Nominal Prices and Output Gaps

This post is my attempt to respond to the SRW’s graphical model of Keynesian and monetarist recessions. In particular, I have a different take on what is a Keynesian recession. I don’t think it has necessarily anything to do with sticky prices or leverage —  although these are all important in the real world, you don’t need them to describe variations in output and employment; you don’t need real productivity shocks, either.

I’m going to try to elaborate on this passage, in which Keynes is addressing Hicks’ IS-LM:

Put shortly, the orthodox theory maintains that the forces which determine the common value of the marginal efficiency of various assets are independent of money, which has, so to speak, no autonomous influence; and that prices move until the marginal efficiency of money, i.e. the rate of interest, falls into line with the common value of the marginal efficiency of other assets as determined by other forces. My theory, on the other hand, maintains that this is a special case and that over a wide range of possible cases almost the opposite is true, namely, that the marginal efficiency of money is determined by forces partly appropriate to itself, and that prices move until the marginal efficiency of other assets fall into line with the rate of interest.

And my approach is an interpretation of Jan Kregel, from whose essay the quote is cited.

Capital Values

Consolidate the financial sector + government into a “bank”. There is no bond market per se — households purchase deposits in the bank and receive the (overnight) government mandated rate, b(t). Anyone can borrow from the bank and pay an overnight rate of b(t) as well. However, there is a capital market, in which capital goods are bought and sold, and these also deliver a (continuous) income stream.

Firms in our model are owned jointly by households that supply capital to them. There are only two firms — consumption goods producing firms and capital goods producing firms. This is a continuous recursive competitive equilibrium. At each point in time, the stock of capital is fixed, but the growth rate of this stock is variable. Assume continuous market clearing at all times (but remember the government is “fixing” the nominal rate).

Households hire managers to operate the firms. There are no intermediate inputs, so earnings are just the difference between revenue and labor payments, and these earnings are passed onto capital suppliers — firm managers have no slush fund in which they can keep earnings for themselves.

The optimization problem of the firm manager is to maximize earnings per unit of capital supplied, r(t), given the quantity of capital supplied by households.

Households, by shifting capital among the consumption goods and capital goods producing firms, ensure that both firms deliver the same earnings. In that case, there is no arbitrage possible by holding a deposit or holding capital, so that

r(t)+ \frac{dq}{dt} = (b(t)+j)q(t)


  • q(t) is the price of a capital good at time t
  • b(t) is the overnight rate
  • j(t) is the (exponential) depreciation rate
  • r(t) is earnings per unit of capital
This is possible as households can borrow (from the bank) to purchase capital and of course they will not pay more to own capital than they would for ownership of the equivalent return from deposits. The above is a simple differential equation with solution:

q(t) = \int_t^{\infty} e^{-(s-t)\left[x(s,t) + j\right]}r(s)ds

where x(s, t) = \int_t^sb(l)dl is the mean rate between periods t and s. The above holds if the integral converges, for example if \liminf_s x(s, t) > 0. If desired, you can add variance terms to b(t), but the specific form of this equation is not important. What is important is that changes in short run earnings are not going to correspond one to one with changes in the price of capital goods.

Let’s combine this with our simple model of firms. For example, suppose that the consumption goods producing firm, F, and capital goods producing firm, G, have production functions:

F(K_c, L_c) = K_c^{1/2}L_c^{1/2}
G(K_k, L_k) = K_k^{2/3}L_k^{1/3}

Note that the capital goods producing firm is more capital intensive — production of consumption goods includes services that are more labor intensive.  In that case, the earnings can be expressed as

(1) r_c = \frac{p^2}{4w} = r_k = \frac{2q^{3/2}}{(27w)^{1/2}}

In particular, the aggregate labor demand curve depends on the proportion of capital allocated towards consumption:

LD = K\left(1 + \frac{K_c}{K}\right)\left(\frac{q}{3w}\right)^{3/2}

What is happening here is that fluctuations in the prices of consumer goods create fluctuations in the ratio of capital goods to consumer goods prices. As households withdraw capital from the consumer goods producing firms and increase the capital supplied to capital goods producing firms , the differences in labor intensity creates shifts in aggregate labor demand and aggregate output. Even though there are no productivity shocks or rigid prices. Markets are clearing at all times.

In graphical terms, capital goods prices are a function of the exponential weighted average of consumption good prices, and so move separately:

But the ratio of returns for capital and consumption producing firms is given by the ratio of capital to consumption goods prices and the square root real wage.

r_K/r_C = A\frac{q}{p}\left(\frac{p}{w}\right)^{1/2}

A 1% increase in the ratio of q to p will force the real wage fall by 2% — the labor demand curve will keep shifting to the right until this happens:

The only way that output and employment could be kept constant would be if the labor supply curve also shifted to the right to match the rightward shift in the labor demand curve. Basically, as households attempt to save by withholding purchases of consumption goods, they must relinquish these savings by lowering their wage demands if they are to remain employed. Even though firms will cut prices in response to a decline in demand, they will shed labor even faster.

Looking at Output Gaps

If we define trade cycle unemployment or conjunctural unemployment as changes in employment due to movements in the labor demand curve not attributable to real productivity shocks, then there is such a beast even under assumptions of full market clearing.

Rather than having interest rates adjust to maintain full output, output and employment adjusts when demand for consumption decreases.

In this simple example, we have an interpretation of Keynes’

the marginal efficiency of money is determined by forces partly appropriate to itself, and that prices move until the marginal efficiency of other assets fall into line with the rate of interest.

The forces “partly appropriate to itself” correspond to expectations of future prices, and the other prices that adjust to match are the decline in the real wage necessary to accommodate the expectation of future return.

Of course, a better analysis would also include changing risk-premiums, and something like a production in advance constraint, in which case other effects would be dragged in. But the purpose of this discussion is to argue that you do not need sticky prices, sticky wages, or real productivity shocks to explain output gaps. The supply side moves in response to nominal profit expectations even under a flexible price model.

A temporary increase in the demand to save is enough of an explanation for cyclical unemployment.

Update: Made some edits for clarity. Added bonus video for those who made it to the end!

Nominal Prices and Output Gaps

12 thoughts on “Nominal Prices and Output Gaps

  1. RSJ — First of all, excellent work (as usual).

    I like this model, and there’s nothing about it I’d much argue with. We’d get into semantic squabbles: I would characterize this as a sticky price model. Your intermediate good is priced according to expectations that do not fluctuate one for one with changes in demand for current output. In the stories I’ve been telling (that you don’t much like), rigidities associated with incentives of shareholders in leveraged firms drive the stickiness. In your story, it is sticky expectations regarding future interest rates that drives the output gap. If shocks to save were correlated with changes in future return expectations (or with changes in the financing market for capital goods, so that the price of capital goods is below your forward-looking no-arbitrage price), then capital goods prices would fall in sync with the demand shock and there needn’t be a fall in output.

    Anyway, I’m perfectly happy with positing moderated changes in capital goods prices as a source of what I call price stickiness. More generally different rates of price adjustment across the universe of goods does the job. I can’t show that in my one good + money model (since there are no capital goods), and I won’t back off from my intuition that price stickiness due to leverage plays a role. But I think sticky expectations about future returns of different (present or future) goods, are likely to play a role as well. I also think nominal prices serve as focal points of coordination equilibria, and are committed to by virtue of interrelated commitments, and then are abandoned only reluctantly. The universe of imperfections is a very big tent.

    I suspect you don’t like characterizing your model as in the generalized family of price stickiness. But if our dispute is just taxonomy, I’d rather not devote a lot of energy to it.

    Anyway, I do think this is very nice, thought provoking. There are gaps in my understanding. You’ve got a lot of moving parts: relative prices change, investors change the fraction of capital devoted to each good per the no-arb condition, piling out of the consumer good, into the capital good until returns are equalized, input prices change, firms choose output that matches final consumer demand, unemployment and output gaps result. It’s not obvious to me how all these gaps are filled in. But that’s probably my deficiency more than the posts.

  2. SRW,

    I am also OK calling this a “sticky” model, as long as we are clear that the stickiness arises from voluntary exchange, and not something that interferes with voluntary exchange.

    If people believe they can sell the house for more next month, then they will not lower the price this month. If they are credit constrained or in a panic, then they might sell the house for a much lower price. In a production in advance model, the speed of decline of capital goods prices will create additional shocks to labor demand.

    But what are the odds that they would sell the long lived good for exactly the same reduction in price as the short lived good? Such a thing can happen, but it would happen by accident. Regardless of what formula you use to calculate “q”, it will not move one for with “p” except by a sheer fluke.

    What is more interesting is asking if the CB can force q to move with p one for one. I don’t think it can, because you run into zero bound problems when p drops quickly. And in any case, the CB isn’t trying to accomplish this.

    What I am trying to argue for — although I am not proving anything — is that there is fundamental stickiness in any system with goods of differing duration, which could be one explanation of why money takes on a life of its own. In that case, governments should do well to be more worried about periods of high profitability and high returns. In my Kalecki moods, I think the last 20 years were a profit fluke, and the biggest problem with the economy right now is that holders of capital are refusing to accept a sustainable return that is much lower than the average return over the last few decades. More than anything else, that might be the stickiness to worry about.

  3. RSJ — All good points. My sense is you’re right about “holders of capital are refusing to accept a sustainable return that is much lower than the average return over the last few decades”. At some level, I think we’ve been struggling even for decades against a crossover, in that the marginal product of undifferentiated capital is no longer greater than the difficulty of storing or reproducing wealth over time. If that’s so, the “natural real interest rate” should be negative. But we have all kinds of theory and institutions bound up in the idea that one should be able to multiply perishable goods simply by failing to consume them and running them through the financial system.

    A bit of a quibble — in a “flexible price” variation of your model, q needn’t move with p “by mere fluke”. One might (and someone more wedded than you or I to the idea of self-equilibriating markets almost certainly would) modify your model with some wrinkle or constraint that ensures this result. The core technique of economics is, after all, teleology. We presume agents are always optimizing, and we (or perhaps they) usually design our “base” models so that somehow optimization by agents composes to a general optimization (by whatever criteria, Pareto efficiency, some social welfare function, or in this case real output). Then we introduce “imperfections” or “distortions” that might upset things. All we’d need to do to shoehorn your exercise into the mold is to invent some story that binds moves in q to moves in p and define that as the “basic model”. Then your construction of q, as a forward-looking no-arb over an exogenous or sluggish set of overnight rates, becomes a twist, an imperfection to be considered by graduate students but never to be mentioned in introductory courses.

    I have a question about your model: Suppose there is an upward shock to demand, a sudden diminishment of consumers’ propensity to save, would that also cause a fall in output? Again, to my discredit, there are pieces of your story I am missing and filling in in general terms. In outline, the effect of a downward shift in output seems like it is due to two components 1) demand for the final good falls; and 2) misalignment of final and capital goods prices means that post-shock, capital is redistributed from final to capital goods providers until returns are equalized, but production consistent with that redistribution is at a lower level than initial production. In essence, there is a demand component and a “structural” component to the loss in output (although the structural mismatch is a result of the demand shock). If there were a positive demand shock, would we find prices misaligned in the opposite direction, so that capital would flow from intermediate to final goods to equalize returns? Your Cobb Douglas production functions include no hard capacity constraints nor do they require inputs in certain ratios, so one might expect that the positive shock to demand would simply be reflected as increased output. But it’s unclear to me whether the sluggishness of the capital goods price would increase or reduce output. I guess intuitively, I think your model predict that a positive shock to demand increases real final goods output by effectively reducing the cost of inputs and so increasing producers’ supply. Is that right?

  4. A positive shock to demand for consumption (there is no “aggregate demand” per se in this model, as the demand for capital goods is horizontal) would increase output in some cases but not in others. There is a “zone” in which this would happen, as the capital goods producing firms are more productive per unit of labor employed, so that compensates somewhat for the decline in labor used when q/p falls, and the reverse is true when q/p increases. There is more labor force participation, but the labor is allocated towards less productive industries. It also depends on how you measure real output — if you divide by income by p(t), then you will get shifts in real output that do not correspond to shifts in quantity produced. I haven’t tried to add a “general” price index due to the aggregation issues. You could add quantities of capital and consumption goods produced, but that doesn’t seem like a relevant measure, as the productive characteristics of the capital goods are varying with quantity.

    I’m going to post more on this model, but not today. It has many problems, but the purpose of the exercise is to point out how individual optimization, specifically arbitrage, prevents the social welfare optimum from being reached. This is possible because we are not in an Arrow-Debreu framework — prices move over time, and it is not possible for someone to store labor or consumption for use in a future period. There are many example of recursive competitive equilibria not producing pareto optimal results. There are many examples of overlapping generation models not producing pareto optimal results. You can get Keynesian style effects in an individual optimizing framework without imposing frictions.

  5. beowulf says:

    “Basically, as households attempt to save by withholding purchases of consumption goods, they must relinquish these savings by lowering their wage demands if they are to remain employed. Even though firms will cut prices in response to a decline in demand, they will shed labor even faster.”
    Of course, that’s aggregate households. Since neoclassical economics is basically a faith-based endeavor, I call and raise them:
    Jesus…said, “this poor widow has given more than all the others who are making contributions. For they gave a tiny part of their surplus, but she, poor as she is, has given everything she had to live on. Mark 12:43-433.

    So when wealthy households save a larger part of their surplus, it means that poor households have to take a cut in everything they have to live on. Hmm, like a Greek play, a happy ending will require a deux ex machina, like a floating payroll tax holiday.
    Say, suspending tax collections at 10% x U3 unemployment rate (correlated, Okun’s Law, to change in GDP). Every month (or every quarter, to minimize paperwork) after BLS reports latest U3, Tsy adjusts withholding schedules. So a 9.1% U3 suspends 91% of baseline payroll tax (in US, 15.3% FICA). For someone earning $30,000, employer and employee would each keep $2088 of baseline $2295, boosting effective demand and lowering wage demand.
    Since full employment is somewhere between 3% and 4% (at the end of Clinton Admin, it hit 3.8%, end of Johnson Admin, 3.5%), tax holiday would never get to 0. That fiscal gap would be partially filled by larger GDP / tax base at full employment (Okun again) and uncapping SS FICA (12.4 out of 15.3 points) above $106K would cover the rest. There’s never a better time to raise taxes than when it can be rebated 91%. Likewise, a floating payroll tax holiday allows for an effective tax cut even if FICA rates were boosted to fund a Medicare for All system (I’d like to think the economist/rabbi quoted above would approve).
    Since part of the neoclassical creed is that full employment is a 5.0% U3 rate (reality and the Full Employment Act mandate of no higher than4.0% notwithstanding), we could accommodate their intelligent design beliefs by changing formula to 5% x U3 rate, so a 45% rebate from current rates.

    1. I like this idea, but would combine it with an increase in income taxes on the top earners. Recessions should be periods when we attack income inequality with full force, rather than just supplying income.

  6. beowulf says:

    Thank you, and I agree with your point about income inequality. I suppose you could uncap SS FICA at the same you limit tax holiday to incomes below, say, $1 million or maybe expand Obamacare’s unearned income Medicare tax rate from 3.8% to 15.3%. Payroll tax holiday was apparently first suggested by James Meade and quickly endorsed by his buddy JM Keynes.

    Speaking of buddies, Sheila Bair must be your soulmate. The new FDIC deposit insurance fee structure look rather like your “tax the bank rents” plan, FDIC shifted base from deposits to assets (and like your plan, adjusts down for long term unsecured debt). Apparently, after the Fed pays out 0.25% interest on reserves, .015% will be taxed away by the FDIC. That’s just ridiculous, either the Fed should be insuring deposits or the FDIC should be setting the policy rate.

    1. I don’t see why this is ridiculous. Rates are set by arbitrage, not tax considerations. For example, taxing capital income does not cause the cost of capital to increase, provided the tax is applied equally to all forms of capital income. In this case, a bank has the option of not lending money to another bank, and getting 25b.p, or lending money to another bank. In that case, it will lend to the other bank at 25b.p, right?

      Now impose a 25b.p. tax on all bank assets — including loans to other banks and reserves. Does the interbank lending rate change? No, it’s still 25 b.p. So is the goal of IOR to supply money to banks, or is the goal of IOR to control interbank rates? If you believe the goal is the latter, then you don’t care about asset taxes.

  7. beowulf says:

    I’m sorry, I fouled up the decimals… 25 basis points paid out to banks by Fed IOR, 15 basis collected from bank by FDIC deposit insurance fees.

  8. beowulf says:

    The ridiculous part is one agency of the US govt so at cross purposes with the other. The Fed was in middle of QE2 cutting interest rates at the same time the FDIC is increasing insurance fees (as Warren notes in linked thread, “FDIC insurance is a bank tax that’s entirely passed through to borrowers as a higher interest rate that’s a function of the fdic levies”).

    I liked your ideas of Tsy capturing seigniorage with a bank asset tax. But it was definitely thinking outside the box, making it hard to visualize and harder to explain. Am I mistaken that Sheila Bair and crew shave moved the ball forward by going to asset-based fees? I do like the thought of 0 interest at the Fed, with the de facto policy rate is set by FDIC adjusting its insurance fee schedule. :o)

    1. No, in principle they are not at cross purposes because an asset tax — as long as interbank loans are counted as “assets” along with reserves, so that both are taxed at the same rate — will have no effect on interbank rates. The policies are orthogonal.

      That said, I don’t know details of the actual FDIC rate schedule, so the policies may be at cross purposes in practice.

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