Interpretation of utility function

Question

I am reading Lucas (1980) and I am a bit confused about the way he formulates the utility function.

So there is one non-storable good that comes in $n$ colours and one unit of labor produces $y$ units of any color. He says:

Consumption is now a vector $(c_{1t},...c_{nt})$ where $c_{it}$ is the consumption of color $i$ in period $t$.

So far I have no problems. But then he defines current period utility as:

$V(c_{1},...c_{n}) = U[\pi_{i=1}^{n}(\frac{c_{i}}{\alpha_{i}})^{\alpha_{i}}]$ where U is a standard utility function. $\sum _{i} \alpha_{i}=1$ and $\sum _{i} c_{i}=c$

What seems to confuse me is first of all the use of lower case pi. Surely Lucas is referring to the multiplication operator? I.e $V(c_{1},...c_{n}) = U[(\frac{c_{1}}{\alpha_{1}})^{\alpha_{1}}...(\frac{c_{n}}{\alpha_{n}})^{\alpha_{n}}]$

Then he says that the relative prices of the goods must be unity which I understand. However, the next part is where my main source of confusion lies, he says:

With these prices, consumers will select color proportions $c_{i}/c=\alpha_{i}$ and given this mix $V(c_{1},...c_{n}) = U(c)$. Without altering the example one can think of all agents having the same $\alpha$-weights, of agents distributed by a c.d.f. $F(\alpha)$ of weights. (...) In each of these cases, the equilibrium output mix (per capita) is $(\bar{\alpha_{1}}y_{1},...,\bar{\alpha_{n}}y_{n})$ each period, where $\bar{\alpha_{i}} = \int\alpha_{i}dF(\alpha)$

I tried equating MRS to the relative prices (i.e 1) but since I'm unsure about the form of the utility function I don't know if my working is correct. Also, I am very confused about the definition of $\alpha$. What does it mean for agents to have weights? Is it just a way of stating that they are equally distributed?

Answer 1

Yes, $\Pi^n_{i=1}$ refers to the product of a sequence.

Regarding Lucas' solution, the Lagrangian of the problem is:

$$ L (c_1,...,c_n) = U(c_1,...,c_n) - (c-\sum c_i) $$

The FOC with respect to $c_i$ and $c_j$ are:

$$ \frac{\partial L}{\partial c_i} = \frac{\partial U}{\partial c_i} +1 = 0 $$

$$ \frac{\partial L}{\partial c_j} = \frac{\partial U}{\partial c_j} +1 = 0$$

From here you get (the MRS):

$$\frac{\partial U}{\partial c_i} = \frac{\partial U}{\partial c_j} $$

Since

$$ \frac{\partial U}{\partial c_i} = \left(\frac{c_i}{\alpha_i}\right)^{-1} \Pi^n_{i=1} \left(\frac{c_i}{\alpha_i}\right)^{\alpha_i} = \left(\frac{c_i}{\alpha_i}\right)^{-1} U(c_1,...,c_n) $$

from the MRS you get that:

$$ \frac{c_i}{\alpha_i} = \frac{c_j}{\alpha_j} $$

Using these, the definition of $c=\sum c_i$, and the fact that $\sum \alpha_i =1$, you can obtain Lucas's solution (I leave the last bit of the solution to you).

Finally, regarding weights, I think that is just to make the model interesting, as it means consumers might prefer some colors more than others (they spend $\alpha_i$% of their income in colour $i$). If not, they would spend $1/n$ of their income on each, to which any loyal disciple of William of Ockham would rightly observe "they why to bother with multiple colours at all"?