Solving Dynamic Stochastic General Equilibrium ModelsEric Zwick ’07

Swarthmore College, Department of Mathematics & Statistics

ReferencesBoyd and Smith. “The Evolution of debt and equity

markets in economic development.” Economic Theory. Vol. 12, pp. 519-560, 1998.

Marimon, Ramon and Andrew Scott, eds. Computational Methods for the Study of Dynamic Economies. New York: Oxford UP, 1999.

Sargent, Thomas J. Dynamic Macroeconomic Theroy. Cambridge, MA: Harvard UP, 1987.

Sumru, Jagjit and Charles Nolan, eds. Dynamic Macroeconomic Analysis. New York: Cambridge UP, 2003.

Walsh, Carl E. Monetary Theory and Policy. Cambridge, MA: MIT Press, 2003.

Acknowledgements

I want to thank Professors Jefferson and Stromquist for

their guidance. Additionally, I want to thank Dan

Hammer for inspiring my poster template and Nick

Groh for sharing.

IntroductionMany economic models begin with the simple idea of

maximizing a utility function subject to a budget constraint

over time. However, the task of solving for an intertemporal

equilibrium is not always an easy one. In particular, it can be

extremely difficult to solve a nonlinear system of equations,

especially when the variables are themselves functions and

even more so when these functions are stochastic

processes.

Fortunately, certain types of models can be simplified

by approximating the stochastics of the system about a time-

invariant equilibrium. The resulting linear system can be

solved using difference equations for a recursive solution,

which can then be analyzed and simulated. Furthermore,

the requirements for this technique agree with stylized

facts across a number of economic problems.

This project outlines the mathematics of the

approximation technique commonly used by economists to

solve nonlinear dynamic stochastic general equilibrium

(DSGE) models. My paper walks through an example

problem using a standard model of the macroeconomy.

Conclusion

There are countless applications of this approach within

a variety of economic contexts. In macroeconomics, It

has been used to simplify models of optimal fiscal

policy, social insurance programs, and the financial

system, to name a few. On the micro side, economists

have used dynamic stochastic models to study asset

valuation, credit and labor markets, and competitive

equilibriums.

Employing this log-linearization technique,

economists can make unsolvable systems of nonlinear

equations tractable, which gives them the option to

improve the explanatory power of their models by

incorporating more theory. DSGE models allow

economists to build, calibrate and simulate economies

while conducting experiments that would be impossible

to run in the actual world.

Step 1: Identify the Model’s EquationsA standard model will consist of a series of equations which

describe the economy in terms of a combination of static (or

exogenous) and dynamic (or endogenous, stochastic)

variables. These equations generally consist of an

expression for the representative consumer’s present

discounted value of utility over time and an expression for

the resource constraint she faces. The inputs for these

expressions are themselves functions of variables that

characterize the economy. By introducing uncertainty about

future outcomes, we consider these variables to be

stochastic and include expectations operations in our

model.

In our example model, consider a consumer who

maximizes utility

and faces the budget constraint

where Ct, Kt, Yt, Zt are consumption, capital, output and

total factor productivity in period t and where δ, β and ρ are

the capital depreciation rate, discount rate and capital

intensity, respectively.

Here, Ct and Kt are pure stochastic variables; Yt and Zt

are endogenously determined by other stochastic variables

in the system; and δ, β and ρ are static, exogenous

variables.

Step 2: Solve for the Steady StateThe first order conditions for this maximization problem

are derived from Lagrangian partial derivatives with

respect to Ct and Kt:

where

and Λt is the Lagrange multiplier.

We manipulate the first order conditions by making

the stochastic variables equal at every time step so that

they form a time-invariant stable equilibrium. The steady

state level of a variable is indicated with a horizontal bar.

So,

and

By fixing δ and β, these expressions can yield exact

values for each of the steady state variables.

Step 3: Log-linearize the equationsWe rewrite each stochastic variable in terms of its steady-state

value and a small random disturbance. So, for example,

where ct is a number close to zero representing the variable’s

log-deviation from the steady-state.

Then, using Taylor expansions to simplify the

exponentials, we rewrite the original nonlinear system in a log-

linearized form. We simplify according to the rules:

Our system now takes the form:

Equations (1), (2) and (3) are the log-linear first-order

conditions. Equation (4) is the log-linear production function.

Equation (5) is the budget constraint and equation (6) is total

factor productivity.

The resulting system of six equations is linear and

contains six unknowns, so it can be solved.

Step 4: Solve for the Recursive Equilibrium

Law of MotionThere are several methods for solving this system of difference

equations. One such method, undetermined coefficients, is

discussed in my paper. The resulting equilibrium law of motion

allows for calibration and simulation of the model to determine

the effects of exogenous shocks on the economy in various

states of the world. Additionally, simulation can be used to

determine the variances, autocorrelations and small-sample

properties of estimators for certain variables.