# 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? – Alecos Papadopoulos Mar 29 '16 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? – FooBar Mar 29 '16 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 $w = qA$ – Alecos Papadopoulos Mar 29 '16 at 8:55
• @AlecosPapadopoulos That suffices as an answer, I think. – FooBar Mar 29 '16 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.