### Background

I do not propose to give a background history of loanable funds within this document. I will summarise it as follows:

- There is assumed to be a market in which economic actors borrow and lend money.
- We assume that there is a composite interest rate that stands in for the entire complex of interest rates.
- The amount of money borrowed for investing decreases as "the" interest rate rises (the demand curve).
- The amount of money entities want to lend increases as "the" interest rate rises (the supply curve).
- The interest rate has to be set at a level so that supply equals demand, since we know from an accounting identity that
*S=I*.

The implication of the above is that increased demand for funds will raise the interest rate. This runs counter to the view that rate expectations set the rate of interest. In particular, expansionary fiscal policy is supposed to raise interest rates, which is a standard complaint of fiscal conservatives.

Nick Rowe has a good article explaining how to teach loanable funds (and hence learn about), by way of comparison with the competing Liquidity Preference theory. Éric Tymoigne (of UKMC) has a pdf which runs through the comparison as well, aimed at a more experienced audience.

I presume that my views would be in the "Liquidity Preference" camp; more specifically, that yields are determined by rate expectations. As the chart above shows, it is hard to line up trends in bond yields with demand for funds, but they track the steady drop in short rates over the years. I have generally ignored loanable funds theory on the basis that it is an anachronism. Accordingly, I am not an expert on the topic.

### A New Keynesian Version Of Loanable Funds

In "The loanable funds fallacy", Lars P. Syll critiqued the translation of loanable funds theory into an "Orthodox New Keynesian" version by Nick Rowe. (Note that Nick Rowe wrote that "I am not an orthodox New Keynesian macroeconomist (ONKM), but I can pretend to be one." Therefore, when I quote him herein, the quotes may not reflect his true views, rather they reflect what he believes a consensus "New Keynesian" position should be.) Once again, I do not want to be drawn into the full details of the discussion they had in the comments on that article. Instead, I want to focus on a few points.

In New Keynesian models, the central bank sets the interest rate. But how does it set the interest rate? The argument is that it is based on loanable funds theory. Nick Rowe writes:

Let output demanded (call it Yd) be a negative function of the rate of interest r, a positive function of actual income Y, and a function of other stuff X.In order to address this, I will have to an excursion into some background theory.

Yd = D(r,Y,X)

And the ONKM central bank wants to set r such that output demanded equals potential output Y*, so that:

D(r,Y*,X) = Y*

Assume a closed economy for simplicity, subtract Cd (consumption demand) plus Gd (government demand) from both sides, remember the accounting identities C+I+G=Y and S=Y-C-G, where I is investment and S is national saving, and we get:

Id(r,Y*,X) = Sd(r,Y*,X)

The central bank sets a rate of interest such that desired investment at potential output equals desired national saving at potential output.Which is precisely the loanable funds theory of the rate of interest.

An inflation-targeting central bank will set a rate of interest equal to the rate of interest predicted by the loanable funds theory.

### The Nature Of Time In DSGE Models

Classical economic models have had great difficulty with time historically. There was always a good deal of handwaving about "the long run" and "the short run". However, not having an idea of what the scale on the time axis represents is not a useful property for time series models. Modern Dynamic Stochastic General Equilibrium (DSGE) models have at least one advantage over earlier generations of models, in that the time steps within calibrated models are associated with a fixed time period. For example, the model may be calibrated for quarterly data.

These models also have a particular convention about how transactions occur - all transactions for the same commodity are all assumed to occur at the same price, and that all transactions are determined simultaneously. You could think of it as all the transactions in a quarter occurring in a single simultaneous burst at the end of the quarter.

This is reasonable enough for a model, although it is obvious that this is not literally true. The convention that all transactions are determined simultaneously underlines the interconnected nature of the economy. However, the focus on simultaneous clearing also makes economists' thinking too complicated on occasion.

In particular, the

*S=I*accounting identity appears confusing when thought about as a condition that is created by an economy-wide equilibrium. (I recently referred to

*S=I*here.) However, it is a constraint that holds for flows over any accounting interval. And that includes accounting intervals which are so short that they include only one transaction.

*The implication is that every possible transaction generates a set of national accounts that preserves the S=I identity*,

*and so there is no danger of the aggregated quantities not equalling each other as the result of some disequilibrium.*

### Does The Central Bank Set Demanded Savings To Demanded Investment?

Nick Rowe observed in the comments of the Lars P. Syll article that since

*S=I*is an accounting identity, the equality of observed savings and investment is not an issue. But what about demanded savings (

*Sd*) and demanded Investment (

*Id*)? That is where the central bank allegedly has to step in.

However, the models do not appear to work that way. Within the models, the actions of the central bank are set in the form of a reaction function. That is, the policy rate is not modelled one period at a time, rather it is a rule that is based on the state of the model. A classic policy rule is the Taylor Rule, or else a constant interest rate. This rule has to be set before the optimising "representative household" determines the optimal solution; otherwise, it cannot determine the solution to calculate the utility.

Therefore, the sequence is like this:

- The central bank chooses a policy rule from within a set of potential policy rules.
- The household then determines the optimal solution to the mathematical problem, based on that policy rule. The solution consists of a sequence of "equilibrium states".

That is, the role of the central bank is to choose amongst a set of possible optimal solutions that will result from its choice. For any (reasonable?) policy rule, the final trajectory of solution will always be a sequence of "equilibrium states".

Equilibrium is defined so that demanded values of state variables* match the realised variables (assuming no unknown shocks). That means in equilibrium, demand savings equals savings, and demanded investment equals investment. Since we know that any feasible state of the system features

*S=I*, it is trivial that*Sd=Id*. No matter what the central bank does, demanded savings equals demanded investment, and so the bank has no role whatsoever in "balancing the loanable funds market".
In summary, if we assume that the economy always transitions between equilibrium states - which is what the DSGE mathematical framework assumes - demanded savings always equals demanded investment. Therefore, there is no reason for a central bank to care about loanable funds.

### What About Disequilibrium?

If we start to include the possibility of "disequilibrium" states, we could see a wedge between demanded savings and demanded investment.

The first problem is that the DSGE mathematical framework is built around the assumption of a transition between equilibrium states, and planning over an infinite sequence of such states. If we admit the possibility of disequilibrium, how do we calculate the state vector? How do we know how much the representative household wants to save, if we cannot calculate its income? Correspondingly, the mathematical description of the system tells us little.

Even in simple cases, it is not clear what happens if we want to be more flexible than the rigid assumptions embedded in DSGE models. For example, assume that the business sector is paying $100 in wages, no dividends, and plans on having a profit of $0 and fixed investment of $0. However, this inconsistent with the plans of the household sector that wants to save $10 out of the $100 income.

In a more sensible Stock-Flow Consistent modelling framework, this deviation of planned values from realised values could be resolved by a number of mechanisms, including:

- the business sector has an unplanned $10 increase in inventories (which raises investment to $10, matching saving);
- the business sector loses $10 and borrows the money from the household sector or draws down its cash balance (dropping aggregate savings to $0).

In the real world, these conflicts of interest appear to be resolved on the basis of which side has market power. But since "market power" has been abolished by decree from DSGE modelling, we are stuck with the markets moving to equilibrium** by assumption - a wave of the "market clearing" wand.

This is just an arbitrary assumption, and it has nothing to do with the level of interest rates. For example, assume the reaction function of the central bank is a Taylor rule, determined by variables from the previous period. The interest rate in the current period is always fixed, and so it cannot adjust to help determine the equilibrium. But since the DSGE model economy is assumed to "move to equilibrium", something else embedded in the other sectors of the economy always has to be available to do the job (the "magic wand").

**Footnotes:**

* Within a mathematical model, the

*state*is the list of variables which describe how the model will evolve (the

*state variables*). In an economic model, the state would include initial balances of stock variables, prices, and flows, as well as expectations of these variables. This makes notation and discussion easier to follow.

**Note that since the DSGE model is only defined at discrete time points, this "movement to equilibrium" has to be conceived of as a computational process that exists outside of time. Historically, this was conceived of changing prices in an auction like environment; only once the prices are consistent across markets do the transactions go through. Although this could work for prices, it will not work for the flow variables within the model, such as income, savings, and investing. There is no mechanism to move from a non-equilibrium set of prices to a set of transactions, as all transactions are assumed to be the result of a planned optimal trajectory, which is only defined in equilibrium.

(c) Brian Romanchuk 2014

So as far as understanding loanable funds, your summary is what an intermediate macro text would suggest. But I don't think it's the proper way to understand loanable funds. Loanable funds should just be understood as a simple supply/demand model of *desired* Saving (S) and *desired* Investment (I) as functions of the interest rate. Any notion of financial lending/borrowing should be separated from the discussion. Textbooks try to connect the model with financial borrowing/lending, but it's a failed analogy.

ReplyDeleteIn any case, I'm not sure I follow your critique of Nick Rowe. In Nick Rowe's model, it's the CB's job to set the interest rate such that Sd=Id. Say such a rate would be 3%. So if the CB sets the rate at 4%, we be in some disequilibrium state where Sd does not equal Id. That's easy to see in a simple supply/demand graph where there's a wedge between the Sd and Id curves.

You seem to be saying that DSGE models don't allow modeling of interest rates that don't lead to Sd=Id? But how is that a critique of Nick's point? If the CB's goal is to set Sd = Id, then it will use tools that will tell it where the rate should be set such that Sd = Id. Re: Taylor Rule, my assumption has been that the Taylor Rule is a more pragmatic tool - back of the envelope essentially - to find the rate such that Sd = Id.

To be clear, I may have understood Nick's point, and I may be misunderstanding the equilibrium condition in DSGE models. Every treatment that I have seen just asserts that the economy is in equilibrium, and so I am unsure how the modern treatments deal with "disequilibrium".

DeleteBut an alternative way of looking at my point: imagine that the central bank can choose between two different "Taylor Rules" (i.e., the parameters are different). Both choices lead to different equilibria, which may or may not be the "full employment" equilibrium. But since both states are equilibria, the market in "savings" clears, and so desired savings equals actual savings (Sd=S), and similarly for desired demand and actual demand (Id=I). And since actual savings always equals actual savings, desired savings equals desired investment.

Therefore, in both cases Sd=Id. The central bank cannot induce a situation to the contrary, unless they set the reaction function so that the economy cannot enter equilibrium (which is somewhat silly and against the rules of the game).

Gotcha. So this seems to be an issue of understanding DSGE and correlating it with the more basic model Nick lays out.

DeleteBrian: In most models, Id=Sd. But that doesn't make it a Loanable Funds model. For example, the simple textbook Keynesian Cross model has Id=Sd, but it is Y, not r, that adjusts to ensure Id=Sd, so it is not a Loanable Funds model of r. The key to a Loanable Funds model is that Y stays at potential output, and r adjusts to ensure Id=Sd.

Delete(Note: I meant to write "I may have MISunderstood Nick Rowe's point...)

DeleteNick - thanks for the response. I guess my point is that the macro models I am familiar with - SFC and new Keynesian DSGE - do not assume that output is at potential, and r is set using some reaction function like a Taylor rule. Intuitively, that makes more sense to me, and so I missed the possibility that Y is assumed to be at potential output. (RBC models have that property, but central bank policy doesn't seem to do anything other than determining the rate of inflation within those models.)

Brian: (Yep, I guessed you meant "misunderstood".)

DeleteYep. Only if the central bank gets monetary policy exactly right at all times will Y always be at Y*. In practice it will make mistakes, because it's not omniscient, so Y will fluctuate around Y*. Which means the Loanable Funds model will only be true up to a (random?) error due to the central bank's forecasting mistakes.

I agree with ATR that the Taylor Rule is just a rough and ready pragmatic "rule of thumb" to try to keep Y at Y* and inflation on target. Central banks don't really follow Taylor Rules. For example, if a central bank sees that the government is planning to reduce the deficit, it will (or should) adjust r downwards in anticipation, without waiting for the Taylor Rule to kick in.

My assumption is that they will hit Y* for about 5 minutes as they pass from one side of the output gap to the other. About all they can hope is that it mean reverts towards that level over time.

DeleteBrian: yes. That would be the orthodox NK assumption too. But in a simple linear model, it would be Y* on average.

DeleteOr, Y* in expectation, would be a better way of putting it.

DeleteNot really related to the post, but it seems to me that the expression "loanable funds theory" conjures different meaning to different people. Also, I don't quite understand why there is so much debate about loanable funds vs liquidity preference theories. One points to flows while the other to the stock of money as determinants of the rate of interest. Perhaps I'm wrong (and may be forgetting something essential from my university days) but, to the extent that the meaning of "money" and "loanable funds" (however they are defined) is sensible, both theories should provide consistent insights about what determines the rate of interest, though from a different point of view.

ReplyDeleteThat said, the notion (put forth not by you, but many others) that loanable funds theory is inconsistent with endogenous money, money creation by banks, and interest rate targeting by the central bank strikes me as misplaced. I honestly don't see anything incompatible with any of these things with the proposition that the rate of interest can be determined by something other than the quantity supplied of money with the quantity demanded, including Sd and Id or even loans and credit. I find it hard to believe that folks like Bernanke, Stiglitz, Blinder, Akerlof, Shiller, etc, would be that way off about how the banking system works (and, consequently, about how best to model it). I get into debates on this point with my offline friends all the time but to no avail. Any thoughts?

Essentially agree. The issue is the use of the word 'determines.' It's too vague and depends on your perspective.

DeleteIf you're talking about what the central bank does *operationally* to target a *given* interest rate in the fed funds market, I suppose this could be a form of 'liquidity preference theory,' although I'm not sure the Feds operations are typically labeled that way. But basically we're not talking about flows of S/I or anything like that, just ceiling/floor rates, the quantity of reserves, reserve requirements, expectations of liquidity shocks to the reserve system, etc.

If you're talking about how the central bank decides to *strategically choose* an interest rate (which it then sets *operationally*), then you're in the realm of loanable funds, if that's the theory the central bank uses.

So depending on if you take an operational or strategic view, one could correctly say rates are determined by liquidity preference or loanable funds.

Well, in re-reading your post, my notion may be different from yours of how the two theories fit together, but that's how I currently see it.

DeleteAs circuit notes, loanable funds means different things to different people. My experience wih loanable funds was that it was a fairly straightforward supply and demand argument about interest rate formation - running a deficit causes interest rates to go up. However, Nick Rowe's understanding of the concept is more complex than that. At this point, I would need a fairly practical example to see what the actual differences in the points of view as an operrational matter. Some obscure-sounding theoretical arguments actually have large differences in predicted real world behaviour, but I am not sure loanable funds is in that category.

DeleteThe difference is that in the real world the central bank has to move the interest rate. Loanable funds as you describe it is how the world would work in a barter economy ("running a deficit causes interest rates to go up"). Clearly, if we run a deficit today, there is no automatic process by which the interbank rate would increase unless the central bank increases it itself. That is, we live in a monetary economy, not a barter economy, so the central bank has to align the money rate of interest with what the theoretical natural rate is.

Delete