(Steady State) Welfare

[MscK]

Dear all,

I want to calculate the welfare differences between two versions of my model ( in one model foreign aid is used for public investment, in the other it goes to households as a lump sum transfer).
In an ideal world I would use a consumption equivalent measure but as I do not know how to do this, and am short in time, I am trying to look at suboptimal ways to calculate welfare. I have never done this before and seem to be unable to find a paper/ lecture sheets/ information on how to set up such an analysis from scratch.

So I wanted to start as easy as possible and calculate the welfare differences of the steady states of the two models. I have the following utility function
U = E_0 \sum_{t=0}^{\infty} beta^t * [C_t^(1-sigma)/(1-sigma) - theta* (L_t^(1+\chi)/(1+\chi))]
This however always returns negative utility as sigma >1. If I then use the value function W = U + bet*W(+1); this will be negative as well. I thought that I used a rather standard utility function. Does this mean that these functions always return negative welfare or am I making a mistake somewhere?

For the next step I would want to start off with the steady state values of model (1) and then switch regimes and use model 2, simulate for more many periods, and "feed" the time series of the variables that come out of this simulation to the welfare function. I hope that this gives me transitional dynamics. As said, I am new to the welfare analysis thing. Is this a logically/ feasible way of looking at the transitional dynamics of switching regimes?

If anyone knows some material where this is done/ explained please let me know as well!

Thank you and all the best!

[StephaneAdjemian]

Dear MscK,

The level of the welfare has no interpretation (that is precisely why, when computing welfare cost of fluctuations, we express the welfare in terms of consumption). This is because the utility functions are defined up to an increasing transform, i.e. if u(c) is a utility function then v(u(c)) is another utility function representing exactly the same preferences if v() is a monotonically increasing function. Only the order matters, i.e. the sign of u(c_1)-u(c_2), not the levels, i.e. the value of the difference. So it's not a problem if steady state welfare is negative.

The consequence is that it is meaningless to compute the difference in welfare levels, only the sign of the difference is relevant.

If I understand correctly your problem, you only have to perform a deterministic simulation to compute the transition from one steady state to another. If in your equations you have the welfare, coded with something like

Code: Select all

Welfare = Utility(Consuption) + beta*Welfare(1);


The sign of the initial jump of the Welfare variable will inform you on the cost or gain of the policy change, integrating all the effects of the transition to the new steady state (because the variable is forward). But the value of this difference is non informative.

Best,
Stéphane.

[MscK]

Dear Stephane,

Thank you for your answer and your explanations.

So if I understand correctly, I only use the steady state value of consumption to calculate welfare and do not use the second (labor) part of the utility function? And to compute the transition from one steady state to another: W= U(c_model1) + beta*U(c_model2)?

Could I ask you a follow up question? Would it be possible to look at a dynamic transition or lifetime welfare changes? My idea was to start from a steady state of model1 and then use the consumption series that model 2 gives after a shock to this model? So W=U(c1) + beta* U(c2_t)? Or is this a rather strange/ stupid idea?

Thanks again, and all the best!

[StephaneAdjemian]

Sorry, if you have endogenous labour supply you have to put it in the definition of the welfare, writing something like:

Code: Select all

Welfare = Utility(Consumption, Labour) + beta*Welfare(1);


If you simulate the perfect foresight model as I described, you will also have the path of Welfare to the new steady state. But all the information you need is in the jump of the Welfare period in first period when the change in policy is announced. Iterating forward on the Welfare equation, we have:

Welfare_t = \sum_{\tau=0}^\infty \beta^\tau Utility(Consumption_{t+\tau}, Labour_{t+\tau})

So that all the transition to the new steady state is included in the level of Welfare at time t=1.

Best,
Stéphane

PS There was an error in the definition of Welfare in my previous post, I fixed it.