Upward-sloping IS curves after Miles Kimball

I hadn't realised until last week that Miles Kimball is way ahead of me on upward-sloping IS curves. (Miles doesn't want to call it an "IS" curve if it slopes up, but that's a pedagogical point.) I also hadn't realised how close Miles is to being One of Us (he calls himself a "new monetarist Ooops! "Neomonetarist", or more recently "Monetarist").

This post is me trying to get my head around Miles' head, and trying to say what he is saying in my own words. (Or in my own parables, which is just the old-fashioned name for "models".) But I'm more interested in whether what I say here is right, and less interested (though not uninterested) in whether I've understood Miles right.

[Update: Miles says on Twitter my model is very close to his model here, and see his graphs here and here and chapter 20 here.]

It's not as clear and simple as I wanted it to be. Sorry, but it was hard. But the punchline is easy to understand: low (real and nominal) interest rates are not a sign of easy monetary policy; they are a consequence of (expected) tight monetary policy.

Start with a simple New Keynesian model. Self-employed monopolistically competitive yeoman farmers with sticky prices use their own land and their own labour to produce different varieties of fruit. Standard neoclassical production function (maybe Cobb-Douglas) Y=F(K,L) where K is land. Each farmer sells his fruit for money and buys other farmers' fruit for money. (He has a chequing account at the Woodfordian central bank which can have either a positive or negative balance, and the central bank sets a rate of interest i it pays on positive balances and charges on negative balances, but because it's a representative agent model where each agent's money income is identical to his money expenditure his balance is always zero in equilibrium.)

Start in full equilibrium where no farmer wants to change his price. For simplicity assume the central bank targets 0% inflation (so nominal and real rates are the same), and that there is zero long-run growth. The rate of interest set by the central bank equals the rate of time preference proper, which we will call r.

Let e be the elasticity of demand for the fruit produced by an individual farmer. The (real) Marginal Revenue Product of land will be [1/(1-1/e)]MPK where MPK is the Marginal (physical) Product of land. There is no rental market in land (because farmers use only the land they own), but we can define the (real) shadow rent on land as R/P=MRPK=[1/(1-1/e)]MPK. There is also no rental market in labour (because farmers use only the labour they own), but we can define the (real) shadow wage on labour as W/P=MRPL=[1/(1-1/e)]MPL.

The price of land will equal the expected Present Value of the shadow rents. In full equilibrium, the (real) price of land Pk/P will be (R/P)/r. Call this the "natural price of land".

So far, there is symmetry between land and labour. But labour has an opportunity cost, if farmers get utility from leisure, while land has none. So if the central bank screws up, by tightening monetary policy, causing Aggregate Demand to fall, and Y to fall, it is labour and not land that will have lower employment. L falls, but K stays the same, so MPK falls, so the shadow rent on land falls too. [Update: the shadow rent will also fall because the marginal utility of leisure falls and MPL rises, since when a farmer is sales-constrained the only advantage of owning more land is that he can produce the same amount of fruit while working even fewer hours.]

So if the central bank screws up, by setting i above r, the real price of land will fall below the natural price of land. It will fall for two reasons: the higher interest rate will mean expected future shadow rents are discounted at a higher rate; the shadow rents will themselves be reduced by the recession. And the size of this second effect depends on how long the farmers expect the recession to last.

Suppose farmers expect a level of (real) demand Ye to last for t periods before the economy returns to full equilibrium Y*. Taking farmers' subjective expectations as given, we can define a relationship between {i,Ye, and t} such that the real price of land is equal to its natural rate. That relationship is (I think) equivalent to Miles' KE curve. It slopes up. If farmers expect a recession, so Ye falls, the central bank would need to cut the rate of interest i to compensate for the fall in shadow rents to prevent the real price of land falling below the natural price.

The IS equation in this economy will be the standard New Keynesian Consumption-Euler equation.

Suppose farmers expect the recession to last for one (very short) period. They expect (real) demand to stay at Ye for one period, then return to full equilibrium Y*. What interest rate would be consistent with that expectation? That interest rate would need to be higher than the natural rate for one period, so that this period's consumption demand would be lower than next period's consumption demand. The "Short Run" IS curve will slope down. And the effect on the price of land of a recession that is expected to be very short will be trivial, because it only affects shadow rents for one very short period.

Suppose farmers expect a permanent recession. They expect (real) demand to stay at Ye forever. What interest rate would be consistent with that expectation? Given standard preferences, the answer is the same interest rate as when the economy is expected to stay at full equilibrium Y* forever (an interest rate equal to the rate of time preference). So the "Long Run" IS curve is horizontal. (This is a standard Consumption-Euler equation result, because it is the expected growth rate of consumption, not the level of consumption, that depends on the rate of interest, and the two are positively related.) But the price of land will rise as we move rightwards along the IS curve, to higher levels of Ye and higher levels of shadow rents.

Now let's bring in Dutch Capital Theory. The farmers figure out a way to create new land, by converting fruit into land. So what was once "land" now becomes "capital". [Cambridge UK types may notice I have rigged it to ensure the investment sector always uses the same capital/labour ratio as the consumption sector, so the PPF between consumption and investment is a straight line with a slope of minus one, which pins down the real supply price of new capital at one.]

If each farmer could use his own fruit to create more land, we would see an investment boom in recessions, for exactly the same reason we see Earth Sciences students who can't get a job going back to grad school to invest in more human capital. If you can't sell the fruits of your labour temporarily, the opportunity cost of using it to create more capital is lower, so investment increases.

But instead we will assume that you need a variety of different fruits to create more land. The investment sector has exactly the same "technology of variety", with the exact same elasticity of substitution between varieties, as the consumption sector's taste for variety. Each farmer buys a variety of fruit from other farmers, consumes some, and invests the rest to create more land. Land, or capital, is just congealed multi-fruit jam.

And to ensure we have a steady state with zero net investment in land, we will assume land depreciates at rate d. So net investment (the change in the stock of land) is gross investment I minus dK.

Suppose farmers expect a permanent recession. They expect (real) demand to stay at Ye forever. What interest rate would be consistent with that expectation? The lower is Ye, the lower are shadow rents, and the lower is the real price farmers would be willing to pay to buy new land, and so the lower is the rate of interest that would be needed to ensure that farmers buy enough new land to keep net investment at zero. The Long Run IS curve slopes up.

So, whether the IS curve slopes up or down depends on how long the recession is expected to last, and on the importance of investment as a component of aggregate demand. It could go either way. But if the IS curve slopes up, then it means that if the central bank screws up, and creates a recession, it would need to lower the interest rate below the long-run natural rate just to prevent the recession getting worse.

Low (real and nominal) interest rates are not a sign of easy monetary policy; they are a consequence of (expected) tight monetary policy.

Someone with an absolute and comparative advantage over me (e.g. any economics grad student) should have no difficulty doing the math to convert my parable into a model. The model will be easier to solve if we assume gross investment can be negative (farmers can consume the congealed jam they own). Alternatively, if we want to create a smooth net investment function at the aggregate level, with putty-clay investment so it's irreversible, assume there are persistent relative demand shocks so some farmers will always want to invest even in a recession.

Comments

[John Handley]

Nick,

"Low (real and nominal) interest rates are not a sign of easy monetary policy; they are a consequence of (expected) tight monetary policy."

Or, you could think of it this way: higher interest rates mean that spending is substituted from the present to the future and vice versa. In this way, defining loose monetary policy as high spending relative to the future, low interest rates are always expansionary - even if they are consistent with recessions that last for a long time. It has everything to do with expectations. If interest rates (real and nominal) are expected to return to normal at some point, then lower interest rates represent loose money in an absolute sense; current nominal spending is higher and nominal spending growth is expected to return to normal in the long run. If, as is probably the case in Japan, interest rates are expected to be permanently lower than they were in the late 80s, then expectations of future nominal spending growth have fallen. This is, of course, rational in the case of Japan because of the demographic decline (perhaps combined with low inflation expectations, at least between 1997 and 2012). I guess, at least in a New Keynesian framework, we should be thinking about the expected path of interest rates to understand the expected path of nominal spending and the current interest rate to see where nominal spending is relative to future nominal spending.

[Nick Rowe]

John: if you were talking about a consumption-only economy, I think what you say would be right. If we are in recession today, so output is lower than normal, and is expected to rise, then the (real) interest rate must be above the long-run natural rate.

But when we add investment to the model, it isn't right. Because the fact that the economy falls into a recession means that the desired capital stock (for a given real interest rate) is less then the actual capital stock. Because the shadow rental rate on capital goods falls when the economy enters a recession. So the real interest rate would need to fall to offset that negative effect on investment demand.

[John Handley]

Nick,

As long as the economy converges to a balanced growth path, it shouldn't matter very much. Lets say there is an economy with a representative agent who maximizes the utility function U = B^t(u(c_t)) where 0 < B < 1 is the discount factor and u(c_t) = 1/(1-a)c_t^(1-a) subject to the budget constraint (1 + r_t-1)k_t-1 + y = c_t + k_t where k_t is the capital stock that will be carried over into the next period, r_t is the next-of-depreciation real interest rate, and y is the constant endowment that the representative agent is given each period. The first order condition for consumption is c_t^-a = B c_t+1^-a (1 + r). Assuming this economy converges to a balanced growth path in which every variable grows at g, the real interest rate will be r = r* + ag where r* = 1/B - 1 is the time preference rate. Low interest rates are always consistent with lower long run growth and vice versa. In the short run, the 'cyclicality' (pretty sure this isn't really a word, but it sounds cool) of the real interest rate should depend on the kind of shock.

[Nick Rowe]

John: what you are describing there is a Robinson Crusoe model. What I am describing is a symmetric Prisoners' Dilemma model. Robinson Crusoe never wishes he would buy more of his own goods. My agents in a recession wish that they would all buy more of each other's goods, but it is not individually rational for them to do so.