## The RG/MV Model

In my last post, I discussed my complaints with the standard approach to teaching aggregate demand in an intro class. I have been trying to come up with a better way of doing it and that has spilled over into IS/LM. I think I have a better model for that too. I will try to describe it here. For the record, I’m not saying IS/LM is “wrong” exactly, just that it is misleading and is not a very clear representation of the relationships it is meant to explain. My model can be turned into the standard IS/LM model with a few assumptions. But forcing you to make those assumptions, I think, helps greatly to understand what is going on.

The point of aggregate demand is to connect the real economy to the monetary economy. All short-run deviations from the long-run equilibrium are due to the monetary mechanism not functioning perfectly. This is what we are trying to model. The threads which connect these two things are the price level and the interest rate. Much like the traditional model, I will divide the demand side of the economy into two sectors: the real goods market and the money market. For the sake of simplicity, I am assuming a closed economy with no taxes or government spending. These other things can be added but it is a little more complicated than just adding a G onto aggregate demand (which, remember, is the whole point). I will leave that for another time. We will start with real goods.

**The RG Curve**

Assume that people have some preferences over consumption now and future real wealth (you can say future consumption if you prefer, but I like this way of saying it better). They also have a budget constraint. The budget constraint depends on income and the real interest rate. Solving the maximization problem is a simple microeconomic exercise. You will get a function for consumption and one for investment, which depend on income Y and the real rate r.

Now, at this point, you can recognize that these functions could have a lot of different properties, but let’s *assume* that they have the following.

I_{Y}(Y,r)>0

I_{r}(Y,r)>0

The response of consumption to changes in either variable is not of fundamental importance. Note that you may assume that the consumption function is linear in Y and independent of r, but this is a restriction on consumer preferences and not necessary. (This eliminates one of my pet peeves about IS/LM.)

Now let this function I(Y,r) be the supply of loanable goods and let the demand for loanable goods be determined by the investment opportunities faced by firms such that the marginal product of investment is always equal to 1+r. Equilibrium in the market for loanable goods, then requires the quantity supplied to be equal to the quantity demanded.

Figure 1

This will determine an equilibrium real interest rate for any level of income. Note that expenditure (C(Y,r)+I(Y,r))is always equal to income. This is just the budget constraint from the consumer maximization problem. So equilibrium in the real goods market (which is made up of equilibrium in the loanable goods market as well as the consumption goods market since, from the consumer maximization we always have I(Y,r)+C(Y,r)=Y) implies a relationship between income and real rates.

Now if we assume that income equals output (Y), then we have a relationship between output and real rates. Note that this assumption, along with the consumer budget constraint, essentially represent the “Keynesian cross” from IS/LM. When income is higher, the supply of loanable goods will increase and the real interest rate will be lower. We can then write a function representing all combinations of income (Y) and real rates (r) which are an equilibrium in the real goods market. Let’s call this the “RG curve” for “real goods.” This will be downward sloping and essentially equivalent to the IS curve.

**The MV Curve**

Now consider the role that money plays. Notice that we did not include the price level or the nominal interest rate in the real goods markets. Also, remember that aggregate demand can be written as PY=MV. Let M be exogenous. Then our goal is to explain V.

Let L(i) be the fraction of total expenditure that people are willing to hold as money. This will depend on the interest rate. The higher the rate, the less cash they will be willing to hold, since the nominal rate is the price of holding money. If people are holding more money than desired, they will try to spend it, *either *on additional consumption or investment (probably by lending it) and this will have to increase *either *the price level or total output and therefore reduce velocity (and vice-versa if they are holding less than desired).

Equilibrium in the money market requires velocity to be 1/L(i) and this must, by definition, be equal to PY/M. This can be seen on a graph of V and Y for a given P and i.

Figure 2

This gives us a level of output which is consistent with the demand for money for any given price level and nominal interest rate. This will be increasing in i (since L_{i}<0), decreasing in P and increasing in M. Call this function the “MV curve.”

**RG/MV**

At this point we have to deal with inflation expectations. One key feature of this model is that it is very explicit about which interest rates it is talking about where and so you have to deal with inflation expectations explicitly as well. This eliminates another one of my pet peeves about IS/LM (the more important one). The simplest way to do this is to assume that they are exogenous, and don’t depend on any of the other exogenous variables. Then, in equilibrium, the Fisher equation must hold.

i=r+π^e

Then we can rewrite the RG curve as RG(i-π^e). Then we have a nice downward-sloping RG curve and upward-sloping MV curve in i/Y space. Equilibrium in both the real goods market and the money market will determine a quantity of total output demanded and a nominal interest rate for any given price level. (Consumption, investment, velocity, and the real rate can all be easily recovered from this.)

Figure 3

**Aggregate Demand**

The aggregate demand curve is then a function giving all combinations of output and price which make up an equilibrium in both the real goods market and the money market. It can easily be seen that this is downward-sloping in price. If the price is higher, PY/M will shift to the left in figure 2 which will cause the MV curve to shift to the left and imply a lower aggregate quantity demanded. Furthermore, we can deduce how the curve will shift when the following exogenous variables change.

Money supply

If the money supply increases, this will shift PY/M to the right for any given P, which will cause the MV curve to also shift to the right and cause the aggregate quantity demanded for any given P to be higher. Note that for a given price level, and inflation expectation, this will lower the nominal rate, increase investment and decrease velocity. The degree to which the increase in money is absorbed by a decrease in velocity vs. an increase in aggregate quantity demanded depends on the slope of L(i).

Inflation expectations

If inflation expectations increase, this will cause RG(i-π^e) to shift to the right since any level of i will represent a lower level of r. This will cause the aggregate quantity demanded to be higher for any given P (an increase in aggregate demand). Note that this will cause an increase in i, but a *decrease* in r since the RG curve will shift up by exactly the amount of the increase in inflation expectations but it will move along the MV curve. This will mean an increase in output, investment and velocity.

Time preferences

If people become more impatient and want to consume more today, this will cause a decrease in investment supply and higher real rate for any given level of income which will cause the RG curve to shift to the right. This will cause the nominal rate, velocity and aggregate quantity demanded to increase for any given P.

Investment opportunities

If a new technology is discovered which increases the marginal product of investment, the demand for loanable goods will increase. This will cause the RG curve to shift to the right and the AD curve to do the same, which will increase investment, output and real and nominal interest rates. Note, that this may crowd out some consumption, depending on the shape of the indifference curves.

“Animal spirits”

If people decide they want to hold less money—L(i) decreases—then 1/L(i) will be higher for any given i which means that the MV curve will shift to the right and the nominal and real rates will be lower, aggregate quantity demanded, velocity, and investment will be higher.

**Remarks**

This model has two main benefits compared to IS/LM. One is that it has a bit more “micro foundations” in that it explicitly incorporates consumer preferences and demand for investment by firms. The other is that it carefully distinguishes between real rates and nominal rates which makes the transmission mechanism for monetary policy much more clear in my opinion. It also fits better with my framework of thinking about AD as Y=MV/P and dividing things into their effects on M and V rather than thinking in terms of C+I+G+NX and dividing things into their effects on those respectively (although note that you can still do that with this model). I see this as a sort of monetarist version of IS/LM, though I don’t know if people with monetarist street cred would agree with me or not.

So far I haven’t explicitly tried to incorporate fiscal policy. To do this would require you to make some assumptions about how it affects peoples’ preferences and where the money comes from. (Is it from taxes or borrowing? If the latter, is it borrowed from the private market or from the central bank increasing the money supply?) Note however, that these questions are central to determining the effect of such policy. You probably could just slap it onto AD if you wanted to assume that there were no effect on preferences (no crowding out) and you could change the household budget constraint to Y-T. If the government borrows, you could add their demand to the demand for loanable goods.

Sorry the figures look like crap. I need to figure out a better way to get them into wordpress