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?