The Lumins Lamp Company, a producer of old-style oil lamps, estimated the following demand function for its product:
Q = 120,000 − 10,000P where Q is the quantity demanded per year and P is the price per lamp. The firm’s fixed costs are $12,000 and variable costs are $1.50 per lamp.
a. Write an equation for the total revenue (TR) function in terms of Q.
b. Specify the marginal revenue function.
c. Write an equation for the total cost (TC) function in terms of Q.
d. Specify the marginal cost function.

a. Total revenue function in terms of Q:
We rewrite the demand curve in terms of Q.
P=(120,000-Q)/10,000;
P=12-Q/10,000;
Then, we multiply by Q to find the total revenue (TR).
TR=P.Q;
TR=(12-Q/10,000)xQ;
TR=12Q-Q^2/10,000;

b. The marginal revenue function:
MR=∂TR/∂Q=12-2Q/10,000;
MR=12-Q/5,000;

c. The total cost function (TC) in terms of Q.
Fixed Costs (FC)=$12,000; Variable Costs (VC)=$1.50.
TC=FC+VC
TC=12,000+1.5Q

d. The marginal cost function:
MC=∂TC/∂Q=1.5

e. Total profits (π) equation in terms of Q: π=TR-TC; π=12Q-Q^2/10,000-(12,000+1.5Q); π=10.5Q-Q^2/10,000-12,000; -To maximize profits, the company would produce and sell enough lamps such that MC=MR and charge the corresponding uniform price. Setting,
MR=MC yields
12-Q/5,000=1.5
(60,000-Q^*)/5,000=1.5
60,000-Q^*=7,500
Q^*=52,500;
At the level of 52,500 lamps, the company would maximize profits.
-Substituting Q* back into the demand equation, we solve for the uniform price P*.
P^*=12-Q^*/10,000
P^*=12-52,500/10,000
P^*=$6.75/unit
The price of output to maximize profits would be $6.75.
- Profit: π^*=P^*.Q^*-(12,000+1.5Q^*) π^*=6.75x52,500-(12,000+1.5x52,500) π^*=$263,625 At the total profits maximized level, the profit would be about $263,625.

f. We are going to check if the marginal revenue is equal to the marginal cost, and to 1.5.
MR=12-Q^*/5,000;