5
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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"?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.