consider a firm\'s quadratic revenue function, R=p.q= a.q 2 +bq , and linear cos
ID: 3004884 • Letter: C
Question
consider a firm's quadratic revenue function, R=p.q= a.q2 +bq , and linear cost function, C=vq+f , where a and b are constant terms, v is unit variable cost and f is total fixed cost. Variables R, C, and p are measured in dollars and q equals the number of units sold. the firm experiences 2 break-even points at (500, $10,000) and (1,800,$12,600).
a. determine the equation of total revenue r=f(q)
b. determine the quantity q* and price p* that maximizes total revenue R
c. Calculate the maximum total revenue R*
d. determine the restricted domain of p and restricted ranges of q and R.
e.determine the equation of total cost C and calculate it at the quantity q*
f. calculate variable and fixed costs at quantity q*
g. calculate average cost, AC , at the quantity q*
h. determine the restricted range of C.
i. write the equation of profit function
j. determine the level of quantity q1, and price p1, that maximize the profit.
k. Calculate total revenue R1 , total cost C1 and profit at the quantity of q1
l.determine the restricted range of profit
Explanation / Answer
a) Break- even occurs when revenue = cost. In other words, R = C and Profit = 0
Using the two points, we can solve for a, b, v and f
pq = aq2 + bq => p = aq + b
10,000 = 500a + b and 12600 = 1800a + b giving us a= 2 and b = 9000
r(a) = 2q2 + 9000q
Similarly, v = 13,600, f = 1,800,000, so, C = 13,600v + 1,800,000
b) Maximum revenue occurs when MR = MC MR = r'(q) and MC = c'(q)
MR = 4q + 9000 and MC = 13,600. Thus, q* = 1150
p* = aq* + b = 2300 + 9000 = $11,300
c) Total Revenue R* = p*q* = 11,300* 1150 = $12.995 Million
e) Already solved in a. Substitute q*
i) Profit = Revenue - Cost = r(q) - c(q) = 2q2 + 9000 - (13,600q + f)
Solve the remaining parts using similar approach.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.