Part 1If the linear inverse demand function LOADING... is pequals=100100minus−22Q,what is the marginal revenue LOADING... function? Draw the demand and marginal revenue curves.Part 2The marginal revenue (MR) function isMRequals=[input]enter your response here . Part 1
01020304050607080901000102030405060708090100Quantity, Q, Units per yearPrice, p, $ per unit Edit coordinates (0,0) interactive graph

Question

Part 1If the linear inverse demand function LOADING... is pequals=100100minus−22​Q,what is the marginal revenue LOADING... ​function? Draw the demand and marginal revenue curves.Part 2The marginal revenue​ (MR) function isMRequals=[input]enter your response here . Part 1
01020304050607080901000102030405060708090100Quantity, Q, Units per yearPrice, p, $ per unit Edit coordinates (0,0) interactive graph

BlackTom AI · Accepted Answer

The problem describes an inverse demand function in the form p = a − bQ and asks for the marginal revenue (MR) function, then to sketch the demand and MR curves. Since no options are provided to choose from, I’ll walk through the standard derivation and interpretation step by step.

First, identify the structure of the given inverse demand. For a linear inverse demand p = a − bQ, the quantity is Q and the price is p. The parameter a represents the intercept on the price axis (the price when quantity is zero), and b represents the slope of the demand curve (the change in price for a one-unit change in quantity).

Next, recall the relationship between total revenue (TR) and MR. Total revenue is TR = p · Q. Substituting the inverse demand into TR gives TR(Q) = (a − bQ)Q = aQ − bQ^2.

To obtain the marginal revenue, differentiate TR with respect to Q: MR(Q) = d(TR)/dQ = a − 2bQ. This formula shows that MR is also linear in Q, with the same intercept a but with slope −2b, i.e., the MR curve is steeper than the demand curve by a factor of two in magnitude.

Apply the specific values given. The inverse demand is stated with a intercept of 100 and a slope b of 2 (since p = 100 − 2Q). Therefore a = 100 and b = 2. Substituting into the MR formula MR(Q) = a − 2bQ yields MR(Q) = 100 − 2·2·Q = 100 − 4Q.

Interpreting the result graphically: the demand curve is a straight line with intercept 100 on the price axis and slope −2, while the MR curve is also a straight line with intercept 100 but a steeper slope of −4. This means MR declines twice as fast as price when quantity increases, and the MR curve lies below the demand curve for all Q > 0.

Regarding the request to draw the curves: on a coordinate plane, plot the demand line p = 100 − 2Q (intercept at p = 100 when Q = 0, crossing the Q-axis where p = 0, which occurs at Q = 50), and plot the MR line MR = 100 − 4Q (intercept at p = 100 when Q = 0, crossing the Q-axis where MR = 0 at Q = 25). The MR curve will be a straight line parallel to the demand line but with a steeper slope, reflecting the relationship MR = a − 2bQ.

If you were to fill in additional blanks or provide numeric checks: at Q = 0, both price and MR equal 100; at Q = 10, p = 80 while MR = 60; at Q = 20, p = 60 while MR = 20; note that MR reaches zero when Q = a/(2b) = 100/(4) = 25, which is half the quantity where price reaches zero (Q = a/b = 50).

Homework:HW10

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