# On Macroeconomic modelling

Since Lucas, macroeconomics have attempted to derive macroeconomic models by extrapolating from microeconomic theory. DSGE models were crowning glory of this endeavour, and they completely failed to anticipate the 2008 economic crisis. Many leading economists are now admitting this, but still think they have to build better models the same way. As Olivier Blanchard put it in “Do DSGE Models have a Future?“:

The pursuit of a widely accepted analytical macroeconomic core, in which to locate discussions and extensions, may be a pipe dream, but it is a dream surely worth pursuing. If so, the three main modeling choices of DSGEs are the right ones. Starting from explicit microfoundations is clearly essential; where else to start from? Ad hoc equations will not do for that purpose. Thinking in terms of a set of distortions to a competitive economy implies a long slog from the competitive model to a reasonably plausible description of the economy. But, again, it is hard to see where else to start from.

As I explain in Can We Avoid Another Financial Crisis?, the belief that “macroeconomics is applied microeconomics” is a fallacy, and there is another way. The models that I used to anticipate the crisis of 2008 can be derived simply by taking macroeconomic definitions and putting them in dynamic form. I illustrate how below.

# Simple Private Sector Only Model Without Prices

1. Take three macroeconomic definitions:
1. The employment rate: the number of people with a job (L) divided by the total population (N). Define $$\lambda$$ = L/N.
2. The wages share of output: the total wage bill (W) divided by GDP (Y). Define  $$\omega$$ = W/Y.
3. The private debt to GDP ratio: Private Debt (D) divided by GDP (Y). Define d = D/Y.
2. Differentiate them with respect to time. Since the definitions are true by definition, so are these dynamic re-statements of them:
1. The (percentage) employment rate will grow if the (percentage) rate of real (inflation-adjusted) economic growth exceeds the sum of the rate of population growth plus the rate of growth of labour productivity
2. The wages share of output will grow if wage growth exceeds the growth in labour productivity; and
3. The private debt to GDP ratio will rise if debt rises faster than the rate of economic growth.
3. As equations, these are:
$$\hat \lambda \equiv {{\hat Y}_R} – \left( {\alpha + \beta } \right)\\ \hat \omega \equiv {{\hat w}_R} – \alpha \\ \hat d \equiv \hat D – {{\hat Y}_R}$$

These statements are true by definition. To convert them into a model, some relationships need to be postulated between them. One of the insights from complex systems mathematics is that the simplest possible assumptions can be used, since the defintion of the system itself incorporates the structural relationships. So I use the simplest possible definitions:

1. Output is a linear function of capital Y = K/v;
2. Employment is a linear function of output L = Y/a;
3. Wage change is a linear function of the rate of employment  $$\frac{1}{w} \frac{ d w}{dt} = \lambda_{S} (\lambda-\lambda_{Z})$$
4. Investment is a linear function of the rate of profit $$\frac{I}{Y} = \pi_{S} (\pi_{r}-\pi_{Z})$$ where $$\pi_{r}= \frac{Y-w L – r D}{K}$$
5. Capital depreciates at a constant rate $$\delta_{K}$$
6. Labor productivity $$a$$ and population $$N$$ grow at constant rates $$\alpha$$ and $$\beta$$ respectively.

This leads to the following three-dimensional, structurally nonlinear model:

## Equations

$$\frac{ d \lambda}{dt} =\lambda\times \left(\frac{\pi_{S}\times \left(\pi_{r}-\pi_{Z}\right)}{v}-\left(\alpha+\beta+\delta_{K}\right)\right)\\ \frac{ d \omega}{dt} =\omega\times \left(\lambda_{S}\times \left(\lambda-\lambda_{Z}\right)-\alpha\right)\\ \frac{ d d}{dt} =\pi_{S}\times \left(\pi_{r}-\pi_{Z}\right)-\pi_{s}-d\times \left(\frac{\pi_{S}\times \left(\pi_{r}-\pi_{Z}\right)}{v}-\delta_{K}\right)$$

Definitions
$$\pi_{s}=1-\left(\omega+r\times d\right), \pi_{r}=\frac{\pi_{s}}{v}$$
Parameters
$$v=3, r=0.04, \alpha=0.02, \beta=0.01, \delta_{K}=0.06$$

$$\pi_{S}=(5;10), \pi_{Z}=0.03, \lambda_{S}=5, \lambda_{Z}=0.6$$

Because of the simplicity of these assumptions, the resulting model uncovers the underlying dynamics of the system: a tendency to a “good” equilibrium as the debt to GDP stabilizes, given a low propensity to invest (a value for $$\pi_{S}=5$$); and a tendency to crisis as debt rises faster than GDP, given a high propensity to invest (a value for $$\pi_{S}=10$$).

Technically, the model has 3 equilibria, two of which are significant (involving non-negative values for $$\lambda$$ and $$\omega$$). One of these has finite values for the debt ratio d and positive values for $$\lambda$$ and $$\omega$$); the other has zero values for $$\lambda$$ and $$\omega$$) and an infinite value for d. Of course this latter outcome would never be achieved in the real world because factors that are omitted in the model–primarily bankruptcy and government spending–prevent that outcome (these are included in more elaborate versions of the model). This simplest possible model simply explores the prospect of a tendency to either equilibrium or breakdown from an initial realistic starting point.

More detail to follow! (Hello David!).