Question

2. Propose and sketch a demand function for a product that currently sells for $500 with quantity sold at 25. Write both regular and inverse demand functions. 3. Representative firm in an industry has total cost function TC(a) -1849+and marginal cost function MC-2q a) What is the fixed cost function for the firm? What is the variable cost function? b) Derive average total, average fixed and average variable cost functions. c) Find the minimum efficient scale (-output that corresponds to the minimum of ATC) of the representative firm in this industry. What is the value of average cost at this output level?

Answer 1

Problem 2
Let the function be represented by
Q=a-bP
Now plug in the given point i.e. P=$500 and Q=25
25=a-b*500
Here we have two constants. We can have a number of options available with it.
Let b=1
So, 25=a-1*500
a=525
Regular demand function is
Q=525-P
On rearranging we get
P=525-Q
Inverse demand function is
P=525-Q

Problem 3
a)
Fixed cost function does not depend upon q (Output) and is fixed irrespective of q
Variable cost function is a function of q.
Fixed Cost=TFC=1849
Variable Cost=TVC=q²

b)
Average fixed Cost=AFC=TFC/q=1849/q
Average variable cost =AVC=TVC/q=q²/q=q
Average total cost =ATC=(1849/q)+q
c)
Average total cost =ATC=(1849/q)+q
We need to find the value of q, where ATC is minimum. It is possible, when d(ATC)/dq=0
ATC=(1849/q)+q
Differentiate with respect to q,
d(ATC)/dq=-1849/q²+1
Put d(ATC)/q=0
-1849/q²+1=0
1849/q²=1
q=43
Minimum Efficient Scale (MES) is at q=43

Average total cost =ATC=(1849/q)+q
Put q=43
ATC=(1849/43)+43=86
d)

q	MC=2q	ATC
10	20	194.90
20	40	112.45
30	60	91.63
43	86	86.00
50	100	86.98
60	120	90.82

In the graph
Economies of scale is observed when q<MES
Diseconomies of scale is observed when q>MES