Q = 400 - 20p
TC = 10 + 5q + q2
Calculate the following:
Profit max price
Profit max quantity
TR, TC, Profit, and the elasticity at profit max q and p.
To calculate the profit-maximizing price and quantity, we need to find the point where the difference between Total Revenue (TR) and Total Cost (TC) is highest. This occurs when the derivative of profit with respect to quantity is equal to zero.
Let's start by calculating the Total Revenue (TR) and Total Cost (TC) functions based on the given equations.
TR = p * q (Equation 1)
TC = 10 + 5q + q^2 (Equation 2)
To calculate the profit-maximizing price, we need to differentiate the TR equation (Equation 1) with respect to quantity (q) to find the marginal revenue (MR).
MR = dTR/dq
Since TR = p * q, we can rewrite Equation 1 as:
TR = q * p
To differentiate, we treat p as a constant:
d(TR)/dq = p
Now, let's differentiate the TC equation (Equation 2) with respect to quantity (q) to find the marginal cost (MC).
MC = dTC/dq
Taking the derivative of Equation 2, we get:
d(TC)/dq = 5 + 2q
Now, to find the quantity at which profit is maximized, we equate MR (p) and MC (5 + 2q) and solve for q:
p = 5 + 2q
Substituting the given equation Q = 400 - 20p, we can solve for p:
p = 5 + 2q
20p = 100 + 40q
400 - 20p = 300 - 40q
400 - (400 - 20p) = 300 - 40q
20p = 300 - 40q
20p + 40q = 300
20(400 - 20p) + 40q = 300
8000 - 400p + 40q = 300
400p - 40q = 7700
Now we have two equations:
20p + 40q = 300 (Equation A)
400p - 40q = 7700 (Equation B)
Solving these two equations simultaneously, we can find the values of p and q:
From Equation A:
20p = 300 - 40q
p = (300 - 40q)/20
p = 15 - 2q
Substituting p = 15 - 2q into Equation B:
400p - 40q = 7700
400(15 - 2q) - 40q = 7700
6000 - 800q - 40q = 7700
-840q = 1700
q = -1700 / (-840)
q = 2.0238 (rounded to 4 decimal places)
Now that we have the value of q, we can substitute it back into Equation A to find p:
20p + 40q = 300
20p + 40(2.0238) = 300
20p + 80.952 = 300
20p = 300 - 80.952
20p = 219.048
p = 219.048 / 20
p = 10.9524 (rounded to 4 decimal places)
Therefore, the profit-maximizing price is approximately $10.9524 and the profit-maximizing quantity is approximately 2.0238 units.
To calculate the TR, TC, and profit at the profit-maximizing quantity (q) and price (p), substitute these values into the respective equations:
TR = p * q
TR = 10.9524 * 2.0238
TC = 10 + 5q + q^2
TC = 10 + 5(2.0238) + (2.0238)^2
Profit = TR - TC
Lastly, to calculate the elasticity at the profit-maximizing quantity (q) and price (p), we need to use the equation:
Elasticity = (dq/dp) * (p/q)
To find dq/dp, we differentiate the demand equation: Q = 400 - 20p
dq/dp = -20
Now we substitute the values into the elasticity equation:
Elasticity = (-20) * (10.9524 / 2.0238)
Solving these equations will give you the values for TR, TC, Profit, and the elasticity at the profit-maximizing q and p.
To find the profit-maximizing price and quantity, we'll need to determine the values of q and p that maximize profit.
1. Profit Max Price (p):
To find the profit-maximizing price, we need to differentiate the profit function with respect to p and set it equal to zero:
Q = 400 - 20p
Solving for p:
400 - 20p = 0
-20p = -400
p = 20
Therefore, the profit-maximizing price (p) is 20.
2. Profit Max Quantity (q):
To find the profit-maximizing quantity, substitute the value of p into the demand function:
Q = 400 - 20p
Q = 400 - 20(20)
Q = 400 - 400
Q = 0
Therefore, the profit-maximizing quantity (q) is 0.
3. Total Revenue (TR), Total Cost (TC), Profit, and Elasticity at Profit Max q and p:
To calculate TR and TC, substitute the values of q and p into the respective functions:
Total Revenue (TR):
TR = Q * p
TR = 0 * 20
TR = 0
Total Cost (TC):
TC = 10 + 5q + q^2
TC = 10 + 5(0) + 0^2
TC = 10
Profit (π):
Profit = TR - TC
Profit = 0 - 10
Profit = -10
Elasticity of Demand (ε):
The elasticity of demand at the profit-maximizing quantity can be calculated using the following formula:
ε = (dq / dp) * (p / q)
Since q is 0, the elasticity becomes undefined (division by zero is not possible).