Lots of econ bloggers have alternatively cited DSGE models or criticized them in posts, but I've never actually seen any attempts to show the lay public what one looks like. My aim is to simply show you what a DSGE model looks like and how it is used in practice. This is not a dumbed-down explain-like-I'm-five post (perhaps that's for a future post, but this is the full unabridged model), nor is this a step-by-step guide to learning how to do DSGE modeling--I present just the model, nothing else. What follows is a DSGE calculator programmed with a log-linear approximation of a standard "Real Business Cycle" DSGE model with complete taxation, followed by a concise definition of the model and all the variables.

How to use this calculator:

1. Plug in whatever parameter values and tax rates you want into the "Control Panel."
2. Set it to either compute an impulse response function (click the "impulse" button) or run a simulation (click the "simulation" button).
3. If you want an impulse response function, you have to specify one or more shocks. When imputing shocks, remember that whole numbers equal percentage points, representing the percent change in that variable. Note that if the tax rate is zero, then a shock to it won't do anything.
4. Click the "Calculate!" button.

When you click calculate, two things will happen: a graph plotting your results over time will appear along with, just below it, the estimated solution to the model (at least, part of it, from which the rest can be easily computed).A simulation generates a random sequence of shocks to total factor productivity (aka "technology shocks") and uses this to compute the time path of all the variables in the simulation. An impulse response function shows the effects of a one-off shock to the economy (you get to pick what kind and how big!) over time. This post comes with a word of caution: the parameter space is incomplete, and there are some parameterizations for which no solutions to our approximation exist. I've detailed how to solve this model here.
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The Model

1   Households

A representative household chooses utility-maximizing quantities of investment, consumption, and labor satisifying her budget constraint by solving where is capital and must be chosen one period in advance because new investment enters the firm's production function with a one-period lag, is labor, is consumption, and are the wage rate and returns on capital, respectively, and , , and are the tax rates on consumption, labor, and capital respectively. In the budget constraint above, is the depreciation rate on capital, and investment is implicitly defined according to the capital accumulation equation . The three constants satisfy , , and respectively.

2   Firms

A representative firm chooses inputs capital and labor to maximize profits given by

In the firm's production function, total factor productivity is a random variable following the stochastic process given by where where .

3   Government

The government lays taxes on consumption, on labor income, and on capital income, and chooses government consumption and investment to satisfy the budget constraint

However it is not possible to consume negative quantities of goods and services, so there is an additional constraint that . Because full Riccardian Equivalence holds in this model, we can assume without loss of generality that the government cannot borrow or save. To model the effect of suprise temporary changes in taxes, we assume choices of taxes consist of a permanent component and a temporary component according to:

4   Equilibrium

We define an equilibrium as an allocation and prices such that households maximize utility, firms maximize profits, and the government satisfies it's budget constraint. Equilibrium is characterized by the following system:

We approximate the solution of this system by linearizing the logs of the equations around the steady state solution, which is defined as the values to which each of the variables would eventually converge in the absence of any shocks. Letting variables without subscripts denote their steady-state values, and defining which can be interpreted as percentage deviations of the variables from their steady-state values. We use the linearized approximations to compute the impulse response functions and simulated data above.