Package Overview#

Representing a DSGE Model#

Like the R package gEcon, gEconpy uses .GCN files to represent a DSGE model. A GCN file is divided into blocks, each of which represents an optimization problem. Here is one block from the example Real Business Cycle (RBC) model included in the package.

block HOUSEHOLD
{
    definitions
    {
        u[] = C[] ^ (1 - sigma_C) / (1 - sigma_C) -
              L[] ^ (1 + sigma_L) / (1 + sigma_L);
    };

    controls
    {
        C[], L[], I[], K[];
    };

    objective
    {
        U[] = u[] + beta * E[][U[1]];
    };

    constraints
    {
        C[] + I[] = r[] * K[-1] + w[] * L[] : lambda[];
        K[] = (1 - delta) * K[-1] + I[] : q[];
    };

    calibration
    {
        # Fixed parameters
        beta  = 0.99;
        delta = 0.02;

        # Parameters to estimate
        sigma_C ~ N(loc=1.5, scale=0.1, lower=1.0) = 1.5;
        sigma_L ~ N(loc=2.0, scale=0.1, lower=1.0) = 2.0;
    };
};

Basic Basics#

A .GCN file uses an easy-to-read syntax. Whitespace is not meaningful - lines are terminated with a “;”, so long equations can be split into multiple lines for readability. There are no reserved keywords* to avoid – feel free to write beta instead of betta!

Model variables are written with a name followed by square brackets, as in U[]. The square brackets give the time index the variable enters with. Following Dynare conventions, capital stock K[-1] enters with a lag, while all other variables enter in the present. Expectations are denoted by wrapping variables with E[], as in E[U[1]] for expected utility at t+1. So I lied, there are a couple reserved keywords (hence the asterisk). Finally, you can refer directly to steady-state values as K[ss].

Parameters are written exactly as variables, except they have no square brackets [].

Anatomy of a Block Blocks are divided into five components: definitions, controls, objective, constraints, identities, shocks, and, calibration. In this block, we see five of the seven. The blocks have the following functions:

  1. definitions contains equations that are not stored in the model. Instead, they are immediately substituted into all equations within the same block. In this example, a definition is used for the instantaneous utility function. It will be immediately substitutited into the Bellman equation written in the objective block.

  2. controls are the variables under the agent’s control. The objective function represented by the block will be solved by forming a Lagrange function and taking derivatives with respect to the controls.

  3. The objective block contains only a single equation, and gives the function an agent will try to maximize over an infinite time horizon. In this case, the agent has a CRRA utility function.

  4. constraints give the resource constraints that the agent’s maximization must respect. All constraints are given their own Lagrange multipiers.

  5. identities are equations that are not part of an optimization problem, but that are a part of the model. Unlike equations defined in the definitions block, identities are saved in the model’s system of equations.

  6. shocks are where the user defines exogenous shocks, as in varexo in Dynare.

  7. The calibration block where free parameters, calibrated parameters, and parameter prior distributions are defined.

## Parameter Values and Priors

All parameters must be given values. In the household block above, all parameters are given a value directly. beta and delta are set fixed, while sigma_C and sigma_L are given priors and starting values. The ~ operator denotes a Prior, while = denotes a fixed value. All parameters must have a fixed value – this is used as the “default” value when building and solving the model. Priors, on the other hand, are optional. At present, the user can choose from Normal, HalfNormal, TruncatedNormal, Beta, Gamma, Inverse_Gamma, and Uniform priors, with more to come as I improve the integration with PyMC. Distributons can be parameterized either using the loc, scale, shape synatx of scipy.stats, or directly using the common parameter values from the literature (such as a mu and sigma for a normal).

As an alternative to setting a parameter value directly, the user can declare a parameter to be calibrated. To do this, give a steady-state relationship that the parameter should be calibrated to ensure is true. The following GCN code block for the firm’s optimization problem shows how this is done:

block FIRM
{
    controls
    {
        K[-1], L[];
    };

    objective
    {
        TC[] = -(r[] * K[-1] + w[] * L[]);
    };

    constraints
    {
        Y[] = A[] * K[-1] ^ alpha * L[] ^ (1 - alpha) : mc[];
    };

    identities
    {
        # Perfect competition
        mc[] = 1;
    };

    calibration
    {
    L[ss] / K[ss] = 0.36 -> alpha;
    };
};

The alpha parameter is set so that in the steady state, the ratio of labor to capital is 0.36. On the back end, gEconpy will use an optimizer to find a value of alpha that satsifies the user’s condition. Note that calibrated parameters cannot have prior distributions!

Lagrange Multipliers and First Order Conditions#

As mentioned, all constraints will automatically have a Lagrange multiplier assigned to them. The user name these multipliers himself by putting a colon “:” after an equation, followed by the Lagrange multipler name. From the code above:

C[] + I[] = r[] * K[-1] + w[] * L[] : lambda[];
K[] = (1 - delta) * K[-1] + I[] : q[];

The multiplier associated with the budget constraint has been given the name “lambda”, as is usual in the literature, while the law of motion of capital has been given the name q[]. If the user wanted, she could use these variables in further computations within the block, for example Q[] = q[] / lambda[], Tobin’s Q, could be added in the identities block.

Interally, first order conditions are solved by first making all substitutions from definitions, then forming the following Lagrangian function: L = objective.RHS - lm1 * (control_1.LHS - control_1.RHS) - lm2 * (control_2.LHS - control_2.RHS) ... - lm_k * (control_k.LHS - control_k.RHS)

Next, the derivative of this Lagrangian is taken with respect to all control variables and all lagrange multipliers. Derivaties are are computed “though time” using TimeAwareSymbols, an extension of a normal Sympy symbol. For a control variable x, the total derivative over time is built up as dL[]/dx[] + beta * dL[+1]/dx + beta * beta * dL[+2]/dx[] .... This unrolling terminates when dL[+n]/dx[] = 0.

The result of this unrolling and taking derivatives process are the first order conditions (FoC). All model FoCs, along with objectives, constraints, and identities, are saved into the system of equations that represents the model.