optimize(): Non-linear flow constraints

[Ulrich Wohak]

Hi everyone,

I'm usually around the Stata forum but I'm trying to pick up Mata generally and the optimizer() in particular.

What I want to accomplish is to reproduce a dynamic optimization problem described in Section 6.1.2 of the Book Adams, Clarke, Quinn (2015) Microeconometrics and MATLAB.

I started with the simpler problem from Section 6.1.1 with linear constraints (flow equations) that has the following strcuture:

1. Household has an endowment of W1 and must decide how much to consume each of T periods (the cake-eating problem).
2. The household maximizes discounted utility over those T periods.
3. The flow equation for capital is given by: k_t+1=k_t-c_t
4. This can be summarized in the following fashion: sum(c_t)+k_t+1=k₁

I managed to solve this problem for given parameter configurations with the following code:

// Rewrite optimization problem 6.1.1 Linear Flow Eqns, p.84 from Adams et al (2015) Microeconometrics for Matlab
mata
mata clear
set matastrict off

// Define flow utility equation
function u(todo, numeric rowvector c, numeric scalar T, numeric scalar beta, u, g, H) {
    
    t = 1..T
    
    u = beta:^(t:-1) * log(c')
    
}

// Set params
beta = 0.9
T = 10
k1 = 100
c0 = J(1, T, 10)

// Set constraints
C = J(1, T, 1)
c = k1

// Set up optimization problem
s = optimize_init()
optimize_init_evaluator(s, &u())
optimize_init_evaluatortype(s,"v0")
optimize_init_params(s,c0)
optimize_init_argument(s, 1, T)

optimize_init_argument(s, 2, beta)
optimize_init_constraints(s, (C,c))
c_optim = optimize(s)
c_optim
sum(c_optim)
end

However, in a slightly modified version of this issue in Section 6.1.2 the flow constraints becomes(Cobb-Douglas-style) non-linear.

That is, the flow constraints now is: k_t+1=f(k_t-c_t, theta)= theta*(k_t-c_t)^alpha

I am not able to come up with a solution for this modified flow equation. After a substantial amount of Googling it seems like that Mata cannot handle such constraints. Is this correct?

Thanks for any insight.

Best,
Ulrich