# The Interest Rate and the Interest Rate

We will return to secular stagnation. But we need to clear some ground first. What is an interest rate?

Imagine you are in a position to acquire a claim on a series of payments  in the future. Since an asset is just anything that promises a stream of payments in the future, we will say you are thinking of buying of an asset. What will you look at to make your decision?

First is the size of the payments you will receive, as a fraction of what you pay today. We will call that the yield of the asset, or y. Against that we have to set the risk that the payments may be different from expected or not occur at all; we will call the amount you reduce your expected yield to account for this risk r. If you have to make regular payments beyond the purchase of the asset to receive income from it (perhaps taxes, or the costs of operating the asset if it is a capital good) then we also must subtract these carrying costs c. In addition, the asset may lose value over time, in which case we have to subtract the depreciation rate d. (In the case of an asset that only lasts one period — a loan to be paid back in full the next period, say — d will be equal to one.) On the other hand, owning an asset can have benefits beyond the yield. In particular, an asset can be sold or used as collateral. If this is easy to do, ownership of the asset allows you to make payments now, without having to waiting for its yield in the future. We call the value of the asset for making unexpected payments its liquidity premium, l. The market value of long-lasting assets may also change over time; assuming resale is possible, these market value changes will produce a capital gain g (positive or negative), which must be added to the return. Finally, you may place a lower value on the payments from the asset simply because they take place in the future; this might be because your needs now are more urgent than you expect them to be then, or simply because you prefer income in the present to income in the future. Either way, we have to subtract this pure time-substitution rate i.

So the value of an asset costing one unit (of whatever numeraire) will be 1 + y – r – c – d + l + g – i.

(EDIT: On rereading, this could use some clarification:

Of course all the terms can take on different (expected) values in different time periods, so they are vectors, not scalars. But if we assume they are constant, and that the asset lasts forever (i.e. a perpetuity), then we should write its equilibrium value as: V = Y/i, where Y is the total return in units of the numeraire, i.e. Y = V(y – r – c + l + g) and i is the discount rate. Divide through both sides by V/i and we have i = y – r – c + l + g. We can now proceed as below.)

In equilibrium, you should be just indifferent between purchasing and not purchasing this asset, so we can write:

y – r – c – d + l + g – i = 0, or

(1) y = r + c + d – l – g + i

So far, there is nothing controversial.

In formal economics, from Bohm-Bawerk through Cassel, Fisher and Samuelson to today’s standard models, the practice is to simplify this relationship by assuming that we can safely ignore most of these terms. Risk, carrying costs and depreciation can be netted out of yields, capital gains must be zero on average, and liquidity is assumed not to matter or just ignored. So then we have:

(2) y = i

In these models, it doesn’t matter if we use the term “interest rate” to mean y or to mean i, since they are always the same.

This assumption is appropriate for a world where there is only one kind of asset — a risk-free contract that exchanges one good in the present for 1 + i goods in the future. There’s nothing wrong with exploring what the value of i would be in such a world under various assumptions.

The problem arises when we carry equation (2) over to the real world and apply it to the yield of some particular asset. On the one hand, the yield of every existing asset reflects some or all of the other terms. And on the other hand, every contract that involves payments in more than one period — which is to say, every asset — equally incorporates i. If we are looking for the “interest rate” of economic theory in the economic world we observe around us, we could just as well pick the rent-price ratio for houses, or the profit rate, or the deflation rate, or the ratio of the college wage premium to tuition costs. These are just the yields of a house, of a share of the capital stock, of cash and of a college degree respectively. All of these are a ratio of expected future payments to present cost, and should reflect i to exactly the same extent as the yield of a bond does. Yet in everyday language, it is the yield of the bond that we call “interest”, even though it has no closer connection to the interest rate of theory than any of these other yields do.

This point was first made, as far as I know, by Sraffa in his review of Hayek’s Prices and Production. It was developed by Keynes, and stated clearly in chapters 13 and 17 of the General Theory.

For Keynes, there is an additional problem. The price we observe as an “interest rate” in credit markets is not even the y of the bond, which would be i modified by risk, expected capital gains and liquidity. That is because bonds do not trade against baskets of goods. They trade against money. When we see a bond being sold with a particular yield, we are not observing the exchange rate between a basket of goods equivalent to the bond’s value today and baskets of goods equivalent to its yield in the future. We are observing the exchange rate between the bond today and a quantity of money today. That’s what actually gets exchanged. So in equilibrium the price of the bond is what equates the expected returns on the two assets:

(3) y_B – r_B + l_B + g_B – i = l_M – i

(Neither bonds nor money depreciate or have carrying costs, and money has no risk. If our numeraire is money then money also cannot experience capital gains. If our numeraire was a basket of goods instead, then -g would be expected inflation, which would appear on both sides and cancel out.)

What we see is that i appears on both sides, so it cancels out. The yield of the bond is given by:

(4) y_B  = r_B – g_B + (l_M – l_B)

The yield of the bond — the thing that in conventional usage we call the “interest rate” — depends on the risk of the bond, the expected price change of the bond, and the liquidity premium of money compared with the bond. Holding money today, and holding a bond today, are both means to enable you to make purchases in the future. So the intertemporal substitution rate i does not affect the bond yield.

(We might ask whether the arbitrage exists that would allow us to speak of a general rate of time-substitution i in real economies at all. But for present purposes we can ignore that question and focus on the fact that even if there is such a rate, it does not show up in the yields we normally call “interest rates”.)

This is the argument as Keynes makes it. It might seem decisive. But monetarists would reject it on the grounds that nobody in fact holds money as a store of value, so equation (3) does not apply. The bond-money market is not in equilibrium, because there is zero demand for money beyond that needed for current transactions at any price. (The corollary of this is the familiar monetarist claim that any change in the stock of  money must result in a proportionate change in the value of transactions, which at full employment means a proportionate rise in the price level.) From the other side, endogenous money theorists might assert that the money supply is infinitely elastic for any credit-market interest rate, so l_M is endogenous and equation (4) is underdetermined.

As criticisms of the specific form of Keynes’ argument, these are valid objections. But if we take a more realistic view of credit markets, we come to the same conclusion: the yield on a credit instrument (call this the “credit interest rate”) has no relationship to the intertemporal substitution rate of theory (call this the “intertemporal interest rate.”)

Suppose you are buying a house, which you will pay for by taking out a mortgage equal to the value of the house. For simplicity we will assume an amortizing mortgage, so you make the same payment each period. We can also assume the value of housing services you receive from the house will also be the same each period. (In reality it might rise or fall, but an expectation that the house will get better over time is obviously not required for the transaction to take place.) So if the purchase is worth making at all, then it will result in a positive income to you in every period. There is no intertemporal substitution on your side. From the bank’s point of view, extending the mortgage means simultaneously creating an asset — their loan to you — and a liability — the newly created deposit you use to pay for the house. If the loan is worth making at all, then the expected payments from the mortgage exceed the expected default losses and other costs in every period. And the deposits are newly created, so no one associated with the bank has to forego any other expenditure in the present. There is no intertemporal substitution on the bank’s side either.

(It is worth noting that there are no net lenders or net borrowers in this scenario. Both sides have added an asset and a liability of equal value. The language of net lenders and net borrowers is carried over from models with consumption loans at the intertemporal interest rate. It is not relevant to the credit interest rate.)

If these transactions are income-positive for all periods for both sides, why aren’t they carried to infinity? One reason is that the yields for the home purchaser fall as more homes are purchased. In general, you will not value the housing services from a second home, or the additional housing services of a home that costs twice as much, as much as you value the housing services of the home you are buying now. But this only tells us that for any given interest rate there is a volume of mortgages at which the market will clear. It doesn’t tell us which of those mortgage volume-interest rate pairs we will actually see.

The answer is on the liquidity side. Buying a house makes you less liquid — it means you have less flexibility if you decide you’d like to move elsewhere, or if you need to reduce your housing costs because of unexpected fall in income or rise in other expenses. You also have a higher debt-income ratio, which may make it harder for you to borrow in the future. The loan also makes the bank less liquid — since its asset-capital ratio is now higher, there are more states of the world in which a fall in income would require it to sell assets or issue new liabilities to meet its scheduled commitments, which might be costly or, in a crisis, impossible. So the volume of mortgages rises until the excess of housing service value over debt service costs make taking out a mortgage just worth the incremental illiquidity for the marginal household, and where the excess of mortgage yield over funding costs makes issuing a new mortgage just worth the incremental illiquidity for the marginal bank. (Incremental illiquidity in the interbank market may — or may not — mean that funding costs rise with the volume of loans, but this is not necessary to the argument.)

Monetary policy affects the volume of these kinds of transactions by operating on the l terms. Normally, it does so by changing the quantity of liquid assets available to the financial system (and perhaps directly to the nonfinancial private sector as well). In this way the central bank makes banks (and perhaps households and businesses) more or less willing to accept the incremental illiquidity of a new loan contract. Monetary policy has nothing to do with substitution between expenditure in the present period and expenditure in some future period. Rather, it affects the terms of substitution between more and less liquid claims on income in the same future period.

Note that changing the quantity of liquid assets is not the only way the central bank can affect the liquidity premium. Banking regulation, lender of last resort operations and bailouts also change the liquidity premium, by chaining the subjective costs of bank balance sheet expansion. An expansion of the reserves available to the banking system makes it cheaper for banks to acquire a cushion to protect themselves against the possibility of an unexpected fall in income. This will make them more willing to hold relatively illiquid assets like mortgages. But a belief that the Fed will take emergency action prevent a bank from failing in the event of an unexpected fall in income also increases its willingness to hold assets like mortgages. And it does so by the same channel — reducing the liquidity premium. In this sense, there is no difference in principle between monetary policy and the central bank’s role as bank supervisor and lender of last resort. This is easy to understand once you think of “the interest rate” as the price of liquidity, but impossible to see when you think of “the interest rate” as the price of time substitution.

It is not only the central bank that changes the liquidity premium. If mortgages become more liquid — for instance through the development of a regular market in securitized mortgages — that reduces the liquidity cost of mortgage lending, exactly as looser monetary policy would.

The irrelevance of the time-substitution rate i to the credit-market interest rate y_B becomes clear when you compare observed interest rates with other prices that also should incorporate i. Courtesy of commenter rsj at Worthwhile Canadian Initiative, here’s one example: the Baa bond rate vs. the land price-rent ratio for residential property.

Both of these series are the ratio of one year’s payment from an asset, to the present value of all future payments. So they have an equal claim to be the “interest rate” of theory. But as we can see, none of the variation in credit-market interest rates (y_B, in my terms) show up in the price-rent ratio. Since variation in the time-substituion rate i should affect both ratios equally, this implies that none of the variation in credit-market interest rates is driven by changes in the time-substitution interest rate. The two “interest rates” have nothing to do with each other.

(Continued here.)

EDIT: Doesn’t it seem strange that I first assert that mortgages do not incorporate the intertemporal interest rate, then use the house price-rent ratio as an example of a price that should incorporate that rate? One reason to do this is to test the counterfactual claim that interest rates do, after all, incorporate Samuelson’s interest rate i. If i were important in both series, they should move together; if they don’t, it might be important in one, or in neither.

But beyond that, I think housing purchases do have an important intertemporal component, in a way that loan contracts do not. That’s because (with certain important exceptions we are all aware of) houses are not normally purchased entirely on credit. A substantial fraction of the price is paid is upfront. In effect, most house purchases are two separate transactions bundled together: A credit transaction (for, say, 80 percent of the house value) in which both parties expect positive income in all periods, at the cost of less liquid balance sheets; and a conceptually separate cash transaction (for, say, 20 percent) in which the buyer foregoes present expenditure in return for a stream of housing services in the future. Because house purchases must clear both of these markets, they incorporate i in way that loans do not. But note, i enters into house prices only to the extent that the credit-market interest rate does not. The more important the credit-market interest rate is in a given housing purchase, the less important the intertemporal interest rate is.

This is true in general, I think. Credit markets are not a means of trading off the present against the future. They are a means of avoiding tradeoffs between the present and the future.

## 5 thoughts on “The Interest Rate and the Interest Rate”

1. Data Tutashkhia says:

I haven't read all this carefully, nor do I understand any of the technical stuff, but just from looking at the graph:
your rent/price-red line there refers to the asset (land) that is scarce and is supposed to appreciate (roughly) with inflation, while your bond-blue line refers to the asset (bond's face value) that is constant.
So, the red line is normalized for inflation, while the blue line goes up and down reflecting the expectation for the inflation at any given moment. And so the blue line really is the interest rate, and the red line is, well, just rent.
Does it make sense, in the layman's world?

2. Philip,

Thanks for the response. It's interesting that Brad DeLong had the same reaction — that there is no reason to think of asset yields as incorporating separate terms for liquidity and risk. But he thinks it's all risk, whereas you think it's all liquidity! personally, I think there are reasons to think about both. For example, future Social Security benefits carry very little risk, but they are very illiquid, since they cannot be transferred to a third party or pledged as collateral.

This does not matter for the point here, though. The point is just that the margin on which the interest rate is et is not consumption today vs. consumption tomorrow, but more dangerous vs. safer balance sheet positions. Whether we think of "danger" here as risk or liquidity or both, is secondary.

One thing I do appreciate is that you describe my post as a restatement of Keynes' liquidity preference theory of interest. That's exactly right. The main problems with JMK's statement, in my opinion, are that he thinks of liquidity only in terms of the asset side of the balance sheet, and he is inconsistent in using liquidity as a generic property of assets and as a synonym for money, in ways that (again IMO) create serious contradictions in his argument and open the door for a reading in terms of what Perry Mehrling calls "monetary Walrasianism." What I'm trying to do here is develop a statement of the liquidity preference theory of interest rates that reflects the fact that liquidity is just as much a property of the liability side as of the asset side of balance sheets.

A third problem with Keynes presentation is that he ignores the transaction demand for money, and the accelerator principle of investment, in order to produce a model with a unidirectional casual structure from financial markets to investment to output, rather than a jointly determined equilibrium. This is the standard critique of liquidity preference as JMK stated it, and the problem that ISLM "corrects." But it is not such a big problem, in my opinion.

So I agree with you that Keynes' argument is a bit of a muddle. I don't think my ideas are muddled (of course not!), but I agree the presentation here is less clear than it could have been. it's a work in progress.

1. 1) I'm not talking about the liquidity of an asset. I'm talking about the fact that liquidity preference determines interest rates. This is entirely different. Of course some assets are riskier than others. But we're asking what determines yields. That's a different question.

2) It does matter because you've included a variable that can give a quanta of risk. Once you allow risk to be objectively quantified then you can model that risk and… poof! Liquidity preference disappears.

After all, why would a rational agent need liquidity preference in a world where all risk is modeled? Rather they would just choose their preferred level of risk (0, 0.5, 0.9999… whatever). We can then start drawing indifference curves and off we go!

2. I'm not talking about the liquidity of an asset. I'm talking about the fact that liquidity preference determines interest rates. This is entirely different. Of course some assets are riskier than others. But we're asking what determines yields. That's a different question.

It looks like the same question to me. Yields, like prices, are relative. To say that liquidity preference determines interest rates, is the same as saying that the relative liquidity of (some set of) assets determines their relative yields.

It does matter because you've included a variable that can give a quanta of risk. Once you allow risk to be objectively quantified then you can model that risk and… poof! Liquidity preference disappears.

I don't agree. All that liquidity preference requires is that not all future cashflows can be mobilized to make payments today. Fundamental uncertainty is one way of motivating that, but it's not logically necessary. Again, Social Security benefits are illiquid not because of uncertainty but because of legal restrictions on their sale or hypothecation.

Of course you are right that a great deal of mainstream work (tho not all of it!) ignores uncertainty and liquidity and thinks all asset prices can be reduced to stochastic risk. That leads to a lot of silliness. But I don't see why we have to go to the opposite extreme. Certainly Keynes was willing to describe future contingencies as being characterized as both risk and uncertainty. Now, it may well turn out to be the case that risk in the conventional sense is not very interesting, but the proposition that no events in the future have calculable probabilities seems hard to defend. Why is this the particular hill you want to die on?

After all, why would a rational agent need liquidity preference in a world where all risk is modeled?

I'm not understanding how you go from some risk to all risk.