Answer to Question #185843 in Macroeconomics for Solen Birhan

Question

Question1. Given bellow are the cost and revenue functions of a perfect competitive firm:

TC= 20+10Q    TR= 40Q-5Q²

Calculate:

a). Profit maximizing level of output

b) Total profit

Question 2. Determine shutdown price and output for the following total cost and total revenue functions:

TR=12Q   TC=1/3Q³-9/2Q²+20Q+50

a). Find MR

b). MC Function

c). The quantity produced at shutdown point

d). The minimum price a firm will sell

Answer 1

Solution:

Question 1:

a). Profit maximizing level of output:

At profit-maximizing level of output: MR = MC

Derive MR:

MR = Derivative of TR with respect to the quantity

MR = ​$\frac{\partial TR_{} }{\partial q_{} }$ = 40 – 10Q

Derive MC:

MC = Derivative of TC with respect to the quantity

MC = $\frac{\partial TC_{} }{\partial q_{} }$ ​ = 10

MR = MC

40 – 10Q = 10

40 – 10 = 10Q

30 = 10Q

Q = 3

Profit maximizing level of output = 3

 

b). Total profit

Total profit = TR – TC

TP = (20 + 10Q) – (40Q – 5Q²)

TP = (20 + 10(3)) – (40(3) – 5(3²))

TP = (120 – 45) – (20 + 30)

TP = 75 – 50

TP = 25

Total profit = 25

 

Question 2:

a). Find MR:

MR = Derivative of TR to quantity

MR = $\frac{\partial TR_{} }{\partial q_{} }$ ​= 12

MR = 12

 

b). MC function:

MC = Derivative of TC to quantity

MC =$\frac{\partial TC_{} }{\partial q_{} }$ ​= Q² – 9Q + 20

MC function = Q² – 9Q + 20

 

c). Quantity produced at shut point:

The shutdown price is equal to the minimum AVC, a point where MC = AVC

Derive AVC:

AVC = $\frac{VC }{Q }$ = $\frac{\frac{1 }{3 }Q^{3} - \frac{9 }{2 }Q^{2} + 20Q}{Q }$

AVC = $\frac{1}{3}Q^{2} - \frac{9}{2}Q + 20$

MC = Q² – 9Q + 20

MC = AVC

Q² – 9Q + 20 = $\frac{1}{3}Q^{2} - \frac{9}{2}Q + 20$

Q² – 9Q = $\frac{1}{3}Q^{2} - \frac{9}{2}Q$

Q² – $\frac{1}{3}Q^{2}$ – 9Q + $\frac{9}{2}Q$ = 0

$\frac{2}{3}Q^{2}$ – 13.5Q = 0

2Q² – 40.5Q = 0

2Q (Q – 20.25) = 0

Q = 20.25

The quantity produced at shutdown point = 20.25

 

d). The minimum price the firm will sell:

P = MC

P = Q² – 9Q + 20

P = 20.25² – 9(20.25) + 20

P = 410.06 – 182.25 + 20

P = 410.06 + 20 – 182.25

P = 247.81

The firm will sell at a minimum price of 247.81