# Determine price and cost of intermediate good

Consider a world where an intermediate good is produced using a linear production function in a competitive market:

$$Y = qAL$$

where $q$ is the price of the intermediate good, $A$ is productivity. Due to perfect competition, profits are given by

$$0 = \pi = (qA-w)L$$

where $w$ are wages. Due to perfect competition, we have that $w = qA$.

Now consider a final production sector where some market power exists. Why is - I believe - irrelevant to the exposition here. Due to some cost of entry to the final sector, we can determine the measure of firms in the final sector, $S$. Each of these final good firms $i$ then will sell the final good at price $p_i$. That is, we have some non-degenerate price distribution in the final sector.

Now, we have a measure $S$ of final sector firms that require the intermediate good.

Assumption 1 Let's assume for simplicity, that they all will purchase the same quantity (although the quantity is irrelevant due to the linearity of the intermediate good's production function).

How would the price $q$ be determined in this setup, with and without assumption 1?

• Can there be market power for the final sector, given that there is a continuum of firms in both sectors?
• If so, how does the price-setting work?
• In particular, if there is market power on one side, the price should be equal to marginal cost.

However, so far, I was using the FOC $w = qA$: taking the price as given and use that as the wage. Due to linearity of the production function, I feel that I cannot determine both $w$ and $q$ at the same time here. Is that correct? What's the way to go?

• What happens in the labor market? Do final good firms shop at the same market place for labor? Could you close the model from this route? Commented Mar 29, 2016 at 8:21
• @AlecosPapadopoulos Final goods dont need labor. On second thought, there's a household with standard KPR preferences. That would give me a correspondence $Y(w) = Y(qA)$, such that $q$ is implicitly given by the total quantity purchased by the final sector - and then $w = qA$, don't you agree? Commented Mar 29, 2016 at 8:44
• If you introduce labor supply $L^s = L^s(w)$, labor demand (in quantities) is determined by the equilibrium in the intermediate sector $D = AL^d$ where $D$ is the total demand for the intermediate good in quantities. Then from labor market equilibrium you get $L^s(w) = D/A$. If $D$ depends on $q$ you have a second equation linking $w$ and $q$, if it doesn't, then from here you determine $w$ and then you determine $q$ from $w = qA$ Commented Mar 29, 2016 at 8:55
• @AlecosPapadopoulos That suffices as an answer, I think. Commented Mar 29, 2016 at 9:06

Let $D_y = D_y(q)$ be the total demand for the intermediate good, in quantities. Since the market for the intermediate good is competitive it will clear, so we have
$$D(q) = AL \implies L^d = D(q)/A$$
Let $L^s(w)$ be labor supply coming from the utility maximization problem of the households and depending also on the wage. Assume a competitive labor market. Then the wage will adjust to clear it. Here we have
$$L^s(w) = L^d = D(q)/A$$
which gives us a second relation between $w$ and $q$, alongside $w=qA$, and it permits to determine both simultaneously.