Monetary Exchange in New Keynesian Models

I was involved in an interesting discussion recently on Stephen Williamson's blog regarding the role of monetary exchange in New Keynesian models.  There were various points to this, but the question that interests me is whether it matters to the conclusions of those models whether exchange is monetary in nature or whether they could be based on barter.  Clearly such models use money as a numeraire, in which the sticky prices are set, but is that its only role?  I'm interested in the question, because to me the predominance of monetary exchange has always been a cornerstone of Keynesian economics.

I want to consider a simple New Keynesian model with households maximising utility from consumption of goods and leisure time.  I'm assuming no government or foreign sector and that all goods are perishable so cannot be stored.  Current output and current household consumption must therefore be equal.

The diagram below shows the marginal utility and the marginal cost for the representative household for different levels of output.

The downward sloping line represents marginal utility.  It is assumed that this declines with increased consumption.  The upwards sloping line represents marginal cost of consumption in terms of labour disutility.  This is given by the marginal disutility of labour time divided by the real wage rate.  The real wage rate here is determined by firms' pricing, which will depend on the marginal productivity of labour and any mark-up due to market power.  The upward slope might be due to either increasing marginal disutility of labour or decreasing marginal productivity.

The equilibrium is determined where these two lines meet.  In a barter situation, households will be directly exchanging labour time for goods.  Where the lines cross is where they maximise utility from giving up labour for goods.

We now want to consider what happens with monetary exchange.  To do this we are going to incorporate the possibility that households and firms can hold money balances.  These will not be physical balances, but rather entries in a central registry (like a bank).  As these balances are not physical, we can allow them to be negative as well as positive.  This is equivalent to having a transaction account at the bank that is permitted to go overdrawn.  Finally, we will assume that the aggregate money supply is zero.  That is, adding up all the balances (positive and negative) always sums to zero.

There are now three possible exchanges: goods for labour; money for goods or money for labour.  In addition, households now have the option to carry positive or negative money balances from period to period.  This enables them in principle to vary the time pattern of their consumption, without changing that of their labour.  Of course, as a group, they cannot do this.  Since we are assuming that goods are perishable, everything produced in a period must be consumed in that period.  However, the individual household decision must be based on the possibility of a mismatch between the timing of consumption and the timing of provision of labour.

As households now have the option to save, the equilibrium position must also balance the marginal utilities of consumption now and in the future.  We can show this with an additional line, horizontal and crossing through the point where the other lines meet.  This line represents the discounted marginal utility of future consumption divided by the relative price (effectively one plus the real interest rate).  This represents the utility of future consumption that would be foregone by spending money now on current consumption.  The line is horizontal because it does not depend on current consumption.

In full equilibrium all these lines must meet at the same point.  What we want to consider now is what happens if there is a shift in preference towards later consumption, with no or limited adjustment in prices.  We can represent this as an upwards shift in the horizontal line.  Future consumption is now valued more highly so the marginal cost of current consumption, in terms of use of money balances, is greater.

The cross-over point between the downwards sloping line and the horizontal line gives us a new equilibrium at a lower level of output.  At this point, households are not prepared to part with additional money balances to buy goods, because they place more value on the future use of that money.

However, we can see that with all prices and wages at this level, there is scope for beneficial barter trade.  Although households will not part with any more money for goods, they would be prepared to trade additional labour for goods.  The marginal utility of consumption exceeds the marginal disutility of the amount of labour that would need to be traded at this real wage.  In fact, barter trade could take us all the way back to the original equilibrium.  At this point, households would prefer to sell the goods obtained by barter in order to increase their money holdings, but they would find no takers and so would consume the goods in preference to wasting them.

So in this instance, we can see that the conclusion that the shift in time preference results in a fall in output depends critically on an assumption that trade has to be monetary and that barter trades are excluded. 

However, this is not always apparent in the New Keynesian model due to the different assumptions about wage and price flexibility.  In general, if some degree of price stickiness is modelled, it is usually assumed that wages are fully flexible.  What that means is that when output and labour requirement falls, the nominal wage rate is bid down until households are indifferent between work and leisure.  In our diagram, this can be represented as a upwards shift in the upwards sloping curve to a point where all three lines cross again, as shown below.

The implication of this is that at the wages and prices that now prevail, there are no longer any beneficial barter trades to be had.  Firms would happily take on more labour at this new real wage rate, but households do not want to supply more labour.  So, it appears that it then makes no difference whether we assume monetary exchange or barter.  However, it is not easy to explain why the real wage would adjust in this way, if the barter trade of labour for goods were allowed.

New Keynesian models do not appear to be monetary models because they do not seem to have a quantity of money in them (another way of looking at this is to say that they do have a quantity of money in them, but that it is always equal to zero).  Nevertheless, the conclusions of those models can be more readily interpreted if we take them to be based on a system of monetary exchange, rather than barter.  There are plenty of things I do not like about these models, but on the whole I'm happy to see them as models of monetary exchange.