Question

The residents of cities A, B, C, D and E consume wi-fi routers, with consumption in each city is 150 routers (see the map below). The firm that produces routers must decide how to set-up production. It could set up five factories, dispersed across each city, with each factory producing 150 routers and supplying to its own local city market. In this case, the firm incurs no cost for shipping output. Alternatively, the firm could locate its factory at centrally located city C, and supply routers to the entire region. The single factory in city C must then produce 750 routers, 600 of which are shipped to the cities A, B, D and E for a shipping cost of $6 per router.

A E
C
B D
(a) Suppose the average cost of producing a router is AC (Q) = 1500/Q, where Q is the number of routers produced in a factory. Calculate AC with Q = 150 and Q = 750, respectively. Note and explain how this production process exhibit economies of scale.
(b) Based on the AC function from part (a), find the optimal arrangement of production for the firm (one central factory or five dispersed factories). The optimal arrangement minimizes total cost for the firm, where total cost is the sum of production cost and shipping cost. Clearly write down all your calculations.
(c) Now suppose the average cost of producing a router is AC = 14000/(Q+1250). Now, repeat the calculation of AC with Q = 150 and Q = 750.
(d) Based on the AC function from part (c), now repeat your calculations to find the cost-minimizing arrangement of production in the case. (e) Explain intuitively the difference is results between responses to part (b) and (d).
(f) Suppose now production costs are those given in part (a) but let shipping cost per router be given by t (in the preceding discussion, we had t = 6, now we assume we don’t know the cost of shipping). What value of t would make the two arrangements for production (centralized versus separate factories) equivalent in terms of cost? i.e. what value of t would make the firm indifferent between a centralized versus a dispersed set-up?

Answer 1

a. The production process shows that the more the quantity produced, the less the average cost of production. It proves that there are advantages arising from economies of scale.

AC with Q = 150 = $10 ($1,500/150) and

AC with Q = 750 = $2 ($1,500/750)

b. The optimal arrangement is (centralized production) to produce the 750 routers at city C and ship to the 4 other cities.

c. AC with Q = 150 = $10 (14000/(150+1250) and

AC with Q = 750 = $7 (14000/(750+1250)

d. The cost-minimizing arrangement of production in this case is decentralized production.

e. The average cost of producing 150 units at the various cities has remained unchanged while the average cost of producing the 750 units at city C has increased from $2 to $7.

f. Suppose now production costs are those given in part (a) but let shipping cost per router be given by t (in the preceding discussion, we had t = 6, now we assume we don’t know the cost of shipping).

The value of t that would make the two arrangements for production (centralized versus separate factories) equivalent in terms of cost is:

t = $10 per router

Therefore, centralized production cost will be equal to $7,500 ($1,500 + ($10 * 600), and decentralized production cost will remain at $7,500 (750 * $10).

Step-by-step explanation:

a) Data and Calculations:

Cities with consumers of wi-fi routers = A, B, C, D and E

Demand for routers by each city = 150

Total number of routers required = 750 (150 * 5)

b) Suppose the average cost of producing a router is AC (Q) = 1500/Q, where Q is the number of routers produced in a factory:

Therefore AC with Q = 150 = $10 ($1,500/150) and

AC with Q = 750 = $2 ($1,500/750)

Cost of Production of routers in city C:

cost of producing 750 routers at $2 per router = $1,500

Shipping cost of 600 routers to 4 cities at $6 per router = $3,600

Total cost of producing at city C = $5,100 ($1,500 + $3,600)

Total cost of producing 750 routers at 5 cities = $7,500 ($1,500/150 * 750)

c) Suppose the average cost of producing a router is AC = 14000/(Q+1250):

Therefore, AC with Q = 150 = $10 (14000/(150+1250) and

AC with Q = 750 = $7 (14000/(750+1250)

Cost of Production of routers in city C:

cost of producing 750 routers at $7 per router = $5,250

Shipping cost of 600 routers to 4 cities at $6 per router = $3,600

Total cost of producing at city C = $8,850 ($5,250 + $3,600)

Total cost of producing 750 routers at 5 cities = $7,500 ($1,500/150 * 750)

d) $7,500 = $1,500 + tQ

where Q = 600 (150 * 4)

Therefore, $7,500 - $1,500 = t600

simplifying

t600 = $6,000

t = $6,000/600 = $10