Question

PC Connection and CDW are two online retailers that compete in an Internet market for digital cameras. While the products they sell are similar, the firms attempt to differentiate themselves through their service policies. Over the last couple of months, PC Connection has matched CDW’s price cuts, but has not matched its price increases. Suppose that when PC Connection matches CDW’s price changes, the inverse demand curve for CDW’s cameras is given by P = 1,225 - 2Q. When it does not match price changes, CDW’s inverse demand curve is P = 775 -0.5Q. Based on this information, determine CDW’s inverse demand function over the last couple of months.

P = ___ - ___Q if Q ≤ 300
    

___ - ___ Q if Q ≥ 300

Over what range will changes in marginal cost have no effect on CDW’s profit-maximizing level of output?

$___ to $___

Answer 1

Solution:

We are given a case of oligopoly market, under which firms face a kinked market demand curve (due to the reasoning given that price cuts are matched and price increases are not). We are given the two forms of such demand curves, which will have a kink at specific quantity. In our example, the kink occurs at Q=300 that is 300 CDW's cameras. We have to simple find which of the two inverse demand functions applies for lower (than 300) units of output and which one applies for higher (than 300) units.

With Q = 300, P = 1225 - 2(300) = $625 (you would get the same value by substituting Q=300 in another inverse demand function as it s the kink point or the intersection point of the two demand curves, and thus, both curves share the point P=$625, Q=300 cameras).

So, as the firm starts from the price of $625, if CDW increases the price any further, PC Connection will not match such price increase. Since, both sell similar products, such price increase will shift a huge consumer base from buying CDW cameras to buying PC Connection cameras. Thus, CDW will lose a greater market then, suggesting that for higher prices (or correspondingly, lower output (<= 300)), firm faces relatively more elastic demand curve.

If CDW decreases it's price, PC Connection will compete by undergoing a price cut itself in order to maintain it's market base (or number of consumers). So, under such scenario, even with price cut, CDW will not be able to attract a huge number of consumers, and price decrease will lead demand to increase by small amount. Due to such small demand responsiveness to price, we can say that for lower prices (or correspondingly, higher output (>= 300)), firm faces relatively inelastic demand curve.

So, to determine which inverse demand function fits which case, we find the price elasticity of the two curves:

ed = ; we have already seen that P = $625, and Q = 300 lie on both curves, so using it for both elasticities.

For P = 1,225 - 2Q, demand function is: Q = 612.5 - 0.5*P

So, = -0.5

And ed = -0.5*(625/300) = -1.042

For P = 775 - 0.5Q, demand function is: Q = 1550 - 2P

So, = -2

And ed = -2*(625/300) = -4.162

So, in absolute terms, elasticity is higher for latter case, and lower (inelastic) for former case.

We can then say that

P = 775 - 0.5Q for Q <= 300

P = 1225 - 2Q for Q >= 300

To find the specified range, we will have to calculate the marginal revenues obtained by two functions at Q=300.

Total revenue = P*Q

For P = 1225 - 2Q, Total revenue = (1225 - 2Q)Q

TR = 1225Q - 2Q²

Marginal revenue =   = 1225 - 2*2Q

MR = 1225 - 4Q

At Q = 300, MR = 1225 - 4*300 = $25

For P = 775 - 0.5Q, TR = (775 - 0.5Q)Q

TR = 775Q - 0.5Q²

MR = 775 - Q

With Q = 300, MR = 775 - 300 = $475

So, the required range is $25 to $475, which is the discontinuity or jump, and thus, intersection of it with the marginal cost (MC) curve will not occur. So, MC has no effect in this range.