# Determine marginal revenue given demand curve and marginal cost

a)Determine marginal revenue curve if firm can only charge 1 price,List demand curve,marginal revenue curve,,marginal cost,average total cost equations.

b)What is optimal price and quantity?

c)If firm fixed cost are 16 dollars, what are the profits?

P=20-q

MC=12

Attempt:

I tried

mc=c/p

12*20-q=x

to find P*X=MR

b)

12=240-12q

q=-228/12

c)

MC=6

6=240-12q/20-q

q=24

The numerical answer is wrong so what mistake did I make? What's the correct equation to solve for test equation?

I assume that the firm is a monopolist.

We know the inverse demand function, the fixed costs and the marginal costs:

• $$P(q) = 20 - q$$
• $$MC(q) = 12$$
• $$F = 16$$

As the marginal costs are constant, we can compute the total costs function as:

• $$TC(q) = MC\cdot q + F = 12 q + 16$$

The average total costs is obtained by dividing by $$q$$:

• $$AC(q) = \frac{TC(q)}{q} = 12 + \frac{16}{q}$$

The demand curve is obtained by inverting the inverse demand function:

• $$q(P) = 20 - P$$

Total revenue is obtained by multiplying together price and quantity:

• $$R(q) = P(q)\cdot q = (20-q)\cdot q = 20 q - q^2$$

Marginal revenue is obtained by taking the derivative of revenue with respect to $$q$$

• $$MR(q) = \frac{\partial R(q)}{\partial q} = 20 - 2q$$.

If the firm maximizes profits, then the marginal revenue equals the marginal costs so:

• $$12 = 20 - 2q \to q = 4$$

The optimal price can be found by substituting $$q = 4$$ into the inverse demand function:

• $$P = 20 - 4 = 16$$

Profits are determined as total revenue minus total costs:

• $$\pi = R - TC = P\cdot q - TC = 16 \cdot 4 - (12 \cdot 4 + 16) = 0$$