Optimal policy welfare function


I would like to solve for the optimal policy under discretion and commitment that maximizes the lifetime utility of the representative household.

\begin{align} \max_{c_{T,t},e_t} \quad &-\sum_{t=0}^\infty \beta^t \mathbb{E}_0\left[ \phi B_t^2 + \sigma{c}_{T,t}^2 + \chi\sigma{y}_{N,t}^2 + \frac{\psi}{\alpha_T}({y}_{T,t}-{a}_{T,t})^2 + \chi\frac{\psi}{\alpha_N}({y}_{N,t}-{a}_{N,t})^2 \right] \nonumber\\ s.t. \quad &c_{T,t} = y_{N,t} -\frac{1}{\sigma}e_t\\ &e_t = \frac{1-\alpha_T}{\alpha_T}y_{T,t} - \frac{1}{\alpha_T}a_{T,t}\\ &c_{T,t} = y_{T,t}+\frac{1}{\kappa}(RB_{t-1}-B_t) \end{align}

The planner chooses two variables, c_{T,t} and e_t to maximize the welfare function. I solve this in Dynare and find that the unconditional welfare loss is the same under discretion and commitment. The code used to derive the welfare loss is attached.

com_cc.mod (946 Bytes)
dis_cc.mod (940 Bytes)

To my understanding, under the discretionary case, we assume that future variables are linear functions of future state variables and assume that the coefficients are exogenous to the policymaker. This would imply that although there are no forward-looking variables in the constraint since the welfare function is forward-looking there should be some effects of the instruments on the future variables that the policymaker does not internalize. Therefore the welfare cost should be different between the commitment case and the discretionary case.

Is this understanding correct?

If so, is the reason that the unconditional welfare cost is the same because perhaps Dynare only computes the welfare for the current period? If so, how should I change the code so that Dynare solves for the lifetime welfare?

Many thanks in advance.

The issue is that there are no forward-looking variables in the constraints. It’s the private sector constraints that the planner needs to take as given and where expected future behavior would matter. With Ramsey, the planner can commit to future behavior and the agents will take that into account. With discretion that would be different. But your model is purely backward-looking, so it does not matter whether policy announcements for the future are credible or not.

Thank you for the reply. I have a follow-up question.

Although the constraints do not have forward-looking variables, the objective function does. The discretionary policy maker at period t will still consider what happens to utility in periods t+1 and onward which depends on savings in period t. Wouldn’t this make a difference between the discretionary policymaker and the Ramsey planner?

Perhaps I am understanding the concept of discretionary policymaker incorrectly. Your reply suggests that it is only the forward-looking variables in the private sector constraints that matter for the distinction between commitment and discretion and that the forward-looking components in the objective function are internalized equally for both cases. Is there some reference that I might be able to look at to learn this definition?

Many thanks.

Have a look at Gali’s textbook. The difference is not about the planner’s objective but the private sector equilibrium. Why is there a difference between commitment and discretion? Because agents’ decisions depend on expected future behavior of the planner. The planner has a temptation to promise one policy (e.g. not to inflate debt away next period) and then once agents have made their choice about debt to renege on that promise and indeed deflate the debt away. With commitment, the planner commits to not doing so. With discretion, agents know about this temptation and alter their behavior accordingly. But if there are no forward-looking variable in agents’ behavior, these future promises do not matter.