Thoughts On This Model

[YoungEconomist]

I saw a model here, and I wanted to hear your input on it: Econ: for comments: GE example with indivisible (dis)utility

I just copied and pasted it below so that you won't have to go to the website if you don't want to.

Title: Econ: for comments: GE example with indivisible (dis)utility

Simple Robinson Crusoe (RC) economy.

There are 2 periods.

Price level p is given exogenously (e.g. through trade).

RC is the "capitalist," he earns competitive profits (=0) so he is indifferent between producing and not producing. He doesn't have to eat to live.

Friday is the laborer, he has a family and a (high) fixed disutility of labor: his disutililty as a function of hours worked H is C1(H) = f + c1 H for f > 0 and c1 > 0.

If the Friday family survive the first period then in the second period, Friday Jr. will be the laborer. He is much more efficient than his father. His disutility is C2(H) = 0 + c2 H where c2 < c1.

Friday's total utility is U = Y - C1 = wH - c1 H - f, where Y = income = consumption = wage x hours worked.

Suppose if Y < C1 then the Fridays cannot survive to the second period.

First-period neoclassical equilibrium is at dC1/dH = c1 = w; suppose c1 = 1 = w. At w = 1, suppose Friday works H = 1 hr. and earns Y = $1.

Further suppose f = $2 > $1.

Given these parameters, Friday works 1 hr. At the end of period 1, all Fridays die. In period 2 there is no production. Their total utility over two periods is then -infinity.

This is an inefficient outcome. Had Friday Jr. survived the first period, production would have folded by a factor of k > 1 in period 2 (i.e., Friday Jr. would have worked many hours more than his father). Total utility over two periods would then have been > 0.

Ends that can be tied together:

1. The minimum level of income necessary to survive period 1 can be tied to p.

2. The linear part of C(H) can be made into a backward-bending labor supply function, c(H), such that there are two equilibrium wage levels, w and W, where W > w. With a downward-sloping labor demand curve, W is an unstable equilibrium but w is stable. If equilibrium selection rule includes "stability," then W would be ruled out. This can further be tied to a low-wage equilibrium that ends up being dynamically inefficient because it does not generate sufficient income to survive the first period.

[trjohnson]

What sort of feedback are you looking for?

I ask because changes to well established models aren't uncommon by any means. If you're wanting feedback on their result, that's one thing. Their method, another. Applications, yet another.

So be a little more specific as to what you want people to analyze. Whole book chapters can be dedicated to models.

[YoungEconomist]

Here is what the poster of that model had to say (in regards to the model):

Not to put too fine a point on it, if the equilibrium wage isn't sufficient for survival then we all die, which is an inefficient outcome, not the least from a personal utilitarian perspective.

The neoclassical, marginalist market equilibrium approach rests on a set of very specific assumptions that are (or should be) taught in any grad-level micro course (under the topic General Equilibrium, or "GE" for short). It was a Frenchman (Gerard Debreau) who first proved the existence of a GE in the 2nd half of the last century; he also realized that the set of minimum assumptions necessary to prove the existence of a GE includes very specific assumptions, even in a static (one-period) model of the economy, as was his. In a dynamic context (e.g. multiple periods), the model (and the assumptions to sustain it) become even messier and even more prone to be violated.

The French education is peculiar in that respect. Elite French colleges are populated with professors and grad students who understand the concept of a general equilibrium in a capitalist (decentralized market) economy probably better than anyone else. They also understand the assumptions that underlie that model very well, and can give 10 different examples in less than 10 minutes which demonstrate that a general equilibrium either does not exist, or is not efficient, and for a different reason in each case. Simply stated, they have written the book on the formal mechanics of a market economy. For this reason, they also understand very well exactly the limits of these models as a realistic description of the economy.

[asquare]

Please don't copy entire posts from other forums. It's fine to provide a link, but if people want to read the whole post, they should visit the other website.

And on the substance, I think trjohnson's post is right on. The model you've asked about is a simple twist on a standard Robinson Cursoe or endowment economy model. Twists like this are very common. What are you asking us to analyze or comment on?

[ssendam]

Friday's total utility is U = Y - C1 = wH - c1 H - f, where Y = income = consumption = wage x hours worked.

This utility function does not take the second period into account, usually there should be a discount factor times second period utility. If the future does not enter into the first period decision at all, then Friday in period 1 might as well be a completely different person from Friday in period 2.

[Antichron]

I don't understand what the inefficiency is. Can you explain how you can feasibly make someone better off without making anyone else worse off? If I understand correctly, the equilibrium the OP described is not inefficient. In order to make Friday Jr. better off, you have to take resources from the capitalist. This is not a Pareto improvement.

[Antichron]

Ooohh... I think I see. If we assume transferable utility (so that a social planner would want to maximize the sum of utilities), then we could conceive of a scheme in which Friday Jr. somehow pays the first period capitalist to let Friday eat enough to survive. (Though it's not clear how we would move resources from the second period to the first, but let's assume that the capitalist lives for two periods and doesn't discount.)

In that case, the reason why the equilibrium is not efficient is that Friday does not internalize the externality that he imposes on Friday Jr. (The model isn't well specified, but I'm assuming that the OP assumed that the utility to Friday Jr. of not being born is -infty) Externalities are a common reason for the first welfare theorem to break down.

[Antichron]

And I don't agree that Debreu thought that it was more complicated to establish existence of equilibrium in a dynamic model. After all, to Debreu (chapter 6 or 7 of Theory of Value), a commodity in a different period is treated as a different commodity. By relabeling, one can effectively reduce the dynamic problem into a static one.

Sorry for the flurry of posts. :-)

[polkaparty]

By relabeling, one can effectively reduce the dynamic problem into a static one.

I've been thinking about this for a while. Particularly I've been trying to see what the precise definition of the commodity space should be. For the most part, all sources I find just give fairly vague intuition about it (Debreu included) and then assume that it's R^n_+ and move on. This issue is compounded when texts talk of a consumption set being some sort of subset of the commodity space.

I could just be worrying about nothing but defining these sets constantly disturbs me, especially at night. So far I have my own working definition based on the nature of constraints imposed at each stage and the meta-commodity definition, but I really just made something up based on what I thought sounded right. I can't find any sources where anyone actually cares about this sort of foundational issue.

Also, how operationally useful is it to allow meta-generalized commodities in the sense of unique time and location and how often is this actually done? Has someone proven some sort of transfer theorem to convert a dynamic problem where the commodity space does not allow for time based definitions into a static problem with time based definitions?

[EnumaElish]

And I don't agree that Debreu thought that it was more complicated to establish existence of equilibrium in a dynamic model. After all, to Debreu (chapter 6 or 7 of Theory of Value), a commodity in a different period is treated as a different commodity. By relabeling, one can effectively reduce the dynamic problem into a static one.

One cannot turn a static model into a (truly) dynamic model simply by relabeling (or the other way around). The relabeled model is a two-period static model, not a dynamic model. For an introduction to dynamic GE models, see http://www.mpsge.org/primer/primer.pdf

And thanks for your comments on my "Robinson Crusoe" model.