
Vertical Integration
Here is the problem:
To illustrate the double marginalization problem, let’s assume that a manufacturer, m, and a retailer, r, use linear pricing
Inverse demand: p = 10  q
Marginal cost of the manufacturer: c = 2
Marginal cost of the retailer: w
The timing of decisions
m sells to r at a wholesale price w
r sells to endconsumers at a price p
This game needs to be solved backward starting with the maximization problem of r
Here is where I'm stuck:
Profit(retailer) = $\displaystyle (10q)qqw$
$\displaystyle = 10q  q^2 qw$
now I need to take the derivative of profit with respect to qantity so I can maximize profit:
$\displaystyle \tfrac{dProfit(retailer)}{dQuantity} = 10  2q  w = 0 $
$\displaystyle q = \tfrac{10w}{2}$
That's as far as I got before the prof started going to fast and lost me. I know that I'm supposed to now find the profit of the manufacturer; which would be the demand multiplied by margin. So, would that be:
Profit(manufacturer) = $\displaystyle (w2)(\tfrac{102}{2}) $ since that would be demand (which I previously found times his margin)?
If that's true, then should I take the derivative of that? I don't know what to do... please help! Thanks