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Walras with Money

As I’ve been saying, in the standard Walrasian model you don’t get absolute prices, you get only relative prices and you have to apply an arbitrary restriction in order to make them look like absolute prices (like all prices sum to 1 or something similar) these relative prices can be multiplied by any scalar (“price level”) without changing the solution. So what if, just for fun, we try to add money in, make it an economy where all goods are traded for money, try to get a price level and see if we can characterize a general glut. This is, I suspect, exactly what most economists have in mind when they imagine a general glut and I assume it has been done before but I don’t recall seeing anyone put it explicitly in this context.

Let’s say you have an economy with n “real” goods and you also have money. The quantity of all of the goods produced as well as the quantity of money are determined exogenously. People only care about the quantity of each good they consume as well as their (average) real money balances (m/P) where m is the quantity of money an individual holds and P is the price level somehow defined. (For instance, we might let P be the sum of all nominal prices or the average nominal price or something along those lines such that we can characterize the price vector as a vector of relative prices–somehow defined–multiplied by the price level). So we have utility functions that look like this.

U(X1,X2,….Xn,m/P)

And assume, for ease of exposition, that this function is separable in money so that we can write:

Ux(X1,X2,…Xn)+Um(m/P)= U(X1,X2,….Xn,m/P)

And everyone has a budget constraint that looks like this.

Sum[Pi(Xi-Xi’)Pi]+ m-m’=0

Where Xi is the quantity of good i consumed, Xi’ is the initial endowment of good i, Pi is the price of good i and m’ is the initial endowment of money (nominal).

Now assume that you have a Walrasian auctioneer calling out nominal prices until every market clears. If you take out the money part and just have Ux() and the Xs in the budget constraints, then you will get a vector of relative prices that clears all markets. If you say that one price is fixed too low, then you get excess demand for that good and excess supply of some other good(s). If you then add to the model by saying that people change their demands for other goods in response to the constraint on their ability to purchase the good with the fixed price and you then have the Walrasian auctioneer call out prices for the other goods until those markets all clear conditional on that constraint, then you have what Nick Rowe has been talking about.

But if you have no money and the Walrasian auctioneer calls out prices which are all too high what happens? The answer is: that question doesn’t make any sense. Without money, he is only calling out relative prices. It’s impossible for them to all be too high. If the supposedly “too high” prices are all exactly half of the supposedly correct prices, then they are the same prices and the markets all clear. If the relative prices change, then you have a case where there is excess demand for some good(s) and excess supply for some good(s) and what happens depends on how you alter the model from the original to account for the persistence of this phenomenon.

In order to even consider the possibility of all prices being “too high” or “too low,” we have to change the model. We have to put money in. Luckily I did that already. So return to that formulation.

With money, the solution will be a vector of prices such that the sum of the excess demands for all real goods equals zero and everyone is holding their desired quantity of money. This means that the marginal utility of a dollar will be equal to the marginal utility of one dollars-worth of each good. This allows us to get an actual set of nominal prices (and by extension, a price level).

So let us assume that the relative price vector called out by the Walrasian auctioneer is the “correct” one (the one which would clear all markets in the case with no money). What if the price level is too low? Even if the real goods are allocated efficiently, the marginal utility of a dollar’s-worth of money balances will be higher than the marginal utility of an additional unit of some good for at least some people and they will try to trade dollars for goods. Since the number of dollars is fixed exogenously, they can’t all do this at once. There will be an excess demand for goods and an excess supply of dollars.

The only way to alleviate this situation will be for the Walrasian auctioneer to call out a higher price level. As he dos this, the quantity of real money balances will fall (the nominal value stays the same but the price level rises) and the marginal utility will rise. At some point, the marginal utility of a dollar will be equal to the marginal utility of a dollar’s-worth of any other good (since we are assuming the equilibrium relative prices) and that will be the equilibrium price level—the level at which people are just willing to hold the quantity of dollars that exist.

Conversely, if the Walrasian auctioneer calls out a price level that is too high, people will want to hold more dollars than there are and the only way to alleviate this is for the price level to fall. This is a general glut. If, for instance, the money supply contracts, prices will need to fall to bring things into equilibrium. If they can’t fall because they are “sticky” for some reason, then you may get a general glut in which the excess supply of real goods is offset by an excess demand for money.

Now does this contradict Walras’ Law? Not exactly. Since we changed the model, we have to change the characterization of the law before we can ask a question like that. If what you mean by “Walras’ Law” in this context is that an excess supply in the market for some real good, measured in dollars, must be offset by an excess demand in the market for another real good, measured in dollars, then no. If what you mean is that an excess supply of goods must be offset by an excess demand for something, potentially money, then yes. Is the latter characterization of the law meaningless? Maybe some would say yes but I think that a lot of people out there could benefit from carefully considering in what sense “Walras’ Law” applies in an economy with money and in my book, that makes it pretty useful.

For the record, this is pretty standard stuff, I don’t think I’m saying anything groundbreaking here. I also think there is more to the story but saying groundbreaking things is hard. I’ll get around to it eventually.

 

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  1. October 3, 2014 at 10:59 am

    Yep. This is a good and clear statement of the argument Don Patinkin made, in his (now unread) classic Money Interest and Prices.

    “an excess supply of goods must be offset by an excess demand for something, potentially money,” that is what Patinkin calls “Walras’ Law. He said it’s right.

    “an excess supply in the market for some real good, measured in dollars, must be offset by an excess demand in the market for another real good, measured in dollars,” that is what Patinkin called “Say’s Law”. He said it is wrong (and leaves the price level indeterminate).

    The next step is to “unpack” the word “market” in your sentence above. Remember there are only n markets, in an economy with n real goods, plus money making n+1.

    • Free Radical
      October 3, 2014 at 4:51 pm

      Thanks Nick. I agree with your characterization of n markets for goods and money (with money making n+1 “goods” if you want to call money a good). That is what I had in mind here and should fit with what I said. Although the one market for n goods where their prices are all quoted in dollars should also work.

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