Consider the Linear demand function Q = 20 - 0.5P
c- Using calculus, find the level of output, Qrmax, where total revenue reaches its maximum value. What price Prmx maximizes total revenue? What is the value of TR at this maximum point.
d- Write the equation for marginal revenue. Using MR, verify that Qrmax derived in part c maximizes total revenue.
e- Calculate the point elasticity of demand at Prmax. Does E have the expected value? Explain briefly.
'cultural appropriation'
ggfdgdg
c- To find the level of output, Qrmax, where total revenue reaches its maximum value, we need to use calculus.
Total revenue (TR) is calculated by multiplying the quantity demanded (Q) by the price (P):
TR = Q * P
In this case, we have the demand function: Q = 20 - 0.5P
To determine Qrmax, we need to find the derivative of TR with respect to Q (dTR/dQ) and set it equal to zero to find the critical point.
First, we express TR in terms of Q:
TR = (20 - 0.5P) * P
Next, we differentiate TR with respect to Q (dTR/dQ):
dTR/dQ = (20 - P) + (-0.5P)
= 20 - 1.5P
Setting dTR/dQ equal to zero:
20 - 1.5P = 0
1.5P = 20
P = 20/1.5
P = 40/3 or approximately 13.33
Therefore, the price (Prmx) that maximizes total revenue is approximately 13.33.
To find the corresponding quantity demanded (Qrmax), we substitute the price into the demand function:
Q = 20 - 0.5P
Q = 20 - 0.5(40/3)
Q = 20 - 20/3
Q = 60/3 - 20/3
Q = 40/3 or approximately 13.33
So, the level of output, Qrmax, that maximizes total revenue is approximately 13.33.
To find the value of TR at this maximum point, we substitute Prmx and Qrmax into the total revenue equation:
TR = (20 - 0.5P) * P
TR = (20 - 0.5(40/3)) * (40/3)
TR = (20 - 20/3) * 40/3
TR = (60/3 - 20/3) * 40/3
TR = 40/3 * 40/3
TR = 1600/9 or approximately 177.78
Therefore, the value of TR at the maximum point is approximately 177.78.
d- To find the equation for marginal revenue (MR), we need to differentiate the total revenue (TR) equation with respect to Q (dTR/dQ):
TR = (20 - 0.5P) * P
Differentiating TR with respect to Q (dTR/dQ):
dTR/dQ = 20 - P - 0.5dP/dQ
Since dP/dQ is the derivative of the demand function, which is -0.5, we substitute it into the equation:
dTR/dQ = 20 - P - 0.5(-0.5)
= 20 - P + 0.25
= 20.25 - P
Using MR, we can verify that Qrmax derived in part c maximizes total revenue. At the critical point (Qrmax, Prmx), we can calculate MR by substituting Qrmax into the MR equation:
MR = 20.25 - P
= 20.25 - 13.33
= 6.92
Since MR at the critical point is positive (6.92), this confirms that Qrmax derived in part c maximizes total revenue.
e- To calculate the point elasticity of demand at Prmax, we need to use the elasticity formula:
E = (dQ/dP) * (P/Q)
Since Q = 20 - 0.5P, we differentiate it with respect to P to find dQ/dP:
dQ/dP = -0.5
Substituting Prmax into the formula, P = 13.33, and Qrmax into the demand function, Q = 13.33, we can calculate the point elasticity of demand:
E = (dQ/dP) * (P/Q)
= (-0.5) * (13.33/13.33)
= -0.5
The expected value for E would be negative, indicating that demand is elastic. In this case, E equals -0.5, which is indeed negative. Therefore, the point elasticity of demand at Prmax has the expected value.