Question

5. A firm needs to make a commitment to one out of two production technologles (and buy only one set of equipment that costs tens of thousands of dollars). The total seasonal total cost of production are glven by C(a)- 3600 + 654 +38q and marginal cost MCr-65+72q for technology 1 If the firm Installs technology 2, the costs will be C(e)-900+900+ with MC,-900+24 What is the minimum efficlent scale of each technology? (Hint you may need to use the quadratic formula at some point) a) b) Which technology should the firm choose (purely from a cost standpoint) if its output is seasonalt the firm expects to sell 30 units in the summer and 10 units in the winter each year? The firm can invest only in one.

Answer 1

Consider the given problem here the cost function of “1^st” technology is given by.

=> “C1 = 3,600 + 65*q + 36*q^2”, => “AC1 = 3,600/q + 65 + 36*q” and the “MC1 = 65 + 72*q”. So, at the efficient scale “AC” must be equal to “MC”.

=> 3,600/q + 65 + 36*q = 65 + 72*q, => 3,600/q = 36*q, => q^2 = 100, => q=10. So, the efficient scale for “technology 1” is “q=10”.

Now, the cost function of “2^nd” technology is given by.

=> “C1 = 900 + 900*q + q^2”, => “AC1 = 900/q + 900 + q” and the “MC2 = 900 + 2*q”. So, at the efficient scale the following relation must hold.

=> 900/q + 900 + q = 900 + 2*q, => 900/q = q, => q^2 = 900, => q = 30. So, the efficient scale for “technology 2” is “q=30”.

b).

Now, the firm want to sell “30 units” in summer and “10 units” in winter each year, => total “40 units”, => the cost under “technology 1” is given by.

=> C1 = 36,00 + 65*q + 36*q^2 = 3600 + 65*40 + 36*40^2 = $63,800.

Now, the cost under “technology 2” is given by.

=> C2 = 900 + 900*q + q^2 = 900 + 900*40 + 40^2 = $38,500 < $63,800.

So, here cost under “technology 2” is less compare to “technology 1”. So, firm should invest only in one.