraint:

[6] (1-s)*c1 + s*c2 = x*f(k) + R*b + m - t 

where t = T/P (see condition [3]).

Finally, there is the "cash-in-advance" (CIA) constraint:

[7] s*c2 <= m

Note: the CIA constraint represents the "cash reserves" the bank has to set aside to meet expected redemptions. Because there is no aggregate risk here, the aggregate withdrawal amount is perfectly forecastable. This constraint may or may not bind. It will bind if the nominal interest rate is positive (i.e., R > 1). More generally, it will bind if the rate of return on bonds exceeds the rate of return on reserves. If the constraint is slack, I will say that the bank is holding "excess reserves." (with apologies to Nick Rowe).

Optimality Conditions

Because bonds and capital are risk-free and equally illiquid, they must earn the same real rate of return:

[8] R = xf'(k)

The bank constructs its asset portfolio to equate the return-adjusted marginal utility of consumption across locations:

[9] R*u'(c1) = u'(c2)

Invoking the government budget constraint [3], the bank's budget constraint [6], reduces to:

[8] (1-s)*c1 + s*c2 = x*f(k) + b + m 

In equilibrium,

[9] m = M/P and b = B/P

We also have the bank's budget constraint [4]:

[10] y = m + b + k

Because the  monetary authority is targeting a bond/money ratio z, we can use [2] to rewrite the bank's budget constraints [8] and [10] as:

[11]  (1-s)*c1 + s*c2 = x*f(k) + (1+z)*m 

[12] y = (1+z)*m + k

Finally, we have the CIA constraint [7]. There are now two cases to consider.

Case 1: CIA constraint binds (R > 1).

This case occurs for high values of x. That is, when the expected return to capital spending is high. In this case, the CIA constraint [7] binds, so that s*c2 = m or, using [12],

[13] m = (y - k)/(1+z)

Condition [11] then becomes (1-s)*c1 = xf(k) + z*m. Again, using [12], we can rewrite this as:

[14] (1-s)*c1 = x*f(k) + A(z)*(y - k)

where A(z) = z/(1+z) is an increasing function of z. Combining [8], [9], [13] and [14], we are left with an expression that determines the equilibrium level of capital spending as a function of parameters:

[15] x*f'(k)*u'( [x*f(k) + A(z)*(y-k)]/(1-s) ) = u'( (y-k)/(s*(1+z)) )

Now, consider a "loosening" of monetary policy (a decline in the bond/money ratio, z). The direct impact of this shock is to decrease c1 and increase c2. How must k move to rebalance condition [15]? The answer is that capital spending must increase. Note that since [8] holds, the effect of this "quantitative easing" program is to cause the nominal (and real) interest rate to decline (the marginal product of capital is decreasing in the size of the capital stock).

What is the effect of this QE program on the price-level? To answer this, refer to condition [4], but rewritten in the following way:

 [16] P = D/(y - k)

This is something I did not appreciate when I wrote my first post on this subject. That is, notice that the equilibrium price-level depends not on the quantity of base money, but rather, on the total stock of nominal government debt. In my original model (without capital spending), a shift in the composition of the D has no price-level effect (I erroneously reported that it did). In the current set up, a QE program (holding D fixed) has the effect of lowering the interest rate and expanding real capital spending. The real demand for government total government debt D/P must decline, which is to say, the price-level must rise.

[ Note: as a modeling choice, I decided to endogenize investment here. But one might alternatively have endogenized y (through a labor-leisure choice). One might also have modeled a non-trivial saving decision by assuming that the young derive utility from consumption when young and old. ]

Case 2: CIA constraint is slack (R = 1).

This case occurs when x is sufficiently small -- i.e., when the expected productivity of capital spending is diminished.  In this case, the equilibrium quantity of real money balances is indeterminate. All that is determined is the equilibrium quantity of real government debt d = m + b. Conditions [11] and [12] become:

[17]  (1-s)*c1 + s*c2 = x*f(k) + d 

[18] y = d + k

Condition [15] becomes:

[19] u'( [x*f(y - d) + d]/(1-s) ) = u'( d/s )

Actually, even more simply, from condition [8] we have xf'(k) = 1, which pins down k (note that k is independent of z). The real value of D is then given by d = y - k. [Added July 10, 2014]. 

Condition [19] determines the equilibrium real value of total government debt. The composition of this debt (z) is irrelevant -- this is a classic "liquidity trap" scenario where swaps of two assets that are perfect substitutes have no real or nominal effect. The equilibrium price-level in this case is determined by:

[20] P = D/d

A massive QE program in case (a decline in z, keeping D constant) simply induces banks to increase their demand for base money one-for-one with the increase in the supply of base money. (Nice Rowe would say that these are not "excess" reserves in the sense that they are the level of reserves desired by banks. He is correct in saying this.)

The question I originally asked was: do these excess reserves (as I have defined them) pose an inflationary threat when the economy returns to "normal?"

Inflationary Risk

Let us think of  "returning to normal" as an increase in x (a return of optimism) which induces the interest rate to R >1. In this case, we are back to case 1, but with a lower value for z. So yes, as illustrated in case 1, if z is to remain at this lower level, the price-level will be higher than it would otherwise be. This is the sense in which there is inflationary risk associated with "excess reserves" (in this model, at least).

Of course, in the model, there is a simple adjustment to monetary policy that would prevent the price-level from rising excessively. The Fed could just raise z (reverse the QE program).

In reality, reversing QE might not be enough. In the model above, I assumed that bonds were of very short duration. In reality, the average duration of the Fed's balance has been extended to about 10 years. What this means is that if interest rates spike up, the Fed is likely to suffer a capital loss on its portfolio. The implication is that it may not have enough assets to buy back all the reserves necessary to keep the price-level in check.

Alternatively, the Fed could increase the interest it pays on reserves. But in this case too, the question is how the interest charges are to be financed? If there is full support from the Treasury, then there is no problem. But if not, then the Fed will (effectively) have to print money (it would book a deferred asset) to finance interest on money. The effect of such a policy would be inflationary.

Finally, how is this related to bank-lending and private money creation? Well, in this model, where banks are assumed to intermediate all assets, broad money is given by M1 = D + P*k. We can eliminate P in this expression by using [16]:

[21] M1 = [ 1 + k/(y-k) ]*D

So when R > 1, reducing z has the effect of increasing capital spending and increasing M1. In the model, young agents want to "borrow" banknotes to finance additional investment spending. But it is not the increase in M1 that causes the price-level to rise. Instead, it is the reduction in the real demand for total government debt that causes the price-level to rise.

Likewise, in the case where R = 1 and then the economy returns to normal, the price-level pressure is coming from the portfolio substitution activity of economic agents: people want to dump their money and bonds in order to finance additional capital spending. The price-level rises as the demand for government securities falls. The fact that M1 is rising is incidental to this process.

Excess reserves and inflation risk

Dave Wheelock, my colleague at the St. Louis Fed, points me to this nice article: Repeat After Me: Banks Cannot and Do Not "Lend Out" Reserves (by Paul Sheard). I have noticed a few papers lately making the same general point. I thought I'd throw my own two cents worth in.

To begin, you have probably seen (or heard about) this scary picture (thank you, "Helicopter" Ben):

That's a picture of the U.S. monetary base--the liabilities of the Federal Reserve Bank consisting of either currency or currency-on-demand (held by private, not government agencies). The monetary base can be divided into two broad categories: [1] currency in circulation (currency held by the non-bank private sector); and [2] reserves (bank sector vault cash and credits in reserve accounts held at the Fed).

In light of the "explosion" of Fed money since 2008, it may seem surprising that inflation has averaged considerably less than the Fed's official 2% target:

A common explanation for this is that most of the new money created by the Fed is being held by banks as reserves. Banks would rather earn 25 basis points (IOER) than lend out their excess reserves.

The following diagram depicts the liability side of the Fed's balance sheet:

We see that currency in circulation has increased, but at a modest and steady pace. Most of the increase in base money (remember, green part not included in money base) consists of reserves. The inflation fear expressed by some rests on the question of what is likely to happen once the economy returns to "normal." Sooner or later, things are going to turn around and banks will want to lend out their excess reserves to earn a higher rate of return. What is going to happen when this tidal wave of money begins to circulate?

According to Paul Sheard, this line of thinking is all wrong. That is, while monetary policy may ultimately result in higher inflation (or not), if it does, it won't be through the "banks lending out their excess reserves" channel, as many seem to suggest.

To understand his point, let's begin with how the Fed actually creates money. The Fed is a bank. And like all banks, it buys (or lends against) high-interest assets, which it finances by issuing low-interest liabilities (profits are returned to the Treasury). When the Fed buys a security on the open market, it credits the seller's bank account with newly-issued electronic digits (reserves). Banks then have the option of redeeming their reserves for currency, an option they may exercise depending on their customers' demand for currency.

Now, individuals regularly make deposits and withdrawals of cash into and out of their bank accounts. The net flow of withdrawals minus deposits determines by how currency in circulation grows over time. Banks do not lend out their cash. When a bank makes a loan, it issues a deposit liability that is redeemable for cash on demand. The demand deposit liabilities can be used as a payment instrument (they constitute money, and are counted as part of a broader measure of money supply, e.g., M1). The key observation here is that the way currency enters the economy is through the net withdrawal activity of bank customers--it has nothing to dow with banks lending out their reserves.

Alright, so why is understanding all this important? Well, for one thing, it is an accurate description of the way money and banking actually works (as opposed to the traditional "money multiplier" story that is commonly told in undergraduate textbooks). It is the right place to start when thinking of policy questions.

In terms of thinking about the inflation risk associated with the size of the Fed's balance sheet, it guides us away from examining how bank lending (the money multiplier) may react to various shocks. Banks can try to lend out their reserves all they want (create new loans). But if the public is satisfied with their currency holdings, any money injected into the system in this manner would have no effect on bank sector reserves. Since it is bank customers that determine how much cash is withdrawn from reserves, we should instead think about the type of shocks that may potentially alter this redemption decision.

To begin, we have to think about a world in which the asset side of the Fed's balance sheet matters. In many macroeconomic models, it is implicitly assumed that the Fed has full support of the Treasury (e.g., lump-sum taxes can be used to drain the economy of excess money), so that the Fed balance sheet does not matter. We want to do away with that assumption. In this case, the only "money draining" tools available to the Fed are asset sales. That is, think about the asset side of the Fed's balance as a giant vacuum cleaner. The amount of power this vacuum has is related to the market value of the Fed's asset portfolio. Any shock that would significantly reduce the market value of the Fed's asset portfolio would be like having your vacuum cleaner malfunction (just when you needed it the most).  

So, what type of shock can we think about here that might lend credence to the idea that excess reserves pose an inflation threat? I'm not really sure, but maybe the story goes something like this. Suppose that inflation expectations suddenly become "unanchored." (for whatever reason, people expect higher inflation). Through the Fisher equation, we might expect a large increase in nominal interest rates. The spike in interest rates would imply a capital loss for the Fed. By how much? Consider this formula (an approximation):

1 ppt increase in interest rate = (average duration)% decline in asset price. 

The average duration of the Fed's asset portfolio is roughly 10 years. So a five percentage point increase in interest rates would induce a 50% decline in the value of the Fed's assets (actually, somewhat less than this, but you get the point).

Now, higher inflation expectations on the part of the public may induce people to want to hold more currency (in nominal terms--the demand for real money balances may decline). This may be what could trigger a mass wave of redemptions. As people start withdrawing cash from their bank accounts, the banks start redeeming their reserves for cash to meet their customers' demands. The spike in interest rates unplugs the Fed's vacuum cleaner -- people know that the Fed does not have the tools to buy back all of its reserve liabilities. The wave of redemptions proceeds unchecked, with the flood of currency generating an inflation that becomes a self-fulfilling prophesy.

Well, that's just a story. I'm not sure if it hangs together logically (I've never seen it modeled formally, though perhaps it has been?) And even if it has a logical foundation, I'm not sure how persuasive it is. I am curious to know what other story one might tell. However the story unfolds, it cannot be one of bank lending out their reserves. 

How far are we from trend?

I am always amazed at how well a log-linear trend line seems to fits real GDP (or per capita GDP) in the United States. Through a great depression, sandwiched by two world wars, secular changes in the relative importance of different sectors (agriculture, manufacturing, services), the baby boom, the increase in female labor force participation, etc. Through it all, the U.S. economy just seems to revert to the same log-linear trend. Maybe it was just a fluke. Whatever the case may be, that trend seems to have broken down since the great recession. We've all seen the diagram (the red line in the graph below -- thanks to my colleague Fernando Martin for the nice pictures).

The red line above plots the real GDP per capita (log scale) since 1955. The trend line is calculated over the sample period 1955-2007. The average growth rate is 2.2% per annum. You can see the big "output gap" emerging in 2008.

But given what we know about U.S. demographics--in particular the large rise, then fall in labor force participation, is the red line really the best way to look at things? A rough way to control for demographics is to consider the real GDP as a ratio of the labor force, instead of population. That is what the blue line does above. Once again, the trend line is computed for the same period 1955-2007. The average growth rate here is 1.5% per annum. What this tells us is that a lot of our measured growth over this period was due to nonstationary labor force behavior, and not just productivity. Oh, and the output gap near the end of the sample is considerably smaller. (It is too small to the extent that the recent decline in the labor force is attributable to "discouraged workers" who plan to return once conditions improve.)

Here is the same analysis for Canada:

The average growth rates are 1.7% (red) and 0.9% (blue), per annum. The blue line suggests that the Canadian economy is right on trend.
 
I think it's instructive to compare the U.S. to Canada. Consider, for example, the employment-to-population ratio for males aged 25-54:

There is a modest secular decline in the employment ratio in both countries.The Canadian employment ratio is currently not too far from its historical average, while the U.S. number still has some way to go. Of course, the great recession hit the U.S. much harder than Canada. Canada's great recession began in 1990 and it took about a decade for that economy to recover.

I began thinking about the similarity of the recent U.S. experience with Canada's earlier experience in this post: The Great Canadian Slump: Can it Happen in the U.S.? I followed up with a cross-country comparison of labor market behavior through the two episodes here: Employment Slumps in Canada and the U.S. Let me update the data there for males aged 25-54:

If the Canadian experience through the 1990s (red line) serves as a guide, then employment growth in the U.S. (blue line) will remain in recovery mode for another 4 or 5 years. While much of the gap has already been filled (it is not as dramatic as the first diagram suggests), there is still a considerable way to go when one looks at employment (instead of unemployment). So perhaps we are closer, but not as close to our goals as Jim Bullard suggests here: A Tame Taper.