Getting Analitycal form of the Policy Rules after Ramsey FOCs

Dear all,

I’m trying to find a solution to this problem by days for my research.
I have used in Dynare the Ramsey policy command to have the value of my objective function so to compare the welfare of different scenarios and here everything is fine.
But, in the paper, after having calculate the FOCs of the Ramsey Optimal Policy Problem (by hand) how can i get the analytical form of the optimal ramsey policy rules (with respect to the variables of the system) ? (as e.g. is done for the household maximization problem in finding the euler equation or lucas equation etc.)

Thank you if someone is able to help me with this problem
My best

What do you mean? For most models, analytical policy functions are close to impossible to obtain. You could only do something like the method of undetermined coefficients to solve the problem with pencil and paper.

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Really thank you for you answer Professor.

I know it s the third time I ask this question but I don’t know really how to proceed if not with some advanced advice and more experience than mine.
I explain better. I have 4 different scenarios models (every with a different european fiscal rules on public investments). and i want to compare the welfare in every scenario with Ramsey Policy command.
In the appendix of the paper (for the benchmark scanario) I m deriving by hand the Ramsey Problem with a standard CRRA utility function to maximize and 12 constraints.
So I obtain my FOCs and after I write my Ramsey Steady State. Now i Have to eliminate all the lagrange multipliers and I m trying to finding the functional forms of the lagrangian multipliers from every FOC so to insert them into the other ones FOCs. In this way I can obtain some FOCs of my interest just in function on the Variables of my system and combining them to obtain an Optimal Decision Ramsey Rule for that variables.

exemple here JUST for the part of the FOCs of the systems. There is not the final part of my interest of a optimal decision rule for some of the variable
( e.g.: pp. 12-14 even if is for a Growth model. The procedure of the FOCs is the same)

Tell me if it is clear and if it is possible or not. And if there is a method or a paper that can explain me how to obtain the Ramsey decision rule (the analytical forms of the functions of the variables after the calculation of the FOCs) . I need to extrapolate from the problem an optimal policy rule expression just of the variables of the system (not with lagrangian multiplier inside)

Really thank you for your anwser and help
My best

P.S @jpfeifer Sample.pdf (87.0 KB)
Quite rudimental but to try to explain better.

I still don’t understand what you are trying to do. The paper you referenced above transforms some of the conditions to show an analytical result. That can sometimes be done, but not always. The mentioned paper only does this for a special case.
In any case, they do not solve for the policy rules. but focus on a particular set of FOCs they do not contain dynamics, as far as I can see.

sorry, I try to explain better.
I have 4 different scanarios (4 different mod file each one of them with a different rule on public debt and public spending. The model is basically always the same. Every scenario has just different formulation for some fiscal equation).

When I compute Ramsey Policy (using as my objective function the Utility Function) for every of them Dynare give me with evaluate planner objective a Planner Objective Function Value and so I can rank the welfare of every scenario.

Now, I need to create a Benchmark scenario (without that rules on public debt and public spending) with the optimal decision rule derived by Ramsey (that him uses to calculate that values) so to can say that: “with respect to the optimal rule on e.g. “public spending” founded by the ramsey problem, the first scenario has this welfare, the second one this…and so on” but the optimal rule is always the same one.
In other word, having the optimal policy rule for a variable of my interest and comparing the welfare losses associated in every scenario characterized by different structures and constraints. A sort of Welfare under Optimal Rule (optimal by construction by Ramsey Model) compared with welfare obtained in alternative scenarios.
I basically need to derive an Optimal Policy Rule (by construction)

That is not an explanation. Ramsey optimal policies are very different from optimal simple rules.

Sorry Professor, It’s not easy to explain.

But I think I have understand that get the analytical form of the Policies calculated by Ramsey model is mathematically impossible to obtain if not in some cases and with methods like e.g. the undetermined coefficients. On the contrary just numerical methods are used (by Dynare alghoritms (as for example “Perturbation Method”) to compute that maximum values).