### The Loanable Funds and other theories

### The Loanable Funds and other theories

This is a long and boring post. (Though some will no doubt find it highly controversial, and a very few might even get over-excited). It might be useful for economics students who want to put the Loanable Funds theory into perspective. They are my main intended audience. And maybe for those who teach them too.

Loanable Funds is a theory of "the" rate of interest. (I'm coming back to that "the" later). There are other theories of the rate of interest. That's why we call it a "theory".

Most (all?) economics students will have seen this diagram:

Put the rate of interest on the vertical axis. Put the flows of saving and investment on the horizontal axis. Draw a downward-sloping desired investment curve Id. Draw an upward-sloping desired saving curve Sd. Mark the point where the two curves cross. Draw a horizontal line to the vertical axis, to find the rate of interest predicted by the theory. The Loanable Funds theory says that the rate of interest is determined by the intersection of the desired saving and desired investment curves. (The rate of interest and actual saving=actual investment are co-determined simultaneously by those two curves.)

"Saving" has to be defined as "national saving", to include both private and government saving. And investment has to be defined as "national investment", to include both private and government investment. Otherwise things won't add up right.

Even then, things won't add up right in an open economy, unless we include borrowing from abroad and lending to abroad. Which means you have to bring in expected changes in the exchange rate, otherwise foreign and domestic interest rates won't always be the same even under perfect capital mobility. For now, let's just stick with a closed economy.

With a little bit of national income accounting, there's a second way to look at the Sd=Id equilibrium condition of Loanable Funds. Desired private saving Spd is income minus taxes minus desired consumption. Spd=Y-T-Cd. Add desired government saving Sgd=T-Gd, and we get desired national saving: Sd=Y-Cd-Gd. So we can re-write Sd=Id as: Y-Cd-Gd=Id. And we can rearrange that to get:

Y=Cd+Id+Gd

One of the joys of National Income Accounting is that it lets you see that two theories that sound very different are really the same. "The rate of interest equilibrates desired saving with desired investment" can be re-stated as "The rate of interest equilibrates desired expenditure on newly-produced goods (Cd+Id+Gd) with the output of newly-produced goods" (Y).

And a second joy of National Income Accounting is that by doing this it can reveal an ambiguity in the theory. Because we might want to replace Y (

*actual*output of goods) with Ys (the quantity of output*supplied*, or that people and firms*would like*to sell). So that we could restate Loanable Funds as "The rate of interest equilibrates the supply and demand for newly-produced goods". But that means that "desired private saving" is ambiguous too. Does it mean "Y-T-Cd"? Or should it mean "Ys-T-Cd"? What does "*desired*saving" mean in Loanable Funds? Does it mean "*Actual*income minus demand for consumption goods"? Or does it mean "The income we*would*earn if we*could*sell all the output we*wanted*to, minus demand for consumption goods"? That difference is obviously going to matter, if the economy is in a recession with deficient aggregate demand, so we can't sell as much output as we would like.
Here's a second diagram for the loanable funds theory. Even though it's equivalent to the first diagram, by an accounting identity, you maybe haven't seen loanable funds presented this way:

What exactly does the vertical black curve represent? Is it

*actual*output (

*actual*income)? Is it the level of output

*supplied*, which means the level of output that people and firms

*would like*to sell (and the income they would earn if they did)? Or something else?

Let's put that question aside for now, and consider an alternative theory of the rate of interest.

The Liquidity Preference theory of the rate of interest is the most obvious alternative. The rate of interest is determined by the supply and demand for money. Put the rate of interest on the vertical axis, the stock of money on the horizontal, draw a downward-sloping money demand curve, and an upward-sloping money supply curve, and the rate of interest is determined where those two curves cross.

Almost every Canadian is familiar with a special limiting version of Liquidity Preference theory. It's the theory that gets reported in the news. "The Bank of Canada sets the rate of interest" is a Liquidity Preference theory. You can think of this as a limiting case of Liquidity Preference, where the money supply curve is perfectly elastic at a rate of interest chosen by the Bank of Canada.

Most (all?) economics students have been taught

*both*Loanable Funds*and*Liquidity Preference. And they have been taught*one*way to reconcile the two theories. They have been taught the ISLM model. The IS curve represents the Loanable Funds Id=Sd equilibrium condition, and the LM curve represents the [Liquidity Preference] Md=Ms equilibrium condition. But each is incomplete, because Y affects Sd (and maybe Id too), and Y also affects Md (and maybe Ms too). So you need to put both curves together to get the complete picture. The rate of interest (and the level of output Y) are co-determined by the IS (Loanable Funds) curve and the LM (Liquidity Preference) curve.
But in the long run, when sticky prices have adjusted to the demand and supply of money, we can ignore the LM curve. [Update: because the price level adjusts so the LM curve shifts endogenously to interesect the other two curves.] The level of output is determined by the Long Run Aggregate Supply curve (or the "Full Employment" curve). So in the long run, money demand and money supply have no effect on the rate of interest. The rate of interest is determined by the intersection of the IS and FE curves -- by desired saving and investment at the long run equilibrium level of output determined by LRAS.

Now let me tell you a

*second*way of reconciling the Loanable Funds and Liquidity Preference theories. (It's not inconsistent with the ISLM way of reconciling the two theories, but it does*sound*very different. It's really just a special case of the ISLM way, but it sounds much more real worldy).
The Bank of Canada sets the rate of interest. But the Bank of Canada does not set the rate of interest on a whim. Something determines where the Bank of Canada chooses to set the rate of interest. The Bank of Canada targets CPI inflation at 2%. It tries to set a rate of interest that will deliver 2% inflation.

If the Bank of Canada sets the rate of interest "too high" (as in the above diagram), output demanded will be "too low", so actual output will be "too low", and inflation will start to fall below the Bank's 2% target. And if the Bank sets the rate of interest "too low", output demanded will be "too high", and inflation will start to rise above the Bank's 2% target.

So, what does "too high" and "too low" mean? Relative to what? It means "relative to the rate of interest at which output demanded would be at the right level to keep inflation at the 2% target". In the above two diagrams, I have called that rate of interest "rn", and that level of output "Yn". The Bank of Canada sometimes uses the names "neutral rate of interest" and "potential output". Some economists call them the "natural" rate of interest and level of output. Others use the name "NAIRU". Others use the name "Full Employment". Names sometimes have connotations, so people fight about names. But I'm going to skip that fight here.

Instead, I'm going to go back to a question I left unanswered: what does "Y" mean (and what does "Sd" mean) in the Loanable Funds theory? Because we now have a useful answer: "Y"

*should be interpreted*to mean "that level of output compatible with inflation staying at the Bank's 2% target". Define "desired saving" to mean "what the the level of desired saving*would be*if output (income)*were*compatible with inflation staying on target" and we get a nice way to reconcile loanable funds and liquidity preference theories of the rate of interest:**The rate of interest is set by the Bank of Canada, but the Bank of Canada tries to set the rate of interest equal to the rate of interest predicted by the loanable funds theory, where desired saving equals desired investment at potential/natural/NAIRU/full employment output.**

[

**Digression**:
What I have sketched above represents the "mainstream"/New Keynesian/Neo-Wicksellian way to reconcile the loanable funds and liquidity preference theories. But before passing on I want to note briefly three possible problems with that reconciliation:

1. The Bank of Canada is trying to hit a moving target (the "natural rate of interest") that it can't see (except, barely, looking backwards). So it will miss. But when it misses the target that will have real effects on the economy (if it sets a rate of interest too high, it will cause a recession, for example). Those real effects may themselves have persistent effects on the economy, and so may move the target from where it would have been in future. It's like trying to hit a moving target you can't see and where your misses themselves cause the target to move.

2. The IS curve may slope up, and probably will slope up if tight monetary policy causes a fear of prolonged recession which reduces desired investment and increases desired saving.

3. There are (in this highly-aggregated model) three goods: output; bonds; and money. It is very tempting to think that three goods means three markets, and that one market can be dropped by Walras' Law. But Walras' Law does not apply in a monetary exchange economy. And in a monetary exchange economy with n goods there are (n-1) markets, one for each of the non-money goods, where it is exchanged for money. There is no "money market". In this case n=3. There is an output market where output exchanges for money, and a bond market where bonds exchange for money. There is no market where bonds exchange for output (it would be a barter economy if there were). But the IS curve implicitly assumes people and firms are looking at the trade-off between output and bonds.

**End of digression**.]

Other Alternative(?) Theories:

1. The Irving Fisher diagram. You may have seen this diagram:

Equilibrium is where the Production Possibility Frontier and an Indifference curve kiss and the same budget line is tangent to both. It's exactly like the standard micro diagram where you have quantity of apples on one axis and quantity of bananas on the other. Except here you have "apples today" on one axis and "apples tomorrow (or next year)" on the other.

The slope of the budget line equals (1+the rate of interest). According to the Irving Fisher theory, the rate of interest is determined by intertemporal preferences and intertemporal production possibilities. In equilibrium, 1+the rate of interest (slope of budget line) equals the Marginal Rate of Intertemporal Substitution (slope of indifference curve) equals the Marginal Rate of Intertemporal Transformation (slope of PPF).

It looks very different from the Loanable Funds theory diagram. But really, it's exactly the same. You can derive the Id and Sd curves in the loanable funds diagram by varying the rate of interest (and so varying the slope of the budget line) and watching for the new tangency points on the PPF and Indifference curves. The tangency with the PPF tells you how much firms want to divert resources away from producing output for current consumption towards investment for output of future consumption. The tangency with the Indifference curve tells you how much of their wealth people want to spend on current consumption and how much thay want to save for future consumption.

2. Walrasian/Arrow Debreu General Equilibrium theory.

This theory is exactly the same as the Irving Fisher diagram, only with loads more dimensions. Lots of time periods, instead of just two. Lots of different goods, instead of just one. Lots of different people/firms, instead of just one. Lots of different states of the world (uncertainty), instead of just one (certainty).

3. The "The rate of interest is determined by the marginal product of capital" theory.

This theory ignores intertemporal preferences. It is not really a theory. It's either a misstatement of the equilibrium condition of the Irving Fisher diagram as a one-way causal relationship, or a special case of the Irving Fisher diagram where the PPF is a straight line so preferences don't affect the slope of the tangency. Or it assumes indifference curves are L-shaped. Or it assumes there is only one output good that can be switched costlessly between consumption and investment and that the stock of that good ("capital") does not change much within the period and can be taken as a predetermined variable. (For example, what happens to the Solow Growth model if people suddenly decided they didn't care at all about the future and wanted to eat the whole capital stock right now?)

4. The "The rate of interest is determined by the subjective rate of time preference" theory.

Same as the above theory, only the other way around. This one ignores intertemporal production possibilities. It's either a misstatement of the equilibrium condition as a one-way causal relationship. Or it assumes the indifference curves are straight lines. Or it assumes the PPF is reverse-L-shaped.

5. The "Start with Walrasian/Arrow-Debreu General equilibrium theory, make some simplifying assumptions,

*forget about the intertemporal preferences*, and then say "*Oh Look! the rate of interest is indeterminate, so it must be determined by my pet theory instead*"" theory of the rate of interest.
Again, not really a theory. More of a non-theory. Useful only as a critique of the "the rate of interest is determined by the marginal product of capital" non-theory, which also ignores preferences. (And even then, the simple Irving Fisher diagram can make the same point more simply and in a much more constructive way. Yes, in general you need to look at preferences as well as technology to determine things like interest rates and other relative prices.) An awful lot of ink has been wasted on this subject.

6. All the other theories I've either forgotten or am too tired to talk about.

[Update: 7. Euler equation. Same as Irving Fisher, only in math not pictures.]

Loose ends:

Aside from the problem of how exactly the loanable funds theory works in the short run in a monetary exchange economy, the biggest problem with the loanable funds theory is that it talks about "the" rate of interest.

Even when we are talking about nominal rates of interest, there are lots of different assets, that all differ by term, risk, liquidity, etc.

Let's just look at the term structure.

Most investment projects have costs and benefits that come at many different time periods, not just costs today and benefits tomorrow. When you look at the Net Present Value to see if the investment is profitable, you will have to look at the whole term structure of interest rates. You can't just assume there is one equilibrium rate of interest for all terms. (Not just empirically, but theoretically too, because with one rate of interest an n-period NPV calculation becomes an n-degree polynominal equation with up to n different solutions for the rate of interest at which NPV=0, which is what the "re-switching" debate was all about).

In principle, this is no problem. You just switch from the 2-period Irving Fisher diagram to a mult-period version. But in practice it's a problem. The desired investment curve (and desired saving curve too) will shift if excepted future interest rates change. So you can't solve for what the current period loanable funds diagram is telling you without solving a whole sequence of loanable funds diagrams. Then you have to ask how people form expectations of those future equilibrium interest rates, when some people are planning to invest or save in future but haven't entered into forward contracts. Expectations matter. And if you expect the Bank of Canada might make a mistake in future, it becomes even more complicated.

Now let's look at the distinction between real and nominal interest rates.

Suppose the Bank of Canada had always been targeting 3% inflation instead of 2%, and suppose it were always expected to go on targeting 3% instead of 2%. If money were "super-neutral", then no real ("inflation-adjusted") variable would be affected by the inflation target. The loanable funds theory would then predict that the nominal interest rate would be 1 percentage point higher, but the real interest rate would be unaffected. (Both desired investment and desired saving curves would shift vertically up by 1%, because borrowers and lenders care only about real interest rates, not nominal). Loanable funds would then determine the long run real interest rate, and we would have to add the nominal inflation target to get the long run nominal interest rate.

Empirically, that seems to be roughly true, in that countries with higher inflation rates over long periods do tend to have higher nominal interest rates. But it is unlikely to be exactly true. If it were exactly true, it wouldn't matter what inflation rate the Bank of Canada chose to target.

Even if monetary policy were super-neutral in the sense defined above, so it makes no difference to real interest rates if the Bank of Canada targets 3% instead of 2% inflation, that is not the only sense that matters. The Bank of Canada currently targets the Consumer price Index. It could also target some other price index, or the price of gold, or the price of wheat, or almost any other nominal variable. And it will probably matter, in real terms, which target variable it chooses. Targeting 2% inflation for the price of fresh strawberries would probably have very bad consequences for the economy. A bad strawberry harvest would probably cause a massive recession, as the Bank of Canada tried to bring all other prices down to prevent the price of strawberries rising faster than 2% per year. And God only knows how a stupid monetary policy like that would affect saving and investment.

For any given monetary policy target, there will be a time-path of nominal interest rates set by the Bank of Canada that could hit the target. But when we talk about the real interest rate, since the real interest rate is the nominal interest rate minus inflation, there are many different measures of inflation (because there are many different goods and many different bundles of goods, and the CPI is only one of millions) and so many different measures of "the" real interest rate. Bear that in mind as we return to look at the open economy version of the loanable funds theory.

The simple loanable funds model won't work in an open economy. That's because we can also borrow from abroad or lend from abroad. The Sd=Id equilibrium condition gets replaced by Sd=Id+desired Net Lending to Abroad. And the Ys=Cd+Ig+Gd equilibrium condition gets replaced by Ys=Cd+Id+Gd+desired Net Exports.

The usual fix for this problem is to say that loanable funds determines the world interest rate (the world is a closed economy, so the world interest rate is determined by equilibrium between world desired saving and world desired investment). And then perfect capital mobility ensures each country has the same interest rate as the world.

But if the Canadian nominal exchange rate is expected to depreciate, the Canadian nominal interest rate would be higher than in the rest of the world, by an amount equal to expected depreciation. And if Canada produces different goods from the rest of the world, there is no reason to expect that the real exchange rate between Canadian goods and foreign goods will stay the same over time. So the Canadian real interest rate could be either lower or higher than the real interest rate in other countries, even with perfect capital mobility, and even in long run equilibrium.

If wheat is expected to rise in price relative to oats, then the real interest rate calculated by subtracting wheat price inflation will be lower than the real interest rate calculated by subtracting oat price inflation. If Canada produces and consumes mainly wheat, and if foreigners produce and consume mainly oats, then the Canadian real interest rate (with lots of wheat in the price index) will be lower than the foreign real interest rate (with lots of oats in the price index). You will need to figure out what is happening to the relative price of wheat and oats if you want to say what determines the Canadian real interest rate.

I think that is the major practical problem with the loanable funds theory of the rate of interest. In an open economy, it's not enough to talk about Canadian saving and investment. It's not even enough to talk about world saving and investment. You need to talk about world saving and investment, and Canadian saving and investment, plus how the relative demand for canadian vs foreign-produced goods is expected to change over time.

This isn't complete. But it's already far too long. And I'm tired.