Question

Geoshapes, the magnetic puzzle company can produce any quantity q of magnetic puzzles with the following cost structure:

TC = 450,000 + 20q, where q measures units of output.

The industry demand for magnetic puzzles is

Q = 100,000 – 500P.

Suppose the profit-maximizing quantity for a single firm is split between two puzzle companies, Geoshapes and SnapMag (so market quantity Q = q+q = 2q where q is the quantity of each firm), both of which have the cost structure of TC = 450,000 + 20q, where q measures units of output.

1. In this case, the profit maximizing quantity for each firm is ___________.
2. The average total cost for each firm is $_________.
3. The measurement of economies of scale at each firm's profit maximizing quantity is __________.
4. Geoshapes is experiencing _____________ of scale at the profit maximizing quantity.

Answer 1

The industry demand for magnetic puzzles is;

Q = 100,000 – 500P
500P = 100000 - Q
P = 200 - Q/500

Market quantity;

Q = q+q = 2q

Cost structure of each firm;

TC = 450,000 + 20q

So, total cost in the market is;

TC = 450,000 + 20q + 450,000 + 20q
TC = 900000 + 40Q

A. The profit maximising condition for the market will be where marginal revenue is equal to marginal cost;

MR = MC

Total revenue;

TR = P*Q
= (200 - Q/500)Q
TR = 200Q - Q²/500

MR = d/dQ(200Q - Q²/500)
MR = 200 - Q/250

MC = d/dQ(900000 + 40Q)
MC = 40

The profit maximising condition;

MR = MC

200 - Q/250 = 40
200 - 40 = Q/250
160*250 = Q
Q = 40000

So, profit maximizing quantity for each firm will be;

q = Q/2
q = 40000/2
q = 20000

B. Given the total cost for each firm;

TC = 450,000 + 20q

Average total cost for each firm is calculated as total cost per unit of outptut;

ATC = TC/q
= (450,000 + 20q) / q
ATC = 450,000/q + 20

q = 20000

ATC = 450000/20000 + 20
= 22.5 + 20
ATC = 42.5

C. Economies of scale exist when increase in output is expected to result in a decrease in unit cost while keeping the input costs constant. It is calculated as cost of elasticity;

E = dC/dq * q/C

= d/dq (450,000 + 20q) * q/C

E = 20q / C

q = 20000

TC = 450000 + 20*20000
C = 850000

E = 20(20000) / 850000
= 400000/850000
E = 0.47

D. Geoshapes is experiencing economies of scale at the profit maximizing quantity. This is because, as calculated above the cost of elasticity is less than 1 which means percentage change in costs is lower than the percentage change in output. In other words, it shows that at cost elasticity of less than 1, costs increase by a lower percentage than output, therefore Geoshapes is facing economies of scale.