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"Had a question on a mid-term paper yesterday (now submitted). One question stumped me a little bit. In particular the "interior Pareto-efficient allocation" part stumped me (I'll explain how I answered after the question). Here's the question:

$$ U_i(c_i)= \sum_{t=1}^{∞} \beta_{i}^{t-1} u_{i} (c_{it}) $$

where $$ 0 <\beta_{^i} < 1 $$ and $u_{i}$ is twice continuously differentiable with $u_i^′(z) > 0$ and $u^{′′}_i (z) < 0$ and satisfies $\lim\limits_{z \to 0} u^′_i(z) = \infty$ and $\lim\limits_{z\to \infty}u^′_i(z) = 0$.

Assume that all $\beta_i$'s are distinct.

Consider that 1 unit of the good is available at each period. Describe what interior Pareto-efficient allocation looks like, and explain why."

So, I think my answer is wrong but I wrote that every period has a Pareto efficient allocation because randomly (we don't know how), one agent receives the good and that means their utility increases and the other n-1 agents utility doesn't decrease. Also, there is no production and no exchange, so Pareto-interior allocation (through the form of marginal rate of utilities being equalised) doesn't come into play?

Those are my thoughts. I was pretty stumped.

(I assumed because they were not in the question that the good is not divisible, exchange is not allowed, and that we do not know how the allocation takes place.)

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  • $\begingroup$ If you want me to write the question without the picture and in text I can do, it just would take me longer (because I haven't done it much). $\endgroup$ – hello1994 Dec 2 '20 at 12:52
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    $\begingroup$ I don't think in this case this is bad enough to be closed outright by mods, but you should still replace the picture with text and equations. There are several reason for this. 1. Text makes the question searchable so other people who have similar question can find it. 2. It help us mods to see if there are duplicates in future. 3. It is against site rules and even though everyone here is reasonable and we won't enforce rules in draconian way, questions that follow the rules are better received by community (get more upvotes - more attention - higher chance of getting good answer) $\endgroup$ – 1muflon1 Dec 2 '20 at 13:01
  • $\begingroup$ Having small portion of question text being a picture does not interfere that much with 1 and 2 so it can be tolerated but if you care about 3 you should put more effort into your question. $\endgroup$ – 1muflon1 Dec 2 '20 at 13:02
  • $\begingroup$ Sorry - I'll edit it now. $\endgroup$ – hello1994 Dec 2 '20 at 13:04
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    $\begingroup$ So if I understand correctly, if savings, exchange are not allowed and the good is not divisible there's not much left really. I think no one allocation is pareto efficient than any other. However, with utilitarian welfare function, allocating to individual with lowest $\beta_i$ is the optimal choice. Anyway, I think question should at least allow for divisibility and/or savings, if not exchange. $\endgroup$ – Dayne Dec 2 '20 at 15:42
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A Pareto optimal allocation can be found by maxmising a weighted sum of the individual utilities. Let $\alpha_i$ be the weight on individual $i$. Then we have the following maxmisation problem: $$ \max \sum_i \alpha_i \sum_{t = 0}^\infty \beta_i^{t-1} u_t(c_{i,t}) \text{ s.t. } \sum_i c_{i,t} = 1\,\,\, \forall t $$ The first order conditions for this problem are: $$ \begin{align*} &\alpha_i \beta^{t-1}_i u'_i(c_{i,t}) = \lambda_t,\,\, \forall i\forall t,\\ &\sum_i c_{i,t} = 1 \,\, \forall t \end{align*} $$ There are no corner solutions (given the Inada conditions)

We know that $u_i'(.)$ is strictly decreasing. Let $\gamma_i$ be the inverse, then: $$ c_{i,t} = \gamma_i\left(\frac{\lambda_t}{\alpha_i \beta_i^{t-1}}\right) $$ We know that $\gamma_i > 0$ with $\lim_{x \to 0} \gamma_i(x) = \infty$ and $\lim_{x \to \infty} \gamma_i(x) = 0$.

If we substitute this into the budget constraint, we get: $$ \sum_i \gamma_i \left(\frac{\lambda_t}{\alpha_i \beta_i^{t-1}}\right) = 1. $$ The functions $\gamma_i$ are strictly decreasing. If $\lambda_t \to 0$ then the left hand side converges to $\infty$ while if $\gamma_t \to \infty$ then the left hand side converges to 0.

As such, there is exactly one unique value of $\lambda_t$ that satisfies this condition. Therefore the optimal value of $c_{i,t}$ (given the values $\alpha_i$) will be unique.

Now, taking the first order conditions for two individuals and taking their quotient gives: $$ \frac{\alpha_i \beta_i^{t-1} u_i'(c_{i,t})}{\alpha_j \beta_j^{t-1} u_j'(c_{t,j})} = 1,\\ \to \frac{u_i'(c_{t,i})}{u_j'(c_{t,j})} = \frac{\alpha_j}{\alpha_i} \left(\frac{\beta_j}{\beta_i}\right)^{t-1} $$ Then: $$ \frac{u_i'(c_{t+1,i})}{u_j'(c_{t+1,j})} = \frac{\beta_j}{\beta_i} \frac{u_i'(c_{t,i})}{u_j'(c_{t,j})} $$ This shows that if $\beta_j > \beta_i$, i.e. individual $j$ is more patient than individual $i$, then the marginal utility of individual $i$ should be increasing relative to that of individual $j$. So in relative terms the more impatient individuals see their consumption decreasing while the more patient individuals see their consumption increasing.

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