The Blue Dragon Restaurant is a new Chinese Restaurant in town. As the only Chinese restaurant in the area, it faces the following daily demand curve:
Q = 500 – 25 P
Where Q is the number of meals it serves per day and P is the average price of its meals.
The cost function of the restaurant has been estimated as follows:
TC = 200 + 8 Q + .02 Q^2
a. Determine the profit-maximizing price of each meal assuming The Blue Dragon is behaving as a monopoly.
b. Determine the profit of the Restaurant.
c. If the company were to produce as a perfectly competitive firm, how much would it produce?
d. What price should it charge as a competitive firm?
e. Would it still make a profit if it behaved like a competitive firm?
As a result of the success of the Blue Dragon other Chinese restaurants start appearing in the area. As the Blue Dragon’s customers gradually start trying other (new) Chinese restaurants, its demand curve gets flatter (more elastic) and shifts to the left. In reaction, The Blue Dragon lowers its price and adjust its output to the point that, eventually, its (economic) profit disappears; It becomes equal to zero. At that point the slope of its demand curve becomes -0.03 .
f. Determine the new (equilibrium) average price The Blue Dragon charges for its meals.
g. Write the equation for this new (zero profit) demand curve.
Hint:
MC = 8 + .04 Q
Slope of ATC = -200/Q^2 + .02
On Wed., Dec 13, Heather asked a very similar question. See my post.
To determine the profit-maximizing price of each meal for The Blue Dragon as a monopoly, we need to find the point where marginal revenue (MR) equals marginal cost (MC).
a. To find the MR, we start with the demand equation Q = 500 - 25P. We can rearrange it to solve for P in terms of Q: P = (500 - Q) / 25.
To get MR, we differentiate the revenue equation wrt Q: MR = d(500Q - 25Q^2) / dQ = 500 - 50Q.
Now, we need to find the MC by differentiating the total cost equation wrt Q: MC = d(200 + 8Q + 0.02Q^2) / dQ = 8 + 0.04Q.
To find the profit-maximizing quantity, set MR = MC: 500 - 50Q = 8 + 0.04Q.
Simplifying and rearranging the equation, we have:
50Q + 0.04Q = 500 - 8
50.04Q = 492
Q = 492 / 50.04
Q ≈ 9.82
Now substitute the obtained value of Q into the demand equation to find the price: P = (500 - Q) / 25
P = (500 - 9.82) / 25
P ≈ 19.93
So, the profit-maximizing price of each meal for The Blue Dragon as a monopoly is approximately $19.93.
b. To determine the profit of the restaurant, we need to subtract the total cost from the total revenue.
Total Revenue (TR) can be found by multiplying the price (P) by the quantity (Q): TR = P * Q.
Total Cost (TC) is given as TC = 200 + 8Q + 0.02Q^2.
Profit (π) is calculated as: π = TR - TC.
Substituting the values:
π = (P * Q) - (200 + 8Q + 0.02Q^2).
π = (19.93 * 9.82) - (200 + 8 * 9.82 + 0.02 * 9.82^2)
π ≈ 107.24 - 311.38 ≈ -204.14
The profit of the restaurant is approximately -$204.14.
c. If the company were to produce as a perfectly competitive firm, it would maximize profit by producing where marginal cost (MC) equals the market price. In this case, MC is given as 8 + 0.04Q.
To find the quantity produced by a competitive firm, we set MC equal to the market price, which is the same as the average price (P) since the firm takes the price as given.
8 + 0.04Q = P
Solving for Q:
0.04Q = P - 8
Q = (P - 8) / 0.04
d. To determine the price charged by a competitive firm, we substitute Q into the demand equation Q = 500 - 25P and solve for P:
(P - 8) / 0.04 = 500 - 25P
P - 8 = 0.04(500 - 25P)
P - 8 = 20 - P
2P = 28
P = 14
So, the price charged by the competitive firm would be $14.
e. To determine if the competitive firm would make a profit, we need to evaluate whether the price covers the average total cost (ATC) at the quantity produced. The ATC equation is given as -200/Q^2 + 0.02.
Substituting the competitive quantity (Q = (P - 8) / 0.04) into this equation and solving for ATC:
ATC = -200/((P - 8) / 0.04)^2 + 0.02
Calculating the ATC at P = 14:
ATC = -200/((14 - 8) / 0.04)^2 + 0.02
ATC = -200/(6 / 0.04)^2 + 0.02
ATC ≈ -200/(150)^2 + 0.02
ATC ≈ -200/22500 + 0.02
ATC ≈ -0.0089 ≈ -$0.0089
Since the ATC is negative, it indicates that the competitive firm would make a profit.
f. When the demand curve becomes flatter and the slope of the demand curve is -0.03, we can write the new demand equation as Q = 500 - 0.03P.
To determine the new equilibrium average price charged by The Blue Dragon, we set MR equal to MC:
MR = 500 - 0.06Q (derive from the new demand equation),
MC = 8 + 0.04Q (given in the question).
Setting MR equal to MC and solving for Q:
500 - 0.06Q = 8 + 0.04Q
0.1Q = 492
Q = 492 / 0.1
Q = 4920
Substituting Q into the new demand equation to find P:
4920 = 500 - 0.03P
500 - 4920 / 0.03 = P
P ≈ $10.67
So, the new (equilibrium) average price The Blue Dragon charges for its meals is approximately $10.67.
g. To write the equation for the new (zero profit) demand curve, we need to find the new slope. The formula for slope is slope = -dQ/dP.
Given that the slope of the new demand curve is -0.03, we can write the equation as:
-0.03 = -dQ/dP
dQ/dP = 0.03
To find the equation, we integrate this expression:
dQ = 0.03 dP
∫dQ = ∫0.03 dP
Q = 0.03P + C
By substituting any known point on the curve, we can determine the constant of integration, C.