Production Function
Important: Formulae and relations:
TP=L x AP L=∑ MP L
d (TP)
MP L =
dL
TP
APL =
L
Where APL, MPL are average productivity or marginal productivity of any factor input say labour.
10. Consider the following production function.
Where:
Y = total real output
L = labour input (the total number of person-hours worked in a year)
K = capital input (the real value of all machinery, equipment, and buildings)
A = total factor productivity
α and β are parameters determined by available technology.
Further, A, α and β >0 and α + β =1
Show that marginal and average productivity of labour and capital are functions of factor ratio.
11. Fill in the blanks
Quantity of Total Marginal Product Average Product
Variable Input Output of Variable Input of Variable Input
0 0
1 225
2 600 375 300
3 900 300
4 240 285
5 1365 273
6 1350 -15
Solution:
Quantity of Total Marginal Product Average Product
Variable Input (L) Output(Q) of Variable Input of Variable Input
(dQ/dL) (Q/L)
0 0 ----- -----
1 225 225 225
2 600 375 300
3 900 300 300
4 1140 240 285
5 1365 225 273
6 1350 -15 225
12. a. Question: Marty’s Frozen Yogurt is a small shop that sells cups of frozen yogurt in a university
town. Marty owns three frozen-yogurt machines. His other inputs are refrigerators, frozen-yogurt
mix, cups, sprinkle toppings, and, of course, workers. He estimates that his daily production function
when he varies the number of workers employed (and at the same time, of course, yogurt mix, cups,
and so on) is as shown in the accompanying table.
Quantity of labor (workers) Quantity of frozen yogurt (cups)
0 0
1 110
2 200
3 270
4 300
5 320
6 330
What are the fixed inputs and variable inputs in the production of cups of frozen yogurt?
b. Draw the total product curve. Put the quantity of labor on the horizontal axis and the quantity of
frozen yogurt on the vertical axis.
c. What is the marginal product of the first worker? The second worker? The third worker? Why
does marginal product decline as the number of workers increases?
13. Determine the return to scale of the following production function
Q=aL+ bK , where L and k are the inputs.
Determine the return to scale.
14. Suppose that output q is a function of a single input, labour (L). Describe the returns to scale
associated with each of the following production functions.
a. q=L/2
b. q = 2L
c. q = log(L)
d. q= f(L)=L
Cost of Production
Important: Formulae and relations:
TC=TVC +TFC =AC X Q
TFC=AFC X Q
TVC=AVC X Q ¿ ∑ MC
TC
AC= = AVC + AFC
Q
d ( TC ) d (TVC )
MC= =
dQ dQ
MC= Minimum AVC
MC =Minimum AC
but Minimum AVC ≠ Minimum AC
Here, Q is output.
15.
Note: Average cost (AC) is also called average total cost (ATC)
16.
17. The average cost of a firm is given as follows:
C=200 + 100Q-1 - 20Q + 2Q2 ,
where Q is output.
(a) Determine Total Cost, Total Fixed Cost, Total Variable Cost Function.
(b) Determine Average Fixed Cost, AVC and MC function
(c) Determine level of output when AVC is minimum and also show that
MC=Minimum AVC.
18. A firm has the following total cost function
C=Q3 -7Q2 + 2Q + 16, where Q is output produced.
(a) Determine AFC, AVC, AC and MC
(b) Show that when AVC is minimum, AVC=MC.
(c) Derive AC and check that when Q=4, the average cost is minimum.
Revenue, Break-Even and Profit
Important: Formulae and relations:
Profit(∏ ) =Total Revenue – Total Cost
At breakeven point, TR=TC
At maximum profit,
MR= MC and MC must be rising at the point of equilibrium
Or,
19. A monopoly firm has the following average cost function
C = 150 Q-1 + 5
The demand curve facing the firm in the market is
AR= 150 – 0.5Q, where Q is output.
Determine the profit maximizing output and price of the product. Justify your
answer by checking the sufficient condition for maximum profit.
20. In a perfectly competitive market, the market price of a commodity is Rs. 3 per
1 3
unit of output (Q). The total cost function of a firm is C= Q -4Q2 +10Q +10
3
Find (a) Profit maximizing output
(b) Maximum profit.
21. Given the demand function and supply function for a market as
P = 80 - 3x and P = 2x + 20 respectively, determine the market equilibrium price and output.
Here X is output and P is price.
22. Find the level of output yielding maximum profit by MC and MR approach. Also find the break
even output.
Output 1 2 3 4 5
AR 10 9 8 7 6
TC 10 11 14 18 25
