Q(P)=50-P/10 C(Q)=Q^3-20Q^2+125Q Q is greater or equal to 0. (a) Caculate the fi
ID: 1192904 • Letter: Q
Question
Q(P)=50-P/10 C(Q)=Q^3-20Q^2+125Q Q is greater or equal to 0.
(a) Caculate the firm's inverse demand function
(B) Calculate the firm's marginal and average cost function.
(c) over what range of Q does the firm have economies to scale? Over what range of Q does it have diseconomies to scale? what is the firm's lowest possible average cost of production?
(d) Does the firm's profit maximization problem satisfy the global SOC?
(e) Find all values of Q that satisfy the first order condition for the firm's problem.
(f) calculate the firm's profit maximizazing price and quantitiy. '
(g) Calculate the firm's mazimized profit and revenue and cost that produce that profit
(H) Calculate the elasticity of demand at the profit maximizing point
(i) What is the firm's markup at the profit maximizing point? confirm that this markup has the expeccted relationship to the elasticity of demand calculated in part (h)
(K) Let change the demand funciton by assuming demand is half of what it was before, in this new situation calculate the firm's inverse demand function, profit maximizing point and maximized profit.
Explanation / Answer
(a)
Q = 50 - P/10
P/10 = 50 - Q
P = 500 - 10Q
which is the inverse demand function.
(b)
C = Q3-20Q2+125Q
MC = dC/dQ = 3Q2 - 20Q
AC = C/Q = (Q3-20Q2+125Q)/Q = Q2-20Q+125
(c)
The average cost function is
AC(Q) = Q2-20Q+125
AC'(Q) = 2Q - 20
At optimum,
AC'(Q) = 0
2Q - 20 = 0
Q = 10
When Q < 10,
2Q < 20
2Q - 20 < 0
AC'(Q) < 0
This means the firm has economies of scale when Q < 10.
When Q > 10,
2Q > 20
2Q - 20 > 0
AC'(Q) > 0
This means the firm has diseconomies of scale when Q > 10.
The firm's lowest possible average cost of production is when Q = 10. That is,
AC(10) = 102 - 20(10)+125
AC(10) = 100 - 200 + 125
AC(10) = 25
The firm's lowest possible average cost of production is $25.
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