Question

Question 1: Frantic Francine runs one of the most popular restaurants in Manhattan. Francine knows on average how many customers to expect on a given day, but some days just end up busier and some days slower. To prepare for the lunch rush, Francine schedules enough cooks (L) to meet the expected number of customers. Francine’s production function is given by q = K 0.5 L 0.5, where K represents kitchen equipment. Cooks and kitchen equipment are imperfect substitutes (Francine could get by with microwaves to a point). Suppose the kitchen equipment is rented at a price of r = 1 and cooks are paid a wage of w = 1. Francine currently has 16 units of kitchen equipment, and this is fixed in the short run, but Francine has many cooks “on call” in the case of an emergency (they live above the restaurant in a cramped apartment).

(a) Suppose Francine expects 16 customers to arrive. How many cooks should she schedule? What does she expect her total costs to be?

(b) Suppose instead that 20 customers arrive. How many cooks must Francine call in at the last minute? What are her actual total costs?

(c) Suppose Francine knew that 20 customers were going to arrive and could have adjusted the amount of kitchen equipment ahead of time. What would her total costs have been?

(d) Explain why you should have known in advance that the total costs in (c) could be no higher than in (b)?

Answer 1

Solution:

q = K^0.5*L^0.5, q is the expected number of customers, K is the kitchen equipment, L is the number of cooks

On squaring on both sides, q² = K*L ... (1)

Wage rate, w = $1 and rental rate, r = $1

Total cost, TC = w*L + r*K

TC = L + K

K = 16 units of kitchen equipment (in short run)

(a) With q = 16, K = 16

(16)² = 16*L

L = 16

Thus, Francine should schedule 16 cooks. And total cost = 16 + 16 = $32

(b) With q = 20, that is additional 4 customers. (K is still 16 units)

(20)² = 16*L

L = 400/16 = 25

Thus, Francine should have 25 cooks, so call in (25-16=) 9 more cooks in the last minute. Actual total cost = 25 + 16 = $41

(c) Had Francine known that 20 customers would arrive, and accordingly set the kitchen equipment, so K is not fixed but variable now. So, the cost minimization will occur where Marginal rate of technical substitution, MRTSlk = input-price ratio

Input price ratio = w/r = 1/1= 1

MRTSlk = $\frac{\partial q/\partial L}{\partial q/\partial K}$ = (0.5K^0.5L^-0.5)/(0.5L^0.5K^-0.5) = K/L

So, cost minimization occurs where K/L = 1, or K = L

Then, using the production function, and that q = 20, we have the following

q = L^0.5K^0.5

20 = K^0.5K^0.5 (since, L = K)

20 = K

So, L = K = 20

Total cost in this case = 20 + 20 = $40 (note that this cost is indeed lower than the cost of $41, as obtained in the previous part)

(d) Since, in (c), Francine can adjust the amount of kitchen equipment to be employed, she could adjust its quantity beforehand more efficiently and reach a lower cost. In case of (b), since kitchen equipment is fixed, to sustain any more customers, only cooks could be hired. With fixed kitchen equipment used by more and more number of cooks, the marginal productivity would have decreased and thus, resulted in a higher cost. In (c), proper usage of inputs is being undertaken, thus achieving a higher efficiency.