Monday, 13 April 2015

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.


  1. I need a model to go with a figure. The model that Figure 1 engendered was of my wife deciding what to take to the church potluck dinner. Remembering that this is a representation of a barter transaction, Figure 1 is her bartering with herself.

    The labor line is easy. The greater the output, the greater the cost in labor whether measured in wages, dis-utility, or energy expended. She would need to spend more labor if her potluck entry was bigger in size or complexity.

    The marginal utility line is a little harder. It is the decision line. How much should she take? How expensive might the ingredients be? If she takes too much, she brings some home, making even more work for herself. Clearly, she needs to do something but she can create problems for herself. The marginal utility line slopes down but where it contacts the labor line is nothing more than a choice of the decision maker.

    The potluck model works for Figure 1 and seems to work for Figure 2 when we add money to the model. For Figure 2, the horizontal line for money is the value of the labor AT THE POINT OF CHOICE. It is also the value of materials, trips to the store, heat on the stove, and all the other inputs into the final product. Therefore, the horizontal line would have it's own scale and would be unique for the entry that my wife was making.

    This potluck model breaks down on Figure 3. Instead, with the potluck model, we could get two monetarily significant events:

    1. My wife trades with a friend. My wife builds TWO entries for the near event, the friend builds two entries for the distant event. A fair exchange is measured by the money value assigned.

    2. My wife brings the fixings for the entry, the friend brings her labor. The balance between the two components is measured in monetary terms.

    The potluck model kind-of-works, but it leaves me thinking that these figures are astonishingly arbitrary. A line that is basically representing a choice-made-by-a-decision-maker is a difficult thing to translate into monetary exactness.

    1. It's difficult to interpret this with just one person. You really need at least two people (your wife and the friend, say) to have trades.

      If you want a model, look again at my post from last year: . This is closely related to what I am saying here. Any point between Y1 and Y* in the third diagram corresponds to the position in the table I set out in that older post. Each line shows the value each household puts on the different commodities. The horizontal line is the value it puts on money. The downwards sloping line is the value placed on goods it would like to consume (fish for the farmer). The upwards sloping line is the disutility from giving up the produce of its own labour (corn for the farmer). The relative heights of these lines across that range correspond to the relative preferences.

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  3. I took a look at "The Problem with Monetary Exchange".

    There are two decision makers in an exchange but each makes two decisions. Each decision maker is both a consumer and a producer.

    An exchange of two products would have TWO different evaluations, each represented by a version of Figure 1. Each party to the exchange would value his own product. An exchange could happen in a barter situation if both parties agreed on the same level of utility (or some multiple of the lower valued such as 3:2).

    I think we could blend the fish-corn and potluck models. We could add the utility line of fisherman (fish) with the utility line of farmer (corn) and find the average utility line. Do the same thing with the dis-utility line to create a second line, the average dis-utility line. The intersection of these two average lines would be the blended maximum utility of the exchange.

    Now introduce money into the models. The question here would be where to place the money scale?

    I think you have utility and dis-utility on the vertical scale. Output is on the horizontal scale. For the barter situation, output would be numbers of items; for one exchange, the maximum utility point would describe X fish and Y corn so the output scale would carry units of X * Y.

    We could convert the X*Y scale to a money scale. We would need to find a fish-to-corn ratio for a single money price.

    If we made the above adaptations, the money line would be vertical, not horizontal.

    I hope this style of discussion is what you were seeking when you made this post!

    1. You can't add different peoples' utilities. There is no useful meaning thta can be attached to such a concept. In fact, the absolute values of the utilities do not matter - all that matters are the preferences. We have to look at who prefers what to work out what kinds of exchange will take place. As in the table in my previous post. Maybe the farmer's utility is 100 times that of the fisherman - it makes no difference. Ony the order of preferences matter.

      In terms of my diagrams the scale of the vertical axis is irrelevant - only the relative positions of the lines matter. The diagram represents the position for any one person, but in the NK model, households are identical, so we know that if one household is a particular position, they all will be.

  4. When I look at these models, it seems to me that fiscal policy is under-appreciated.

    No matter what the household sector does, the next period's holdings of financial assets has to equal the current holdings, plus the fiscal deficit. So why does the household sector care about it's financial assets? It's holdings will be determined by fiscal policy (plus interest), so it is completely out of its control. If you have no control over your money balance, and nothing you do affects it, why should you care about your financial assets within a utility optimisation problem?

    1. I totally agree on the issue of fiscal policy. I find the treatment of fiscal policy in these models to be most unsatisfactory. I've been meaning to write something on it but I haven't quite tied down my thoughts yet. It's something to do with expectations. The policy rules that are used in these models shape expectations of future outcomes, but it strikes me that these rules are wholly unrealistic. They might figure on a theoretical economist's wish list, but they just do not describe what happens in the real world where long term policy is shaped by political considerations. As such, even a perfectly rational individual would not expect the the government to act in that way. The models would make much more sense to me with long-term fiscal policy that conformed to political reality.

      I'm not sure I follow your main point though. The household sector's budget constraint must coincide with that of the public sector (assuming just the two sectors), so the household sector as a whole cannot save unless there is a deficit. But, in principle, an individual household can save and must base its decison making on the possibility of doing so.

    2. Let's take a case where the interest rate is greater than the growth rate of the economy,so that the IGBC converges.

      The government may decide to eliminate all debt (and money) in the next period. If the tax is a poll tax (which is what these models assume), and all households have equal monetary endowments, all households know that they cannot spend a penny more than what they earn, as otherwise they cannot meet the poll tax bill. So no matter what happens, their factor income has to be exactly equal to their spending. Since everybody knows this, their financial assets are entirely superfluous to their optimization choices.

      If initial financial endowments are unequal, the households with surpluses have the other households over a barrel. They could demand practically anything, and the other households have no choice but to accept, since breaking the household budget constraint is not conceivable. In such a case, the notion of a representative household seems quaint.

    3. I had a longer comment which was lost. To fully explain my point might require a few hundred words, so I might as well turn it into a rambling post. One underlying issue seems to be that the representative household assumption basically implies that the representative household always remains representative, as otherwise you then have to determine the solution where you have households with distinct "class interests".

      A second problem is that I am looking at the optimization problem like a mathematician, and not a hand-waving economist. The optimal solution in all of these problems (assuming money is not in the utility function) is the full employment solution trajectory, regardless of what prices are doing. Economists seem to believe that by invoking marginal arguments, the optimal solution is somehow not the solution to an optimization problem. I cannot wrap my mind around that concept.

    4. I'd be interested to see that post, because I'm afraid I'm still not entirely sure I know what you're saying - my fault, no doubt.

      With regard to the optimal solution not being the solution to an optimization problem, isn't that related to what I've been looking at in this post (and in the related post )? All parties would be better off with more trade, but no one person is prepared to part with money, in case the others then don't.

  5. "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."

    I do not think that is a very good/realistic assumption. The demand deposit account can't go negative. If an entity gets below zero, it has to borrow by issuing a bond or have the transaction denied.

    Using that assumption, you can use currency (a physical balance) too.

    1. Actually it's not unusual to have transaction accounts that can go overdrawn. I did it that way because it seems the most simple. But you could have loans ( or bonds ) and currency instead. But to be consistent with the NK model, all currency would have to be returned to the issuer by the end of the period.

  6. Market Fiscalist18 April 2015 at 09:23

    Great post. I have a question that I hope makes sense.

    In the model you describe, if you debar barter exchange then when people's preference for future over present goods increase:

    - As all goods produced are for present consumption and demand for present consumption has reduced, then demand for labor (and output) will fall

    - you describe a re-equilibrating mechanism where wages but not consumption prices fall until the labor market is back is cleared.

    - After this adjustment the value of goods in terms of labor will have increased since wages have fallen but not other prices.

    - If I have understood the model correctly this means that even if the workers consume all their wages buying consumption goods there will a bigger excess of consumption goods than before assuming productivity is the same.

    My question is: Who is consuming this excess ? Is it the owners of businesses ?

    1. Thanks.

      Yes. You can set this up different ways, but the way I have done so, it would generate profits which would accrue to and fund consumption by the business owners (which are also the households of course ).