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Monday, January 13, 2014

DSGE Models: Without Capitalists, You Cannot Model Capitalism

One of the key defects of macro Dynamic Stochastic General Equilibrium (DSGE) models is the use of the utility optimisation by a single Representative Household to drive the solution. If there is only a single household in the economy, it has own the equity of the business sector as well as being the source of labour. Therefore, there is no conflict between labour and capital, and the solution to the household optimisation problem is trivial - it is the "full employment" solution (always).

This article explains this rather academic point. However, there is an important implication for the analysis of bond markets. (It is no accident I have been writing about DSGE models.) Properly understood, rising income inequality will cause government debt levels to rise regardless of the wishes of the government sector. This effect cannot appear in a model with a representative household. It is literally impossible to correctly model government debt dynamics using models within this framework.


Update: Academic mathematical version here.

A Libertarian Fairy Tale


I will now tell you a libertarian fairy tale to give a verbal explanation of my point. I give a semi-mathematical description at the end of this article in the technical appendix.

Imagine that you got sick of government intrusion into your life, and you decided to go move to an isolated island. (“You” in this story may include other household members, who are assumed not to be involved in the economy.) Being a good capitalist, you create a corporation (that you own 100% of), and have a central bank (that you presumably staff). Since you could not find any gold yet, the money stock is zero. As for fiscal policy, ha!

You came to the island with some form of capital, and you use it to generate a magical single good that covers every possible consumption or investment need (but it cannot be stored for the next day). It requires time to operate this capital base, and the longer you work, the more you produce. And since you are a firm believer in Free Market Economics, you have determined your Utility Function, which you are going to Optimise. This utility function increases the more you consume each day, but is depressed by the more time you work. And of course, you solve this equation on the basis that you will live forever, because that makes so much sense.

What determines the wages the corporation will pay yourself, or the price the corporation charges for the good? Who cares – you own 100% of the corporation! You can pay yourself whatever wages you want and yet still buy all the goods produced that day, as you can pay yourself a dividend to purchase whatever output that is left over after using up all of your wages.

Dividends = Profits = (Corporate Sales) – Wages,
which implies:
(Corporate Sales) = (Household Consumption) = Dividends + Wages.

The net cash flow between the corporate sector and the household sector is zero, and so this process can be continued forever. (You do not get to accumulate any gold, but that makes sense, as you are not producing any.)

As is readily seen, the optimal solution is determined by the “real” factors in your utility function – the tradeoff between increasing production versus the “disutility” of labour. No factors can break the optimal solution away from this “full employment” trajectory, including price movements, interest rates or whatever. Since prices do not matter, they are indeterminate. Fiscal policy could move the solution away from this solution if it existed (the government could consume some of the output), but this will only add another real factor to consider within the optimisation.

A Look At DSGE Model Solutions


This triviality of solution was sort of intended by the Real Business Cycle enthusiasts – they wanted to prove that money was neutral. However, the mathematical framework is more extreme than they probably intended; no monetary variable including relative prices can cause the the real variables to move away from the "full employment" solution (unless money is added into the utility function for some bizarre reason, or else fiscal policy is specified in dollar terms).

Solution triviality does not appear to be widely discussed. In fact, deviations from "full employment" was the justification of the development of “New Keynesian” DSGE models. Because the models are rarely solved in their original nonlinear form, I do not have a firm grasp on what people think the true solutions look like.

I see two explanations as to why people believe the solutions could depart from the full employment trajectory, or that prices matter in these models:
  1. Stock-Flow Consistency is broken by ignoring dividend payments in the model economy.
  2. Microeconomic reasoning is incorrectly applied, which leads to a suboptimal solution. (This is similar to the concept of the fallacy of composition.)

As there are hundreds (if not thousands) of these models, I have no idea which explanation holds.

Implications For Fiscal Policy


I do not have space to discuss this point in detail in this article, but a unitary Representative Household poses problems for the analysis of fiscal policy.

It seems fairly safe to say that richer households have a higher propensity to save out of income; in fact, this result could probably be generated by another class of DSGE models (overlapping generation models). Stock-flow considerations tell us then that if richer households have an increasing percentage of national income, the stock of financial assets must increase to accommodate the increased savings. If a modern welfare state is properly modelled, those increasing assets will probably come in the form of increasing government debt, except in the case that some other part of the private sector is in the process of issuing a lot of debt.

The ongoing rise in the government debt-to-GDP ratio in the developed countries is readily understood if you accept these plausible assumptions. However, this explanation is incompatible with a model that says that we only have a single actor that is representing the entire household sector: there is only a single propensity to save out of income. (In DSGE models, the propensity to save out of income is not a fixed parameter, rather its value has to be calculated.)

A standard defense of DSGE models is that they aid understanding the economy, even if their forecasting record leaves a lot to be desired. Since they cannot hope to model fiscal dynamics correctly, that defense seems weak to me.

Technical Appendix


UPDATE: See this article for the full maths.

Let’s start off with the basic RBC model from Chapter 2 of the text Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework (affiliate link) by GalĂ­.

For simplicity, assume that fiscal policy is non-existent, and the initial money stock is zero.

I really only need the utility function and production function from that text. Noting my simplifications above, and fixing a typo in the text, the household budget constraint at time 0 is:

(Price of Goods)*(Quantity of goods) = (Hourly Wages)*(# of Hours Worked) + (Dividends).

Prices can be flexible (as in Chapter 2), or you can use Calvo pricing.

Lemma 1: The profits of any sub-sector of the business sector are greater than or equal to zero.
Proof: (I introduce the notion here of a subsector, as they exist in some DSGE models. The model in Chapter 2 does not have subsectors.) Assume the contrary, that profits can be negative.
However, looking at the equations describing the business sector in the text – in which there are no fixed costs - it is left as an exercise to the reader to show that profits are zero if that sector produces nothing. (No wages are paid, nor is there output to sell.) Since a solution exists with zero profit, rational expectations indicates that a solution cannot be chosen with negative profits (it is a suboptimal choice). Therefore, the original assumption that profits are negative must be incorrect. Q.E.D.

Lemma 2: The profits of the aggregate business sector must be greater than or equal to zero.
Proof: By inspection, applying Lemma 1.

Theorem For any level of wages (W(t)) at a given time, all goods prices (P(t)) above a certain level P*(t) represent a valid solution to the Household Utility maximisation problem. The level of production Y(t) is given by the maximising the composed function: U(Q(N(t),t),N(t)), where
U = one-period utility function;
N(t) = number of hours worked at time t.
Q(N,t) = production function at time t [NOTE: this my addition to the notation].

Proof: Determine the optimising solution N(t) to the composed one-period utility function; this determines the output Y(t) = Q(N(t),t).

Then if we fix W(t),
P*(t)Y(t) = W(t)N(t).
If P(t) = P*(t), then aggregate profits are zero, and all production is paid for out of wages. (By Lemma 2, P(t) cannot be less than P*(t).)

If P(t) > P*(t), then the corporate sector can pay a dividend of (P(t)-P*(t))Y(t).  In this case, profits will equal the dividend payments, and the net cash flow between the household sector and the business sector is zero. Since we optimise U(t) for all t, the full optimisation problem over the infinite horizon is satisfied. Q.E.D.

Remark: If a cash-in-hand constraint is imposed to prevent the dividend payment, the household sector can borrow the deficient funds from the business sector to arrive at the same solution. Any interest payments this generates will be returned as dividends in the future, and so they can be ignored.

Remark: Since the price level has no effect on the solution, price stickiness cannot affect the real variables in the solution. Additionally, since we can find the optimal production level at each time independently, the time preference parameter has no impact on the solution.

(c) Brian Romanchuk 2014

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