Question

6. Lenny's Car Wash uses two variable inputs: workers and materials (cloth, soap, etc). Lenny's production function, in units of cars washed per hour, is shown below. The cost of a worker in wages and benefits is $20/hr, and cost of materials is $10 per materials. In addition to these variable costs, Lenny pays fixed costs of $50 per hour on rent and machinery Lenny's Production Function # Materials 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 4 4 4 2 0 1 2 3 4 5 5 5 30 1 2 3 4 5 5 5 3 4 0 1 2 3 4 5 6 6 #5 0 1 2 3 4 5 6 6 6 0 1 2 3 4 5 6 6 7 0 1 2 3 4 5 6 7 Workers (a) What's the most efficient way for Lenny to clean 3 cars per hour? How about 5 cars per hour? Worker (b) Fill in Lenny's cost function below. # Cars Material Fixed Cost 0 20 Material Cost Fixed Cost Total MC ATC AVC Total Cost 50 0 $0 (c) Plot the MC, ATC, and AVC curves in Excel.

Answer 1

6. (a) For the production level Q= 3, we have the input combinations as . The most efficient combination would be what minimizes the W and M. As can be seen, combination requires least amount of inputs to produce Q=3, since $(1,3) \preceq (W,M)$ . Hence, the efficient way to produce Q=3 would be .

For the production level Q=5, we have the input combinations as . The efficience combination in this case would be since $(2,5) \preceq (W,M)$ . Hence, the efficient way to produce Q=5 would be .

(b) The table would be as below.

Q	W	M	FC	TC	MC	ATC	AVC
0	0	0	50	50	-	-	-
1	1*20=20	1*10=10	50	80	30	80	30
2	1*20=20	2*10=20	50	90	10	45	20
3	1*20=20	3*10=30	50	100	10	33.33	16.67
4	1*20=20	4*10=40	50	110	10	27.5	15
5	2*20=40	5*10=50	50	140	30	28	18
6	4*20=80	6*10=60	50	190	50	31.66	23.33
7	7*20=140	7*10=70	50	260	70	37.14	30

We have $MC = \frac{\Delta TC}{\Delta Q}$ , $ATC = \frac{TC}{Q}$ and $AVC = \frac{VC}{Q} = \frac{(20*W) + (10*M)}{Q}$ . The W and M here are the combination of minimum input required to produce the given output.

(c) The plot would be as below.

C18 Τ Α 1_(Cars) Ι MC - Β D | E | F | G | H Ι Ι Ι Κ Ι ι Μ Ν Ι ο Ι Ρ Ι ! 30 30 WNP = C ATCAVC 80 45 10 33.33 10 27.5 30 28 50 31.66 70 37.14 20 16.67 15 18 BOUWN- 23.33 30 13 14 15 17 18 Cost MC 20 21 ATC AVC 22 23 27 35 36 2 לכ K + Sheet1