<p>Yes, $\Pi^n_{i=1}$ refers to the <a href="https://en.wikipedia.org/wiki/Multiplication#Capital_Pi_notation" rel="nofollow noreferrer">product of a sequence</a>.</p>

<p>Regarding Lucas' solution, the Lagrangian of the problem is:</p>

<p>$$ L (c_1,...,c_n) = U(c_1,...,c_n) - (c-\sum c_i) $$</p>

<p>The FOC with respect to $c_i$ and $c_j$ are:</p>

<p>$$ \frac{\partial L}{\partial c_i} = \frac{\partial U}{\partial c_i} +1 = 0 $$</p>

<p>$$ \frac{\partial L}{\partial c_j} = \frac{\partial U}{\partial c_j} +1 = 0$$</p>

<p>From here you get (the MRS):</p>

<p>$$\frac{\partial U}{\partial c_i} = \frac{\partial U}{\partial c_j} $$</p>

<p>Since</p>

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

<p>from the MRS you get that:</p>

<p>$$ \frac{c_i}{\alpha_i} = \frac{c_j}{\alpha_j} $$</p>

<p>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).</p>

<p>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 <a href="https://en.wikipedia.org/wiki/Occam%27s_razor" rel="nofollow noreferrer">William of Ockham</a> would rightly observe "they why to bother with multiple colours at all"? </p>

CC BY-SA 3.0
2017-08-16T08:27:37.380Z
2017-08-16T08:32:56.217Z
2017-08-16T08:32:56.217Z
luchonacho
<p>I am reading <a href="https://www.minneapolisfed.org/~/media/files/pubs/books/models/cp131.pdf?la=en" rel="noreferrer">Lucas (1980)</a> and I am a bit confused about the way he formulates the utility function.</p>

<p>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:</p>

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

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

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

<p>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 $V(c_{1},...c_{n}) = U[(\frac{c_{1}}{\alpha_{1}})^{\alpha_{1}}...(\frac{c_{n}}{\alpha_{n}})^{\alpha_{n}}]$</p>

<p>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:</p>

<blockquote>
  <p>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  $\bar{\alpha_{i}} = \int\alpha_{i}dF(\alpha)$</p>
</blockquote>

<p>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?  </p>

<mathematical-economics><utility>
Interpretation of utility function
user11767
# Interpretation of utility function

(Asked by: username_0 on 2017-07-26)

I am reading [Lucas (1980)](https://www.minneapolisfed.org/~/media/files/pubs/books/models/cp131.pdf?la=en) 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?

**Tags:** mathematical-economics, utility


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## 1 answers

### Answer by username_1 on 2017-08-16. Score: 3 | Accepted answer

Yes, $\Pi^n\_{i=1}$ refers to the [product of a sequence](https://en.wikipedia.org/wiki/Multiplication#Capital_Pi_notation).

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](https://en.wikipedia.org/wiki/Occam%27s_razor) would rightly observe "they why to bother with multiple colours at all"?


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