Production Function
Fun Calculations
Curves
Equally as Fun Calculations
100
Q1: "In a Leontief production function, capital and labor can be substituted for each other but the rate of substation is not constant" (True or False)
False
Keywords:
- C &L CAN be subbed (Leontief or cobb)
- NOT constant rate (Only cobb)
100
A firm has the following prod. Function. Q= 30k+81L. The firm's MPK is 30 (T/F)
True
100
An average curve that is continually downward sloping
AFC
100
Q7:"A firm produces 5,200 units of output. Its total cost is $192,400. Its average fixed cost is $16. What is the firm's total variable cost?"
$109,200
Q: 5200
TC: $192,000
AFC: 16
TVC:?
AFC = (TFC/Q) --> 16 =(x/5200)-->TFC =83200
TC = TFC + TVC --> 192,400= 83,200 + x --> x=$109,200
200
K and L are used in fixed proportions in the Leontief production function (T/F)
True
200
Q3: "A firm's production function is as follows:
Q= 40L(3/4)K(1/4) ....If K=1, L=16, the firm's MPL is__"
15
Q= 40L(3/4)(1)(1/4) -->40L(3/4)
MPL= (3/4)(40)L(3/4 -4/4) = 30L(-1/4)
30(16)(-1/4)= 15
200
This curve does not touch either axes
MC,ATC,AVC,AFC
200
Below is a Cobb Douglas Function: Capital =25, Wage=$20, Price of Output =$5,
Q= 400L1/4K1/2
A) Calculate the marginal product of labor for the equation above
(1) Plug in K
Q= 400L1/4(25)1/2= 200L1/4
(2) Take Derivative of L
MPL= (1/4) 200L(1/4 - 4/4)
MPL= 500L(- 3/4)
300
Given a production function. Q= 4k +3L... K=2, L=3
a) what kind of production function is this. How can you tell?
B) What is Q?
C) Find MPL and MPk
A) linear (bc there is a plus sign)
B) Q= 4(2) +(3) = 17
C) MPL= 3, MPk= 4
300
An economist estimated that the cost function of a single-product firm is:
C(Q) = 100 + 20Q +15Q2 + 10Q3
Based on this, determine:
a) The fixed cost of producing 10 units of out
b) The total cost of producing 10 units of output
a) Fixed cost = 100
b) TC = VC+ FC = $11,700 + $100 = $11,800
VC = 20(10) + 15(10)2+ 10(10)3 = $11,700
300
Inverse S shaped and upward sloping
TVC and TC
300
Below is a Cobb Douglas Function: Capital =25, Wage=$20, Price of Output =$5,
Q= 400L1/4K1/2
Determine optimal amount of labor the firm should use
(1) Find VMPL
VMPL = MPL * P
= 500L(- 3/4) * ($5) = 2500L(- 3/4)
VMPL= 2500L(- 3/4)
(2) Set VMPL = Wage
2500L(-3/4)=20 or 2500/(L3/4)
2500 = 20(L3/4 )
2500/20 = L3/4
(125)4/3 = (L3/4)4/3
L = 625
400
Given a production function Q= (10K0.5)(L.5), K=9 L=16
a) what kind of Production Function is this. How can you tell?
b) What is Q?
C) Find MPL and MPK
A)Cobb Douglas, exponents
b) Q= (10(9)0.5)((16)0.5) --> Q=10(3)(4) --> Q=120
c)
MPL= (0.5)(10)K(0.50)L(0.50 -1.00)= 5K0.5L(-0.5)
MPk=(0.5)(10)K(0.5-1.00)L0.5= 5K-0.5L0.5
400
An economist estimated that the cost function of a single-product firm is:
C(Q) = 100 + 20Q +15Q2 + 10Q3
Based on this, determine:
(d) The average fixed cost of producing 10 units
(F) The average total cost of producing 10 units of output
d) AFC= FC/Q= 100/10= $10
f) ATC = TC/Q = 11,800/10 = $1180
400
The curve passes through the minimum points of the AVC and ATC curves
MC
400
Q4: "A manager hire labor and rents capital equipment in a very competitive market. Currently the wage rate is $22 per hour and capital is rented at $32 per hour. If the marginal product of labor is 81 units of output per hour and the marginal product of capital is 115 units of output per hour, should the firms increase, decrease, or leave unchanged the amount of capital used in its production process? "
a) the firm should increase its use of capital
b) the firm should decrease it use of capital
c) the should leave unchanged the amount of capital used.
MPL/w ? MPK/r
81/22 ? 115/32
3.68 > 3.59
--> the firm should decrease its use of capital
500
Which production function allows capital and labor to be substituted for each other with ratio changes?
Cobb-Douglas
500
An economist estimated that the cost function of a single-product firm is:
C(Q) = 100 + 20Q +15Q2 + 10Q3
Based on this, determine:
g) The marginal cost when Q= 10
TC = 100 +20Q + 15Q2+ 10Q3
MC (Q) = 20 + 30Q + 30Q2
MC(10)= 20 + 30(10) + 30(10)2 = $3320
500
Q8: "The Total cost curve and total variable cost curve have identical slopes at every quantity of output" (T/F)
True
500
A firm can manufacture a product according to the production function Q= 5k.5L.5
a) calculate the average product of labor when the level of capital is fixed at 30 units and the firm uses 6 units of labor (round to the nearest 2 decimal place)
(1) Find Q
Q = 5(30).5(6).5 = 67.11
(2) APL = 67.11/6 = 11.18