Suppose that the demand functions for price increases and for price cuts facing an oligopolist are, respectively, Q1=560-80p1
and
Q2=200-20p1
Suppose that the firm’s total cost function is TC=4Q+0.050Q^2
3.1 Derive the MR1, MR2 and MC functions facing the oligopolist.
3.2 Determine the price and the output at the kink on the demand curve.
3.3 Determine the upper and lower limits of the MR gap and prove that MC falls in the MR gap.
3.4 Draw a well labelled diagram to capture the kinked demand curve.
3.5 Calculate the total profits of the firm at output Q.
3.1 To derive the marginal revenue (MR) functions facing the oligopolist, we need to take the derivative of the total revenue (TR) function.
Total Revenue (TR) is given by the product of quantity (Q) and price (P): TR = Q * P.
For the first demand function Q1 = 560 - 80p1, we can rearrange it to solve for price:
p1 = (560 - Q1)/80.
Substituting this price expression into the TR function, we get:
TR1 = Q1 * [(560 - Q1)/80].
Differentiating TR1 with respect to Q1, we obtain:
MR1 = d(TR1)/dQ1 = (560 - Q1)/80 - Q1/80 = (560 - 2Q1)/80.
Similarly, for the second demand function Q2 = 200 - 20p1, rearranging it for price yields:
p1 = (200 - Q2)/20.
Substituting this price expression into the TR function, we get:
TR2 = Q2 * [(200 - Q2)/20].
Differentiating TR2 with respect to Q2, we obtain:
MR2 = d(TR2)/dQ2 = (200 - Q2)/20 - Q2/20 = (200 - 2Q2)/20.
Now, let's find the marginal cost (MC) function from the total cost (TC) function.
TC = 4Q + 0.050Q^2.
Differentiating TC with respect to Q, we get:
MC = d(TC)/dQ = 4 + 0.100Q.
Therefore, the MR1 function is (560 - 2Q1)/80, the MR2 function is (200 - 2Q2)/20, and the MC function is 4 + 0.100Q.
3.2 To determine the price and output at the kink on the demand curve, we need to find the quantity (Q*) where MR1 = MR2. Setting MR1 equal to MR2, we have:
(560 - 2Q1)/80 = (200 - 2Q2)/20.
Cross-multiplying and simplifying, we get:
560 - 2Q1 = 4(200 - 2Q2).
Distributing, we have:
560 - 2Q1 = 800 - 8Q2.
Simplifying further:
2Q1 - 8Q2 = 240.
To find the output Q*, we can use the equation system of demand:
Q2 = 200 - 20p1,
Q1 = 560 - 80p1.
Substituting Q2 from Q1 into the equation above:
2(560 - 80p1) - 8Q2 = 240.
Simplifying and rearranging:
-80p1 = -560.
Solving for p1, we find:
p1 = 7.
Substituting p1 = 7 into Q2, we get:
Q2 = 200 - 20(7) = 60.
Therefore, at the kink on the demand curve, the price is 7 and the output is 60.
3.3 To determine the upper and lower limits of the MR gap and prove that MC falls in the MR gap, we need to compare the MR1 and MR2 functions.
The upper limit of the MR gap is the maximum MR value among the two functions, which occurs when Q = 0. To find it, substitute Q = 0 into both MR1 and MR2:
MR1(0) = (560 - 2(0))/80 = 560/80 = 7.
MR2(0) = (200 - 2(0))/20 = 200/20 = 10.
Therefore, the upper limit of the MR gap is 10.
The lower limit of the MR gap is the minimum MR value among the two functions, which occurs at the kink. From 3.2, we know that the price at the kink is 7. Substituting p1 = 7 into MR1 and MR2:
MR1(7) = (560 - 2Q1)/80 = (560 - 2(560 - 80(7)))/80 = 7.
MR2(7) = (200 - 2Q2)/20 = (200 - 2(60))/20 = 7.
Therefore, the lower limit of the MR gap is 7.
To prove that MC falls within the MR gap, we need to evaluate the MC function at Q = 60:
MC = 4 + 0.100(60) = 4 + 6 = 10.
Since 7 < MC < 10, we can conclude that MC falls within the MR gap.
3.4 Drawing a labeled diagram to capture the kinked demand curve:
The diagram should have quantity (Q) on the horizontal axis and price (P) on the vertical axis. Plot two separate demand curves with different slopes, intersecting at the kink. Label the kink as (Q*, P*) where Q* = 60 and P* = 7. Then, label the MR1 and MR2 functions and the MC function accordingly.
3.5 To calculate the total profits of the firm at output Q, we need to subtract total costs (TC) from total revenue (TR).
Total revenue is given by the product of Q and P, where P is the price at output Q. From the demand functions, we know that at Q = 60, p1 = 7. So:
TR = Q * P = 60 * 7 = 420.
Total cost is given by the TC function:
TC = 4Q + 0.050Q^2.
Substituting Q = 60 into the TC function:
TC = 4(60) + 0.050(60)^2 = 240 + 180 = 420.
Therefore, the total profits at output Q = 60 is 420 - 420 = 0.