# Welfare in growth models

May I ask what is the acceptable way to measure welfare in growth models? For example, in OLG models, Docquier, F., Paddison, O., & Pestieau, P. (2007) define the social welfare function of the social planner as follows:

W=\sum_{t=0}^{\infty} \beta^t U\left(c^1_t, c^2_{t+1}\right) \;\;\;\;\; (1)

And then solves for a modified golden rule: \frac{\beta f^{\prime}(\bar{k}_t)}{1+n}=(1+g)^{1-b}, where \bar{k}_t = \frac{k_t}{h_t}. So, here, k_t, c^1_t, c^2_{t+1}... grows in steady-state but \frac{k_t}{h_t}, \frac{c^1_t}{h_t}, \frac{c^2_{t+1}}{h_t}... are constants in steady-state, and U(\cdot) is assumed to be homogeneous of degree b. Although a log-utility function is not homogeneous, it seems it can be interpreted as having this property by assuming b=0 and applying some manipulations as discussed in (Del Rey, E., & LOPEZ‐GARCIA, M. A. (2012)).

In my project, however, I am not interested in finding optimal \bar{k}_t in the social planner’s problem. I only want to see if some policy is welfare-improving (in steady-state). So, I guess I can use the following welfare function, right?

\bar{W}=\sum_{t=0}^{\infty} \gamma^t U\left(\frac{c^1_t}{h_t}, \frac{c^2_{t+1}}{h_t} \right) \;\;\;\;\; (2)

Del Rey, E., & LOPEZ‐GARCIA, M. A. (2012) suggest using the specification in (2) rather than (1) as an alternative social welfare function of the social planner to find optimal \bar{k}_t. Using either (1) or (2) determines which policy should be introduced in the Laissez-faire/decentralized problem to match the allocations in the social planner’s problem.

But if I only care about whether a policy is welfare-improving, then I guess the welfare function specified in (2) will suffice, right?

A planner’s problem forces you to be explicit about the social preferences in your economy, i.e. it forces you to specify the weight attached to different cohorts. Without such a restriction, you still need to be able to defend your choice.

Dear Prof Pfeifer, many thanks for the reply. I do not fully understand your point, though. There are 2-cohorts in my model (old and young), and I have explicitly calibrated the value of \beta. Is there some other restriction that I need to specify?

If I may also ask, is there an acceptable way to calculate steady-state welfare effects in endogenous growth models? I am struggling a little bit with finding papers on the topic.

1. The issue is computing social welfare. It’s not obvious that a social planner should use the time preference that is relevant within cohorts to evaluate welfare across cohorts.
2. What makes the endogenous growth model special? The absence of a well-defined steady state? If yes, then that answers your question.