# Cobb-Douglas production function with continuum of goods

I'm trying to derive the solution to a final producer with unit elasticity of substitution and a continuum of inputs (of measure $x$).

She minimizes input costs, given an output of $q$.

$$\min_{\{y(i)\}_i, 0 \leq i \leq x} \int_0^x p(i)y(i) di + \lambda\left[ \log q - \log A + \int_0^x \log y(i) di \right]$$

Combining two FOCs over $y(i)$ and $y(j)$, the shadow price drops out:

$$y(i) = p(j) \frac{y(j)}{p(i)}$$

Multiply with $w(i)$ and integrate, to get total labor cost on the left-hand-side:

$$\underbrace{\int_0^x p(i)y(i)}_{C(x)} = p(j) y(j) x$$

Taking logs:

$$\log y(j) = \log(C(x)) - \log p(j) - \log x \tag 1$$

Plugging back into the production frontier:

$$\log q = \log A(x) + \log(C(x)) - \log x - \underbrace{\int_0^x \log p(i) di}_{\equiv P(x)}$$

where I denote the price index as $P(x)$. Plugging back (1):

$$\log q = \log A(x) + \log y(j) + \log p(j) - P(x)$$

or,

$$q = \frac{A(x)y(j)p(j)}{P(x)} \\ y(j) = \frac{P(x)}{p(j)}\frac{q}{A(x)}$$

So, demand for each variety increases in total output $q$, decreases in productivity, and increases in the relative costs of variety $j$ compared to the total price index $P(x)$. This all appears to make sense.

However, Acemoglu gets for $x=1$, and $A(1) = 1$

$$y(j) = \frac{q}{p(j)}$$

that is, the price index does not appear in his equation, which you can find in his slides, slide. What am I missing?

• I suspect that follows from his normalisation of the price of the final good to $1$. However, I didn't follow the derivation carefully. (If this turns out to be correct, anyone can feel free to take my comment and turn it into an answer.) Jan 29 '17 at 13:02
• @TheoreticalEconomist it shouldn't. The price of the final good doesn't even enter at this stage. Notice that I only solved the first step of the problem: Given production of $q$, and distribution of prices, how do we combine the inputs? The second step would be "given optimal input combinations and cheapest production costs, and taking into account the price of the good we sell (or its demand function)", how much do we produce? That's when the final good price should enter - so far I haven't even used it. Jan 29 '17 at 14:13
• I understand that -- it just seems to me that in equilibrium, the final good price has to equal $P(1)$ given CRS. It seems to me to be the only possible explanation, other than you or Acemoglu making a derivation error (which, given a cursory examination, I find less likely than the explanation I've offered). It also explains why Acemoglu mentions the price normalisation where he does. Why point it out there if it's irrelevant? Jan 29 '17 at 14:19
• @TheoreticalEconomist if the final good price is $P(1)$, you are absolutely right. Jan 29 '17 at 14:48
• It ought to be if equilibrium imposes a zero-profit condition. Jan 30 '17 at 21:54