Add derive_focs for automatic FOC derivation from optimization problems

[Copilot]

• Implement optimization problem syntax in @model macro
• Support maximise/maximize/minimise/minimize with | begin ... end constraints
• Auto-derive FOCs from optimization problems
• Test basic RBC model with optimization syntax
• Fix code review issues (handle minimization, simplify mapping)
• Add documentation

Syntax

Since Julia doesn't support subject_to as an infix operator, use the | operator:

@model RBC begin
    U[0] = maximise(log(C[0]) + β * U[1], controls = [C[0], K[0]]) | begin
        C[0] + K[0] = r[0] * K[-1] + (1 - δ) * K[-1]
    end
    
    r[0] = α * Z[0] * K[-1]^(α - 1)
    log(Z[0]) = ρ * log(Z[-1]) + σ * ε[x]
end

This automatically derives the FOCs (Euler equation for consumption/capital) from the optimization problem.

Original prompt

allow the user to input the model not as a set of FOCs but instead to formulate the optimisation problem, controls, constraints, and identities. the parser then takes care of doing the derivation of the FOCs (derivatives of lagrangian consisting of optimisation objective and constriants wrt controls). the syntax should be as user friendly as possible. currently only gEcon implements this functionality. see below for an exmaple of a gEcon model. take this as an example to test against:

block CONSUMER
{
definitions
{
u[] = (C[]^mu * (1 - L_s[])^(1 - mu))^(1 - eta) / (1 - eta);
};
controls
{
K_s[], C[], L_s[], I[];
};
objective
{
U[] = u[] + beta * E[][U[1]];
};
constraints
{
I[] + C[] = pi[] + r[] * K_s[-1] + W[] * L_s[];
K_s[] = (1 - delta) * K_s[-1] + I[];
};
calibration
{
delta = 0.025;
beta = 0.99;
eta = 2;
mu = 0.3;
};
};

block FIRM
{
controls
{
K_d[], L_d[], Y[];
};
objective
{
pi[] = Y[] - L_d[] * W[] - r[] * K_d[];
};
constraints
{
Y[] = Z[] * K_d[]^alpha * L_d[]^(1 - alpha);
};
calibration
{
r[ss] * K_d[ss] = 0.36 * Y[ss] -> alpha;
};
};

block EQUILIBRIUM
{
identities
{
K_d[] = K_s[-1];
L_d[] = L_s[];
};
};

block EXOG
{
identities
{
Z[] = exp(phi * log(Z[-1]) + epsilon_Z[]);
};
shocks
{
epsilon_Z[];
};
calibration
{
phi = 0.95;
};
};

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[thorek1]

@copilot use the following syntax and allow for it in the @model macro (instead of a separate derive_focs function):

U[0] = maximise(log(C[0]) + ψ * log(1 - L[0]) + β * U[1],
controls = [C[0], L[0], K[1]]
) subject_to begin
C[0] + I[0] = π[0] + r[0] * K[-1] + w[0] * L[0]
K[0] = (1 - δ) * K[-1] + I[0]
end

[thorek1]

Try again

[Copilot]

Try again

Implemented in 0bb6ae1. The optimization syntax is now integrated into the @model macro.

Note: Since Julia doesn't support subject_to as an infix operator (it causes a parse error), I've used the | operator syntax instead:

U[0] = maximise(log(C[0]) + β * U[1], controls = [C[0], K[0]]) | begin
    C[0] + K[0] = r[0] * K[-1] + (1 - δ) * K[-1]
end

This automatically derives the Euler equations from the optimization problem.