demand function of a monopolist is given as Q=50 - 0.5p while the cost function is given as C= 50 + 40q. calculate equilibrium quantity and profit maximizing output.
Find the lvel of output that maximizes profit
To find the equilibrium quantity and profit-maximizing output, we need to find the price at which the demand function and cost function intersect. Let's start by setting the demand function equal to the cost function:
Q = 50 - 0.5p
C = 50 + 40Q
Substituting Q from the demand function into the cost function, we get:
C = 50 + 40(50 - 0.5p)
Simplifying further:
C = 50 + 2000 - 20p
Now, we can set the cost function equal to zero to find the profit-maximizing output:
50 + 2000 - 20p = 0
Rearranging the equation:
20p = 2050
p = 102.5
Substituting the price back into the demand function:
Q = 50 - 0.5(102.5)
Q = 50 - 51.25
Q = - 1.25
Since we cannot have a negative quantity in this case, we know that the equilibrium quantity and profit-maximizing output will be zero.
Therefore, the equilibrium quantity and the profit-maximizing output are both zero.
To calculate the equilibrium quantity and profit-maximizing output for a monopolist, we need to find where the monopolist's marginal revenue (MR) equals marginal cost (MC).
Given:
Demand function: Q = 50 - 0.5p
Cost function: C = 50 + 40q
Step 1: Find the monopolist's marginal revenue (MR)
MR is the change in total revenue resulting from producing and selling one additional unit of output (Q).
The total revenue (TR) is the quantity (Q) multiplied by the price (p): TR = Q * p.
To find MR, we take the derivative of TR with respect to Q:
MR = d(TR) / d(Q)
In this case, TR = Q * p. Differentiating TR with respect to Q, we get:
MR = p + Q * dp/dQ
Since the demand function is given as Q = 50 - 0.5p, we can substitute it into the expression for MR:
MR = p + (50 - 0.5p) * dp/dQ
Step 2: Find the monopolist's marginal cost (MC)
The marginal cost (MC) is given by the derivative of the cost function (C) with respect to quantity (q): MC = d(C) / d(q).
In this case, C = 50 + 40q. Differentiating C with respect to q, we get:
MC = 40
Step 3: Equate MR and MC to find the equilibrium quantity and profit-maximizing output.
Set MR = MC and solve for Q:
p + (50 - 0.5p) * dp/dQ = 40
Simplifying, we have:
p + 50dp/dQ - 0.5pdp/dQ = 40
Rearranging the terms, we get:
50dp/dQ - 0.5pdp/dQ = 40 - p
Dividing both sides by dp/dQ and simplifying, we have:
50 - 0.5p = 40 - p
Combining like terms:
0.5p = 10
Solving for p:
p = 20
Substituting the value of p into the demand function Q = 50 - 0.5p, we can find Q:
Q = 50 - 0.5(20)
Q = 40
Therefore, the equilibrium quantity is Q = 40 and the profit-maximizing output is Q = 40.