# Does Pareto-efficency take into account growth?

I'll try to state my question clearly:

For a given a "wealth" allocation Wt = (w1, w2, w3, ... ) for individuals I = (i1, i2, i3, ... ).

And a distribution of D= (w1/|W|, w2/|W|, w3/|W|, ... ), that is, as a percentage of the total.

Where Wt is Pareto Efficient at a given time frame  t , and W(t+1) is Pareto Efficient at  t+1 .

When there is growth such that     |W(t+1)| > |Wt|

Is Dt=D(t+1)?

If so, why? If not, how do you allocate say, capital, whose value is intrinsically linked to growth?

I understand many of my assumptions may be plain wrong or badly stated, I am still an undergrad. Sorry about that. And thanks in advance.

Further clarifications:

For each element  w  of Wt = (w1, w2, w3, ... ),

w= P * B

Where P is the price vector of a commodity bundle B.

I hope that my representation of growth as an increase in the magnitude of the W vector is correct. This is what has been bugging me the most. It's safe to assume that growth causes an increase in |W|, right?

• I don't have time at all to look at your equations right now. Here's the kicker: Pareto-Efficiency is in utility terms. If "growth" (of whatever kind) is represented correctly in preferences, then Pareto-efficiency takes that into account. Otherwise it doesn't. – FooBar Sep 24 '15 at 17:13
• You identify wealth with purchasing power, which is not wrong, but by writing $$w_i = \mathbf p\cdot \mathbf b_i$$ you identify it with nominal purchasing power. Note that in this way, total "wealth" may "increase" because prices have increased, while the quantity bundle of goods has remained unchanged. No growth here. Perhaps you should re-think this part of your question. – Alecos Papadopoulos Sep 24 '15 at 20:01
• Oh, you're both right. Thanks about the input. I'm thinking this through. – Pablo Derbez Sep 24 '15 at 22:54
• @AlecosPapadopoulos So it turns out I was making a crass mistake by identifying the influence of time-discounted utility on the present distribution with a unique distribution at any point in time. Anyways, what I'm intuitively grasping at is that, it seems problematic that a distribution at $t$ is determined by future discounted utility, but the distribution itself affects utility at $t$. To avoid the C word (capital), we could think about this in terms of seeds. Say we have corn, wheat, corn seeds and wheat seeds (...) – Pablo Derbez Sep 25 '15 at 18:57
• (...) and the utility value of a corn seed is the value of the utility we can get for the grown produce then (by consuming or trading the corn), time-discounted. But this new allocation affects the quantity of corn at t+1 and consequently, its relative price, and utility (say we are producing corn for trade, as we are interested in a trade economy). I say this seems problematic because it seems difficult to achieve a Pareto-efficient outcome by having each individual maximize his Utility function. For example, if too many people plant corn. – Pablo Derbez Sep 25 '15 at 19:14

It seems to me that as long as every person likes wealth any allocation $W_t$ will be Pareto-optimal.

Even if this was not the case, so supposing that $w_1, w_2,...$ are not real numbers representing wealth but vectors representing bundles of goods the distribution from one time to another can change drastically if you do not specify how growth occurs. At $t=1$ someone might have everything, then at $t=2$ someone else might have everything. Both of these states are Pareto-optimal, but the distributions are quite different.

So no, it does not follow from your conditions that $D_t$ necessarily equals $D_{t+1}$.

• Yup! I see that now. In hindsight, I don't know why I thought Dt = D (t+1) should hold true. – Pablo Derbez Sep 24 '15 at 22:55
• I'm still musing about Pareto Optimality over time, but my particular question was answered. Thanks! – Pablo Derbez Sep 24 '15 at 22:58

Perfectly fair question. Let the price vector be expressed in terms of a numeraire- i.e. a physical good. Stipulate that all economic activity is voluntary, there is no preference revelation problem (i.e. no strategic behaviour), no hedging etc (i.e. the standard perfect competition assumptions) then the answer is no if there is preference and endowment diversity provided some goods are consumed in each period. If agents are identical, then the answer is yes. This is because a Wt is Pareto efficient if you can make a mutually advantageous trade which enables you to consume more- assuming that is your preference- provided you find someone who wants more assets and prefers to be thrifty for the moment. If Preference diversity obtains this happens so Dt will differ from the previous Dt-1. Interestingly, if preference diversity is too great or too little and markets use up scarce resources then no conclusion can be drawn. The same thing happens when you introduce a little uncertainty- i.e. hedging appears- or a bit of information asymmetry- so preference revelation becomes problematic.

I take it you want an answer the question 'what type of allocatively efficient Growth is distributionally neutral'- this was a big topic in the Sixties- turnpike theorems and so on. It's good to see young people pondering these things in our present climate! My tuppence is that it may be that some distributions are better for human beings than others- if so there is a Muth Rational economic theory which, under common knowledge, would be the Schelling focal solution to Society's coordination problem. This means Preferences change so that Society can still get to where it really wants to be- even if the Maths is intractable! There was a Japanese 'peasant sage'- Ninomiya- who redefined what we call saving as 'concession'- i.e. a voluntary foregoing of consumption so the worst off can eat- and what we call Return on Investment- as 'obligation to acknowledge Virtue' which can be discharged collectively. There has been some General Eqbm work based on this sort of thinking. Here there is no 'Paradox of Thrift' nor naughty 'Income effects' to spoil our nice models because people are Muth rational- not zero intelligence rule following droids! This does not mean that basic existential problems- Uncertainty, Concurrency, Co-ordination etc disappear- it's just that mathematical models can give better, more human, insights into profound dilemmas. You asked a good question- that's the point of going to College. The good questions you ask there- not the answers you are required to give- will enrich your life to the end of your days.

• Thanks for the detailed response! I have one more quesiton, if you don't mind: Could we consider capital as a commodity $c_t$ (that is, a capital which matures at time $t$), whose utility is the increase of maximum utility at $t$, discounted for time preference? – Pablo Derbez Sep 25 '15 at 17:54
• Hi Pablo, not sure what is meant by Capital with matures at time t. Suppose Capital item x is a machine which lasts t years producing a certain amount of Consumer good y per annum. Then, you could say its Utility is the Present Value of the consumption stream. The problem is that K is the summation of all Capital goods- some are breaking some are being replaced and there is also 'Capital deepening'- so you get an aggregation problem. Also, you can have 'reswitching' and cases where a consumer good can become a capital good and vice versa. With Uncertainty, things get really scary! – Vivek Iyer Sep 26 '15 at 18:25
• Why no up vote? – Vivek Iyer Sep 27 '15 at 17:27
• Sorry, voted now – Pablo Derbez Sep 27 '15 at 17:29
• Thanks. A lot of bright Indian people wasted their time chasing after Sraffa. My teacher was Morishima who had a 'rational distribution' theory- basically shadow prices do the work. Silly- I know. – Vivek Iyer Sep 27 '15 at 18:53