Theory

Capacity and Expectations


Letter from the Producer


What is it?  It’s a DSGE model that fully incorporates financial markets, and that doesn’t rely on Calvo pricing or an interest rate rule.  It’s a re-imagining of the equation of exchange, with transaction costs and expectations at its center. It’s a macroeconomic theory that explains money, debt, and exchange a little bit better than what's already out there. If these seem like big claims coming from an uncredentialed recent undergraduate, they are; no one, including myself, really expects be me to be right.  But because I've got nothing to lose, and because right now I can’t prove myself wrong, I've decided that I might as well claim the moon.

I’ve tried to keep this exposition as short and as clear as possible, but I'm afraid it's neither short nor clear, and I have had to leave some points out. Feel free to skip around the page or pursue particular concepts in depth from the multitude of hyperlinks. If you have questions or feedback I’d love to get an email from you at smithandkeynes@gmail.com.


Table of Contents

It's the Human Drama, in three acts. Painting: The Railway Station, William Powell Frith. 

The Cast of Characters 


The surest way to build a model is from the ground up. So at first my cast of characters will be small; only two agents, and a market maker. I define my agents simply, not because they are simple, but because I want to leave them free to be anyone.

One Agent





THE GIVEN: These are the three essential limits to which the economy is subject. See Mr. Smith and Mr. Keynes. 

THE CHOSEN: The variables that the agent chooses each period to maximize his utility. These variables will be explained further in Setting the Scene. To understand my choices in the context of modern macroeconomics, see Optimization.
  • The Demand for Borrowing: The agents’ willingness to take on debt. 
  • The Supply of Lending: The agents’ supply of lending. 
  • The Demand for Consumption: The agent’s demand for goods and services. 
  • The Supply of Production: The agent’s willingness to work.  

THE DERIVED: All of these variables will be defined in terms of The Given and The Chosen in Setting the Scene.

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Two Agents

THE GIVEN  Note that expectations are a function and can’t be simply summed and divided like the the other variables.

THE DERIVED  Note these aggregate relationships:
  •  Borrowing = Lending 
  •  Production = Consumption 
In Generalization, the third aggregate relationship, Spending = Wages, will also hold, but in this two agent model part of each agent's spending is siphoned off by the market maker. 

The Market Maker


While the agents defined above produce goods and services that when consumed produce utility, the market maker produces exchange. Where average agents are bakers, butchers, or candlestick makers, market makers are merchants, shopkeepers, and customs officers. The more exchange the market maker produces, the more goods and services can be moved about. (See Velocity).  The market maker can receive his fee from the producer or the consumer, it makes no difference; like a tax, the consumer always pays in the end.

The market maker is like a scaffold used to build an arch; it's necessary to build the model, but when I'm finished I will take away the market maker and the model will still hold (Promise 2).  In reality, most agents are partial market makers; whenever you go to the grocery store, you are playing the role of the market maker; but in the two agent model I will pretend that the market maker performs all exchange and does nothing else. For now think of the market maker like a half agent that makes choices about production but not about consumption, borrowing, and lending:






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Setting the Scene


The relationship between money, it's velocity, price, and production is captured by the Equation of Exchange, developed by David Hume and John Stuart Mill and formalized by Irving Fisher. The next four sections will define P, Y, M, and V.

MV = PY

Price

"The market price of every particular commodity is regulated by the proportion between the quantity which is actually brought to market, and the demand of those who are willing to pay” 
-Adam Smith, Wealth of Nations, Chapter VII 
Following Smith, I define price as demand over supply. I define supply as the quantity brought to market, or as a vertical line on a graph of quantity and price, and therefore graph demand as curved line equal at all points to price times quantity.
  • Price = Demand/Supply
  • Supply = Quantity 
  • Demand = Price*Quantity 
Usually both demand and supply are responsive (as will become clear in Optimization). If demand shifts out, price increases and in response supply shifts out as well. If supply shifts in, price decreases, and demand shifts in as well. By graphing these shifts, I can turn an odd-looking chart into something that resembles the crossing diagonals of classic supply and demand:

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Production


Production, oat, is determined jointly by the agent’s capacity to produce, Κat, and the agents’ supply of production, δat.  δat is best interpreted as the percentage of capacity employed, a number from 0 to 1:
  1. oat δatΚat  
Limited by their capacity to produce, agents face a tradeoff between leisure and consumption (See: The Utility Function.)  If agents work more (if δ is closer to 1), they can consume more, but they have less time for leisure.  Usually, the returns to both leisure and consumption are marginally decreasing, ie as demand shifts out by equal amounts, supply shifts out by increasingly smaller increments, as illustrated in the graph to the left.  Below, I've drawn supply and demand in the "value markets" of the two agent model; on the right agent a is the producer, and on the left the consumer. Note oat = ybt. 











Money


Traditionally money is defined as three things:
  1. A unit of account 
  2. A medium of exchange 
  3. A store of value. 
Unfortunately, these three characteristics aren’t particularly helpful when it’s time to specify the money supply, which is usually defined in four increments:
  1. M0: currency 
  2. M1: M0 + demand deposits at commercial banks 
  3. M2: M1 + “close substitutes" like money market deposits 
  4. M3: M2 + “long term” deposits like certificates of deposit 
To find a simpler definition, I turn to the history of money. (The truth is often found at the beginnings of things). Most people imagine that before money agents bartered to exchange goods. That would have been a horribly inconvenient; the shoemaker would have had to wait until the baker wanted a new pair of shoes to get anything to eat, a problem known as the coincidence of wants. Historical and anthropological evidence suggests that before money agents exchanged more often by credit and debt than by barter.  An IOU fits the definition of money: it is a unit of account (“IOU 2 pigs” or “IOU 4 chickens”) it is a medium of exchange, and it is a store of value.

The problem with debt is that it requires trust. IOUs work well enough in small communities, but accepting credit becomes much riskier when you don’t know the debtor. So early societies started using gold or other valuable commodities instead of IOUs to exchange long-distance. Kings realized that this was a good idea, and struck coins to make money easier to count and carry. Counterfeiters, coin scrapers, and desperate governments gradually decreased the value of coins, but as more agents accepted them, their actual value became less important; they began to function like universally trusted IOUs, so old and often traded that the original debtor hardly mattered. Eventually, Kings realized that the world no longer needed to back currency with actual value at all, and the world began to trade entirely on trust. Therefore I define money as:
  1. Credit
This definition is not nearly as revolutionary as it sounds, or as this wikipedia article makes it seem. Fractional reserve banking creates money by lending deposits. Cash is categorized as an asset, along with stocks and bonds (loans to companies or governments), personal loans (loans to individuals), and all of the demand deposits located somewhat fuzzily in M2 or M3 (loans to banks). Cash you hold is a loan to yourself, saved now to spend tomorrow.

The level of debt is determined by the supply and demand of each agent, or the loanable funds model, where the debt of one party is equal to the credit extended by the other. To keep things simple I will make two more assumptions until Generalization:  I will pretend that cash does not exist (Promise 3), and that the debt markets are "frictionless" (Promise 4).







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Velocity


In the two agent model, the market maker facilitates all exchanges and is the only one moving money about.  Therefore the velocity of money is determined by the capacity to produce, Κmt, and supply of production, δmt, of the market maker. You can imagine that δm(a)tΚm(a)t represents the portion of exchange supplied to agent a, and δm(b)tΚm(b)t to agent b.  But like light can at any point be measured either as a mass or a wave, money can at any point be measured either as a stock or a flow, in the possession of single agent, or in transit to somewhere else.  In the eyes of the agent, the only quantity of interest is how much they expect to be able to spend in a given period; money and velocity are indivisible.  Therefore I define the capacity for exchange as the product of the market maker's capacity to produce and the agent's available debt, M times the capacity of V:
  • φbt = λatΚm(a)t   
  • φat = λbtΚm(b)t   
Like any agent, the market maker's supply of production is responsive to demand. Therefore I can draw a third set of markets, with cat, spending on goods and services in the value markets on the x axis, and qat, the market maker's price, on the y axis. qat is best understood as the percentage of every dollar spent required to facilitate exchange; it's the part of the price of the loaf of bread that pays for shipping, packaging, and the upkeep of the bakery.




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Expectations


If the table above defined the drama, I wouldn't have a play. When the scene is set, neither agent knows what choices the other agent will make, or what constraints the other agent faces; they make decisions based on their expectations, which in economies of exchange take the form of expected prices. As in any tragedy or comedy, the drama unfolds when expectations aren't met. 


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Act I: Optimization



In this act I've tried to stick as closely as possible to the standard plot of a DSGE model and noted differences, to emphasize that all the important innovations were introduced when the characters were cast and the scene was set.

The Utility Function



Agents choose Ψat, δat, αat, and λa to maximize their utility. Utility can come from two things: consumptionyat and leisure, (1-δat). I define consumption and leisure as loosely as possible to leave agents free to want what they wish:
  • Consumption: Purchase of goods or services produced by another agent
    • Consumption does not require the agent to consume the good or service him/herself; for example, an agent's purchase of food for his/her child is consumption, and an agent's charitable donation to stranger is also consumption. 
    • Usually utility increases with consumption, although in most cases the marginal utility of consumption is decreasing. 
  • Leisure:  Time not spent producing for other agents
    • Leisure may be spent producing for oneself (for example, cooking), enjoying consumption, or resting. 
    • Usually utility increases with leisure, although like consumption in most cases the marginal utility of leisure is decreasing, and it may at some point become negative (For many agents there is such a thing as too much R&R).
Agent a maximizes expected utility over a stream of infinite periods:
Differences:  
  • For simplicity, most DSGE models limit the types activities that produce utility, often leaving out leisure altogether. 
  • Most DSGE models incorporate a discount factor to model time preference. I prefer to incorporate the discount factor into the expectations function, because following John Rae I consider intertemporal discounting to primarily reflect irrational expectation of lower utility in the future, and rational expectation of the possibility of death. (Morbidity not intended.) 

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The Budget Constraint


Maximization of the utility function in each period is limited by a "flow" budget constraint that can be defined in terms of capacity to produceΚat, capacity to exchangeφat, expectations, E, and Ψat, δat, αat, and λa. Unfortunately an equation with all of those variables is rather hard on the eyes, so here's the flow budget constraint in terms of derived variables, with cat equal to spending on consumption and  catqat equal to spending on the market maker:


Differences:
  • No DSGE that I know of models a market maker 
  • For simplicity, many DSGE models define borrowing and lending as one variable, positive for credit and negative for debt. I define them separately to allow the agent to take on debt and make loans at the same time, a utility-increasing choice if the agent expects to profit from the interest rate spread, and possible because my model does not require an interest rate rule.
  • Many DSGE models begin by defining credit and commodity money as two separate variables, although most eventually condense the two into one. I'll separate them in Generalization (Promise 3).
  • Some DSGE budget constraints include taxes. I lump taxes into consumption; they are, after all, payments to purchase services (security, safety nets, public goods), even if the price is extremely distorted by market power
  • Many DSGE models include some stochastic factor somewhere in the utility function or budget constraint to explain changes in productivity (Promise 1) or preferences and create enough drama for a business cycle. The only drama in this story is unmet expectations
  • Most DSGE models include a "solvency" or "no ponzi scheme" condition limiting long term accrual of debt. I also prefer to incorporate the solvency condition into the expectations function; lenders have spent centuries inventing things like credit scores, debtor's prisons, and shame to prevent borrowers from expecting unlimited credit; to cap debt arbitrarily with a "no ponzi scheme" condition is an insult to their effort.



Equilibrium


The next step is to solve this system of equations for Ψat, δat, αat, and λat, or rather, since without specifying the utility function or each expectations function solving is impossible, to show that the solution to a problem of this form exists.  Now I reach my own capacity, because learning the math to prove beyond a doubt that the solution exists would delay this project for longer than I want, but I've learned enough to be reasonably confident that one can go about finding a solution in this way: 
  1. Transform the multi-period utility maximization problem into a Euler Equation  that equates the derivative of consumption in period t with the derivative of expected consumption in period t+1. "By the theory of the optimum, if a time-path of the control is optimal, a marginal increase in the control at any t, must have benefits equal to the cost of the decrease in t+1 of the same present value amount."
  2. Transform the flow budget constraint into an intertemporal budget constraint by setting loans made today equal to loans received (with interest) tomorrow, given the flow budget constraint in t and t=1. 
  3. Solve for agent a, solve for agent b, solve for the market makerΨat, δat, αat, λat, Ψbt, δbt, αbt, λbt, and δmt are all defined, and bam, I've got a general equilibrium in the two agent model.   
If you're an economist, you probably don't need more than that to see that it works, and if you're not an economist, you probably don't care (In fact, you probably stopped caring a long time ago). Now the drama: the optimizing agent plans for the equilibrium predicted by their expectations, but actual equilibrium can be quite different.  When reality doesn't meet expectations but the budget constraint is binding, Ψat, δat, αat, λat has to adjust; the agent needs to loan less, spend less, work more, or borrow more.  Usually, agents cut spending last, especially if consumption is low to begin with.  

Differences 
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Act II: Generalization 


Now that the model works in an economy of two, it's time to kick away the scaffolding. In the two-agent model the agents can sell and buy only from each other.  More agents introduce choice, and choice introduces competition.  In an n-agent economy, the composition of each chosen variable, Ψat, δat, αat, and λat, is determined by a sub-problem, where agents choose between the offers of other agents.   For example, the sub-problem of the demand for consumption, Ψat is the simple utility maximization problem taught in all introductory microeconomics courses, solving for the optimal bundle of consumption given a set of prices, although here prices are expected, not given.  Expectations play an especially important role in the sub-problem of the supply of lending, λat. 

In The Cast of Characters and Settting the Scene, I made four limiting assumptions four promises to take them away.  I apologize in advance for not taking them away terribly cleanly; check back at the blog for better explainations: 
  1. Assumption 1: The Capacity To Produce is Exogenous  
    • So far I've modeled intertemporal choice as black and white; consumption delivers utility today, and loans deliver it in the future. In reality, of course, most choices, like the choice to purchase a car, or a house, or even a box of chocolates, deliver some utility today and some tomorrow. Investment in technology, education, health, machinery, or anything else that increases the capacity to produce, is a form of consumption that delivers utility in the future by allowing agents to increase their wages or increase their leisure. Generalizing the period of utility delivery does not threaten the validity of the equilibrium as long as all future utility is expected. 
  2. Assumption 2: The Market Maker is a Half Agent 
    • Simply make the market maker a full agent (I'm a real boy now!). The production of the market maker, once special, can now be counted with all other production, cat becomes total spending on both goods and services and on exchange by agent a, not just spending on goods and services. qat remains the percentage of every dollar spent required to facilitate exchange, but is incorporated into the budget constraint via cat. If this model is actually right, then this weird double counting explains better than anything else why no one has thought this up before. The flow budget constraint becomes:  
  3. Assumption 3: Cash Does Not Exist 
    • I mentioned in Money that cash is essentially a loan to yourself. It is; in the generalized model holding cash is simply one option in the lending sub-problem.
  4. Assumption 4: Debt Markets Are Frictionless 
    • Frictionless markets are markets are markets subject to only one price; secondary costs are generally interpreted as transaction costs. So far, I've modeled each element of the budget constraint as subject to only one price, except for Ψ, subject to both q and p. Generally, δαand λ can all incur (or earn) multiple prices. For example, borrowers typically pay both interest to the lender, and fees to the banker, and producers pay menu costs when they change prices. Like q, transaction costs in debt markets, menu costs, and all other secondary costs influence budget decisions, are paid to some market maker, and are counted in c.
One more thing: in the generalized model, agents can be anyone, or anything. That's right, firms are people too; Ψ becomes wages, δ revenues, and α and λ stay more or less the same. So are governments (Hey, Big Spender), and so are banks, even central banks (Hey, Big Lender).

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Act III: Aggregation


For the last act, I will add everything up. Note that while you can model agents, firms, and governments with the set up above, you can't add up all three when you aggregate; then you would be double counting. So assume that I'm adding up agents. The three aggregate relationships that didn't quite hold in the two agent model now hold in the generalized model:
  • Borrowing = Lending 
  • Production = Consumption
  • Spending = Wages

The first graph to the right shows the aggregate value market, more or less the same as aggregate supply and demand. Note that Production (Y) is equivalent to Real GDP, and that the asymptote, aggregate Capacity to Produce, is the equivalent to potential output.

The second graph shows the aggregate exchange market that determines the level of spending. Note that Spending (C) is equivalent to Nominal GDP, and that the equilibrium of the aggregate loanable funds market (not shown) determines λ in the supply curve. The aggregate exchange market is not terribly different in its form or implications than the IS/LM model.  If expectations improve, demand for spending increases, and, because of the relationships in the budget constraint, either the supply for lending decreases or the demand for borrowing increases, pushing the interest rate up. Thus, the interest rate usually moves in the same direction as the price of exchange, so a graph of GDP and the interest rate (ie, the IS/LM) would yield correct predictions most of the time.

The last graph captures the movement of the four quantities in response to changing aggregate states of expectation. Aggregate spending is most sensitive (or elastic), followed by the capacity to exchange, followed by production, followed by the capacity to produce.  In this last graph the choice of the word "capacity" comes to fruition; spending is the actual realization of the potential capacity to exchange, and production is the actual realization of the potential capacity to produce.  Finally, it clearly shows that more than anything else, the economy looks like lasagna.

The End.

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