Believe those who are seeking the truth. Doubt those who find it. Andre Gide

Wednesday, June 25, 2014

Excess reserves and inflation risk: A model

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. 

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.


  1. The reason I prefer a fiscal approach is your second to last paragraph. The IOR is the same as Treasury issuing short term interest paying bills. A Fed capital loss is the same as a larger Treasury deficit. What does either have to do with "money"?

    I think what you are really saying is that the fiscal authority may finance capital losses by issuing short term bills at below-market yields (i.e. zero or negative real). When debt to gdp is large, a broad swath of agents has an incentive to avoid holding those bills and purchase goods instead. Pull-forward buying of goods causes inflation. Lending comes into play because, once inflation gets going, speculators have an incentive to finance an increase in goods inventories. It is all about pull-forward.

    In terms of your model, an exogenous increase in "R" occurs when agents fear that the fiscal authority will persistently finance capital-loss deficits at below-market rates. This sets the whole thing in motion. All that is needed for the feedback loop to work is 1) fear and 2) duration risk accumulated by the fiscal authority.

    1. What does either have to do with "money"?

      The question is whether the larger deficit is to be monetized or not. So clearly, monetary and fiscal policy are intertwined here.

    2. "Monetization" is a concept complicated by the IOR, as are all other monetary concepts (money, monetary base, etc). The government issues liabilities to fund deficits. It can issue them at market rates (Treasuries) or coerce acceptance at below market rates (interest-paying reserves). If the it does the latter, velocity rises and inflation ensues. The only thing it takes to spark inflation is the fear of losses from future coercion. The risk of loss is magnified when the taxpayer also holds a great deal of duration risk. The worst-case scenario is one in which taxpayers suffer both real wealth losses AND duration losses as a behind-the-curve Fed raises nominal rates only to see inflation rise further.

  2. This is the common case for hyperinflation. The more people are not rolling over their bonds, the more money the central bank makes to buy bonds. But the more new money there is, the less people want hold bonds. So you get a death spiral. The central bank can not stop buying bonds because the government can not operate without the help of the central bank. Government does not have the money to pay off bonds coming due except by selling new bonds, but the only buyer is the central bank. So really they end up printing and spending money. So once the spark is lit, this blows up your currency.

    It happens again and again and yet most economists don't appreciate the danger of having huge amounts of debt. The excess reserves can be converted to cash and will in a hyperinflationary scenario. So they will end up adding to the hyperinflation just as the debt does.

  3. I'm having some trouble following this, in particular how you are changing M and R independently. In your case 1 for example, when you keep D fixed but change the money/bond ratio, doesn't a change in the price level violate your first constraint (y = m/p + b/p)? I would expect that R would have to change here to accommodate the change in money.

    1. Yes, Nick, I think you may be correct. Let me check over my calculations. Thank you!

    2. The Nicks (Rowe and Edmonds) will set you straight every time! :D

  4. "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."

    Presumaly the Fed doesn't need treasury permission to simply reduce the amount of profit it remits, thus the first few increases could be funded out of the interest earned from the asset holdings and thus wouldn't involve the creation of new government debt,so wouldn't be inflationary.

    Only if the rate of IOR goes above the average running yield of the asset portfolio does the fed need to ask the treasury for money.

    That does however bring up an interesting question that you may know the answer to. Usually the Fed can only create resereves to buy assets or lend against collateral, they can't just create them and give them out.

    So, when they were given permission to pay interest on reserves was this done with a treasury promise to fund the interest if the Fed couldn't do so out of earnings or was it a case of the Fed being able to create reserves and just give them away without receiving anything in return?

    1. (A) IN GENERAL- Balances maintained at a Federal Reserve bank by or on behalf of a depository institution may receive earnings to be paid by the Federal Reserve bank at least once each calendar quarter, at a rate or rates not to exceed the general level of short-term interest rates.

      That's all Congress had to say on the matter.

      Some economists argue deferring the loss means "never having to say you're sorry" because future profits will cover it. This is simply wrong. The CBO already includes projected Fed profits in its baseline budget projection, which, in turn, is the basis for revenue decisions. Thus, if the Fed's remittance comes up short of baseline, it must be funded by either tax increases or higher Treasury issuance. This is why the Fed's balance sheet creates a contingent tax liability anyway one slices the accounting.

    2. I'm not entirely sure I understand this, Diego.

      Booking a deferred asset essentially permits the Fed to finance the carrying cost of government debt (whether in the form of interest-bearing reserve accounts or US treasuries makes no difference economically) by printing money. In principle, this does not mean higher taxes or Treasury issuance, although it could mean higher inflation. Am I wrong?

    3. I think having the Fed in the equation confuses the matter (no offense!).

      Let's say the fiscal authority decides to fund itself short duration and its asset is long duration (future tax revenues). If interest rates rise, the resulting duration loss produces a tax liability. The fiscal authority may decide to prevent interest rates from rising by coercing banks into buying its liabilities at below-market rates. This leads to inflation. The only way to stop inflation is to issue liabilities at market rates. This, eventually, requires tax revenue to cover the government's duration losses (higher deficits). Thus, the bigger the fiscal authority's duration bet, the bigger the taxpayer's contingent tax liability. This only makes sense. Seigniorage revenues are really irrelevant to this whole dynamic.

      This paper on a fiscal theory of inflation is relevant, although it doesn't mention the duration-loss dynamic that I focus on. My view is that the "spark" for the inflation dynamic is duration aversion on the part of the public in the presence of a large government duration bet.

  5. I think I just realized what you meant by "deferred asset", it's claim on it's own future profits that would otherwise be remitted to the treasury but instead are kept to retire the asset.

    So, if IOR can't be funded out of current asset income it is funded with a claim on future asset income. Correct?

    1. Correct. I think you can get more details here:

  6. David thanks for the update after Nick E's comment.

  7. David, O/T: Where's the best place to find a curve derived from a macroeconomic theoretical model of the price level (P) vs M (using any measures of M and P) with, for comparison, empirical data plotted on the same chart? Here's an example of what I mean for Canada and for Japan. Those both use the same model, by the way, and the author of the model also created a related pair of curves for the interest rates in those two countries (and others). He's trying to compare his model results with professional examples but he can't find any professional examples (he says when he does a Google image search for "price level model" he only finds his own plots!). Where should he be looking?


    1. Excuse me, vs year, not M!

    2. Hi Tom,

      First, I'm not even sure how this fellow defines a "model."
      Second, even if I did have a model that made conditional predictions for the path of the price-level, why would I only want to focus on the price-level and not on the joint-behavior of ALL the model's variables? It is possible to fit well along one dimension, while missing on others.

    3. David, thanks for your reply. I passed that along. The "hard core" of his theory is comprised of just two items, one of which includes a simple definition of price (p). That differential equation (in the link) has both an endogenous solution and an exogenous one. The endogenous price level (P) model follows from that, and looks basically something like this:

      log P ~ (1/kappa - 1)*log M

      with kappa = log M / log NGDP.

      So perhaps the answer to your second question is that the simple core of his theory is concerned with price, and thus perhaps it's one of the most natural things for him to model. Regarding fitting some things well and others less well, I know the model does produce other outputs such as interest rates. And also, although this doesn't address your concern directly, but if we just look at a couple of things: P and NGDP vs M say, granted these are a limited number of items to model, but the same model seems to match data from a number of economies (different countries) pretty well, for example this plot. The cluster of blue model curves on the right there are intended to give a feel for the theoretical variance/distribution.

    4. Hi David,

      My admittedly snarky challenge was issued because I wanted to see how well I was doing relative to "real" models and it doesn't seem like many are available out there. It is certainly possible to get one variable right and all the rest terribly wrong ... but I'd even be interested in seeing that :)

      Additionally, predictions are hard (the economy is a stochastic system), so I'm more interested in models that describe the existing time series with a (hopefully) small number of variables. That kind of defines what I mean by model ... a theoretical equation (or differential/difference equation) with a few parameters that encompasses the behavior of data.

      This works (see Figure 5):

  8. David, yesterday Scott Sumner had a look at Jason's model, and had this question:

    "Jason, Does the model forecast P better than alternative approaches, like TIPS spreads?"

    This prompted Jason to investigate, and he produced these three posts in response:

    Sumner's response:

    "Interesting. I hope people with more time and skill that I have will investigate your model."

    Do you think this model has some promise?