p1=14-2q1and p2=10-q2 in which demand function determine the profit maximizing price and quantities in each monopolistic separet market?
Profit is maximized at the quantity of output where marginal revenue equals marginal cost. Marginal revenue represents the change in total revenue associated with an additional unit of output, and marginal cost is the change in total cost for an additional unit of output.
Therefore, both marginal revenue and marginal cost represent derivatives of the total revenue and total cost functions, respectively. To determine marginal revenue and marginal cost; setting them equal to one another maximizes total profit.
The monopolist’s demand function for both markets are
P1=14-2Q1
P2=10-Q2
TR= PQ, so we can generated the total revenue for both markets equation.
TR1= (14-2Q1) Q1
=14Q1-2Q21
TR1 = (10-Q2) Q2
TR1=10Q2-Q22
The monopolist total cost function is the same for both markets that is TC= 2Q
The profit-maximizing quantity of output is determined for both markets through the following steps:
1. To determine marginal revenue by taking the derivative of total revenue with respect to quantity both markets.
MR=dTR/dQ1=14Q1-2Q21 =14-4Q1
MR=14-4Q1
MR= dTR/dq2=10Q2-Q22
=10-2Q2
2. To determine marginal cost by taking the derivative of total cost with respect to quantity both markets.
MC= dTC/dQ1 =2Q1
Q1=2
MC= dTC/dq2 =2Q2
Q2=2
3. Set marginal revenue equal to marginal cost and solve for Q1 and Q2.
MR=MC
14-4Q1 = 2
14-2=4Q1=16=4Q1=
Q1=4
And for market two
10-2Q2 = 2
10-2 =2Q2
Q2= 4
4. Substituting 4 for q1 and q2 in the demand equation enables to determine price.
P1=14-2Q1=14-2*4=14-8=6
P1=6
And for market two
P2=10-Q2=10-4
P2= 6
The profit-maximizing quantities for both markets are 4 units and the prices are $6 per unit. This means the value of price and quantities are the same for both markets because the total cost function is the same
B. To get the greater profit results from price discrimination as follows
TR1 = 14*4-2(4)2 =56-32
TR1 =24
TC1 =2Q1=2*4=8
TC1 = 8
∏1 = TR1-TC1 = 24-8
∏1 =16
We can calculate the profit price discrimination for market two
TR2 = 10*4-(4)2 =40-8
TR2 =32
TC 2=2Q2 =2*4= 8
∏2 = TR2-TC2 = 32-8
∏2 =24
From this we can conclude the greater profit results from price discrimination is market two