[Update 4 December: after long arguments with Adam P., which finally started to bear fruit in the comments of December 4th, I'm now clearer on what's essential in this model about the difference between "monetary exchange" and "barter exchange". It's this: In monetary exchange, there is no restriction on what the seller can do with the proceeds of a sale. He can spend the proceeds on anything, or save them. In barter exchange, there is a restriction on what the seller can do with the proceeds of a sale; he must spend all the proceeds on the buyer's goods. That is what is driving the results in this model. I think that is one important distinction between monetary and barter exchange. I have left the rest of the post as is.]

Keynesian macroeconomics in general, including New Keynesian macroeconomics in particular, makes absolutely no sense whatsoever in a barter economy. If people could trade goods and labour directly at zero transactions costs, without having to use monetary exchange, all Keynesian macroeconomics would be total rubbish. All Keynesian macroeconomics, either explicitly or implicitly, assumes monetary exchange. It's not just sticky prices that generate Keynesian results. It's sticky prices plus monetary exchange.

I used to think that the above paragraph was totally uncontroversial. I thought everybody understood this. I have learned they don't. So I'm going to do my best to explain why it's true.

I show that if we introduce barter into an otherwise nearly standard New Keynesian model, the solution immediately reverts to the perfectly competitive equilibrium, regardless of the degree of imperfect competition in the original model, regardless of any mistake made by the central bank in setting the nominal rate of interest, and regardless of sticky prices.

Just to be clear, this is in not a criticism of Keynesian macroeconomics. Barter is costly, and people do use monetary exchange. It's entirely reasonable that a model should assume people always use monetary exchange and cannot barter. And Monetarism would also be total rubbish if people could easily use barter instead of monetary exchange.

And just to be clear again, I am not saying that the particular way in which Keynesian models incorporate monetary exchange is unproblematic. I think it is problematic. But that's not the (main) point of this post. It's just an aside.

What I am going to build is a model of a model. Just as a model is an interpretation of the world, my model is an interpretation of a model. It's a model of the canonical New Keynesian model.

Here's my model.

There are n people, where n is a large enough number that each individual ignores the effect of his choices on the aggregate. Each person is a worker/firm, who produces one type of fruit. Each person produces a different fruit. The apple producer gets disutility from working L hours to produce A apples, with a production function A=F(L), and utility from consuming a Dixit-Stiglitz basket of fruit C with elasticity of substitution e (different fruits are imperfect substitutes). Each person maximises an infinite horizon subjectively discounted sum of present and expected future utilities U(C,L) etc.

In other words, exactly the same set-up as in the standard New Keynesian model, except I have suppressed the labour market for simplicity. Each firm is a worker. (If the labour market were perfectly competitive, and money wages were perfectly flexible, it would be exactly like my model.)

I will make one other small simplifying change to the standard New Keynesian model. I will assume that all firms must set prices before the central bank announces the nominal rate of interest. The firms that can change prices in a given period can only do so before observing the nominal rate of interest. This allows me to consider a symmetric equilibrium, where all firms have set the same price, and this makes everything much simpler when the central bank sets the "wrong" rate of interest. (Under the standard Calvo assumption, some firms would change their prices immediately when the central bank sets the wrong interest rate.)

I want to take this same underlying model, and impose two different trading structures. First I will impose "monetary exchange"; then I will add "barter". I want to show that adding barter makes a difference.

In what follows, remember that the Walrasian auctioneer is a great fiction. The auctioneer imposes a trading structure, and also solves the equations. I will introduce auctioneers. Like the Walrasian/Edgeworthian auctioneer, they will use "tatonnement with recontracting" to solve the equations. (Groping towards the solution, and all deals are re-negotiable until the auctioneer finds the solution). Unlike the Walrasian auctioneer, first there will be a tatonnement over prices, prices are then fixed, and only then will there be a tatonnement over quantities.

"Monetary exchange".

There is a central bank that also acts as auctioneer.

First, the central bank holds a tatonnement with recontracting on prices. Each firm announces a price, to maximise expected profits/utility. The auctioneer then announces all those prices, and asks if any firm wishes to change its price (recontract), now that it has seen all the prices set by the other firms. When no firm wishes to recontract, the auctioneer then closes the auction, and those prices are then fixed for the period.

Second, the central bank announces a nominal rate of interest.

Third, the auctioneer asks each firm how much fruit it would like to produce and sell.

Fourth, the auctioneer conducts a tatonnement on quantities. The auctioneer announces a tentative initial vector of quantities of fruit sold. That initial vector can be arbitrary (quantities, not prices, criee au hazard). Each firm is assigned monetary credits equal to the quantity of fruit sold times the price it set. The firm then decides how to spend those credits. It can spend them on fruit, or save them. Any unspent credits earn interest and can be spent next period, or saved again. (And any negative credit balance must pay interest, then be deducted from next period's balance.) The auctioneer checks to see if the amount of apples demanded equals the initial guess at the amount of apples sold, then checks bananas, and each fruit in turn. If the initial guess is wrong, all the initial demands are nullified, and the auctioneer guesses again, and repeats the auction. Only when the auctioneer guesses right are the contracts to buy fruit binding.

Fifth, all people then buy what they have contracted to buy, and work to produce and sell the quantity that others demand.

(In principle, the quantity of apples demanded might be larger than the quantity of apples supplied, and the auctioneer might have to ration buyers of apples. But this could only happen in equilibrium if the central bank set a rate of interest that was much too low. In general, because of imperfect competition, so that firms set prices above the competitive equilibrium, the quantity of apples sold will equal the quantity demanded, and will be less than quantity supplied. I will return to this point later.)

The solution to the above model will be identical to the solution to the standard New Keynesian model. Each firm will set price as a markup over marginal cost, where the markup depends on the elasticitity of its demand curve.

For simplicity, I want to consider only the solution where all firms set the same price. This would happen in a world where there were no real shocks, where the central bank had always targeted zero inflation, and where it had always set the nominal interest rate at exactly the right level in the past, and was confidently expected to set exactly the right interest rate in future.

In that long-run symmetric zero inflation equilibrium, the solution for the representative firm would be where the marginal rate of substitution between labour and the consumption basket (the shadow real wage) would equal (1-(1/e)) times the marginal product of labour (where e is the elasticity). Call the solution to that equation C*, or the natural rate of output.

MRS(C*,L*)=(1-(1/e)).MPL(L*) and C*=F(L*) if you want it in math.

(In the third stage of the auction, the firm asks each firm how much it would like to produce and sell. Since prices have already been set, the firm will answer this question exactly like a perfectly competitive firm, which takes the price at which it can sell its output as given. Because that price is now a pre-determined given. The answer to that question will be where the marginal rate of substitution between labour and the consumption basket (the shadow real wage) equals the marginal product of labour. Call the solution to that equation C^, or output supplied. C^ will be greater than C*, unless the labour supply curve is perfectly inelastic, or the firm's demand curve is perfectly elastic (e is infinite). Actual output and sales will be whichever is less, equilibrium quantity demanded, or quantity supplied. Since the natural rate C* is less than C^, the supply constraint will only be binding if a very big positive shock to demand causes demand to rise a long way above C*.)

MRS(C^,L^)=MPL(L^) and C^=F(L^) if you want it in math.

The above defines the solution if the central bank sets exactly the right rate of interest.

Now suppose that, after all firms have set prices, the central bank announces a nominal interest rate that is too high. And, to keep it simple, let's suppose that everyone is confident that this is a one-time mistake by the bank, so that output will return to the natural rate C* in the following period, and firms will choose to keep their prices constant next period. What happens?

Each firm will regret not having set a lower price. But by the time they get the news, it's too late to change price. The higher nominal and real interest rate gives every person an incentive to postpone consumption.  In the fourth stage of the auction, if the auctioneer initially guesses that sales will equal the natural rate C*, and tentatively announces credits accordingly, he finds that demand is less than C*, because verybody wants to save some credits. So he learns his initial guess is wrong, and guesses again.

The solution will be where the Consumption-Euler equation is satisfied. The marginal rate of intertemporal substitution between current and future consumption, evaluated where future consumption equals C*, must equal (one plus) the (real and nominal) rate of interest set by the central bank. Call that solution Ci. Ci will be less than the natural rate C*.

(And if the central bank had set too low an interest rate, Ci would instead be above C*. And if the central bank had set a very low rate of interest, Ci would be so far above C* that C^ would be a binding constraint on supply.)

If you want math, the solution Ci is defined by MU(Ci)/MU(C*) = (1+r)B

where MU is marginal utility of consumption, B is the subjective discount factor, and r is the real rate of interest. Or something like that.

So far, my model is exactly like the standard New Keynesian model (except for a couple of trivial simplifying assumptions.) Now I'm going to introduce "barter".

"Barter".

Let's introduce a second auctioneer, who only opens for business after the first auctioneer has closed his books. (We can assume, if you like, that there is some trivial cost of using the second auctioneer, so everybody will try to do as much business as possible with the first auctioneer before resorting to the second auctioneer.) The second auctioneer is exactly like the first, except that he enforces Say's Law, at the individual level. No individual can carry a positive or negative balance of credits. Each individual can only swap fruit for fruit. He cannot sell fruit without buying fruit of equal value. He cannot buy fruit without selling fruit of equal value. (And by "value" I mean at the same prices that each firm had previously set, so I am not introducing price flexibility in this second auction.)

Again, let's assume for simplicity we start in a symmetric equilibrium, where all firms have set the same price. When the first auction closes, all firms have contracted to sell Ci, which may be above or below the natural rate C*, but is below the competitive equilibrium C^.

The apple producer maximises U(C,L) subject to the production function A=F(L) and the budget constraint P(C-Ci)=Pa(A-Ai), where Pa, the nominal price of apples, is pre-determined, and is equal to P, the price of a basket of fruit, in symmetric equilibrium.

Each firm will sell (and buy) an additional (C^-Ci) units in the second auction. The equilibrium after the second auction closes will be the competitive equilibrium C^, regardless of the rate of interest seat by the central bank. The proof is obvious. If the marginal rate of substitution of labour for fruit is less than the marginal product of labour, and it is less for any Ci less than C^, the firm/worker would prefer to eat more fruit even if it means working longer to produce it. Now the apple producer doesn't want to consume just apples, but he can swap his apples for bananas and all the other fruits in the second auction. If C is less than C^, there are unexploited gains from trade.

So, at the end of the second auction, C=C^. The economy gets to the competitive equilibrium, regardless of imperfect competition, and regardless of any mistake made by the central bank in setting the rate of interest too high.

Let me try to give the intuition.

First, Say's Law applies in barter. Both the apple producer and the banana producer want to sell more fruit. Each wants to sell more and save part of the proceeds. But neither is willing to borrow from the other. Barter allows both to undertake the mutually advantageous trades that can be made. "I will buy your bananas, but only if you buy my apples in return".

Second, under imperfect competition, each firm sets price above marginal cost. But if there's perfect symmetry (as there is here), that means the relative price of apples and bananas is equal to their relative marginal costs. So we still get the competitive equilibrium volume of trade in direct barter of apples and bananas. The (relative) price is right.

Just to avoid potential misunderstanding, I am not saying that money is the root of all problems. What I'm saying is that barter is very costly, and using monetary exchange is much less costly, so people use monetary exchange. The root of all problems is not money; the root of all problems is the cost of using barter, which means we have to use money.