PC Connection and CDW are two online retailers that compete in an Internet marke
ID: 1193924 • Letter: P
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 = 1,250 - 2Q. When it does not match price changes, CDW’s inverse demand curve is P = 800 -0.5Q. Based on this information, determine CDW’s inverse demand function over the last couple of months. Over what range will changes in marginal cost have no effect on CDW’s profit-maximizing level of output?
Explanation / Answer
In its standard form a linear demand equation is Q = a - bP. That is, quantity demanded is a function of price. The inverse demand equation, or price equation, treats price as a function g of quantity demanded: P = f(Q). To compute the inverse demand equation, simply solve for P from the demand equation. For example, if the demand equation is Q = 240 - 2P then the inverse demand equation would be P = 120 - .5Q, the right side of which is the inverse demand function.
The inverse demand function is useful in deriving the total and marginal revenue functions. Total revenue equals price, P, times quantity, Q, or TR = P×Q. Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 - .5Q) × Q = 120Q - 0.5Q². The marginal revenue function is the first derivative of the total revenue function; here MR = 120 - Q. Note that the MR function has the same y-intercept as the inverse demand function in this linear example; the x-intercept of the MR function is one-half the value of that of the demand function, and the slope of the MR function is twice that of the inverse demand function. This relationship holds true for all linear demand equations. The importance of being able to quickly calculate MR is that the profit-maximizing condition for firms regardless of market structure is to produce where marginal revenue equals marginal cost (MC). To derive MC the first derivative of the total cost function is taken. For example assume cost, C, equals 420 + 60Q + Q2. Then MC = 60 + 2Q. Equating MR to MC and solving for Q gives Q = 20. So 20 is the profit maximizing quantity: to find the profit-maximizing price simply plug the value of Q into the inverse demand equation and solve for P.
When CDW inverse demand curve is given by P = 1,250 – 2Q R = 1250Q - 2Q^2 Marginal revenue (MR1) = 1250 - 4Q Here you have given a demand function in the form of Q(p) - Q as a function of P. The inverse demand function will be found with P(q), P as a function of Q.
Q = 2000 - 100P
Q - 2000 = -100P
20 - (Q/100) = P
If you graph P = 20 - (Q/100), that is the inverse demand curve.
The inverse demand function expresses px as a function of x (rather than quantity as a function of price, as above). We simply solve the above equation for px: x = 10, 000 10px becomes 10px = 10, 000 x, or px = 1000 x 10. (b) Total revenue is price multiplied by quantity. To get total revenue as a function of x, use the inverse demand function: T R = pxx = (1000 x 10)x = 1000x x2 10 . Marginal revenue is the derivative of total revenue with respect to x, so MR = TR x = 1000 x 5 . (c) To maximize total revenue, we find the value of x where marginal revenue is zero. Setting marginal revenue equal to zero, we have 1000 x 5 = 0. Solving, we have x = 5000, and plugging into the inverse demand function, we have px = 500. (d) The formula for the elasticity of demand is d = dx dpx ³px x ´
From the demand curve, x(px) = 10, 000 10px, we calculate dx dpx = 10. Therefore, we have d = 10 ³px x ´ . At the point of maximum total revenue, x = 5000 and px = 500, the elasticity is d = 10 µ 500 5000¶ = 1.
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