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 MP_{L} is__"

15

Q= 40L^{(3/4}^{)}(1)^{(1/4)} -->40L^{(3/4)}

MP_{L}= (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= 400L^{1/4}K^{1/2}

A) Calculate the marginal product of labor for the equation above

(1) Plug in K

Q= 400L^{1/4}(25)^{1/2}= 200L^{1/4}

(2) Take Derivative of L

MP_{L}= (1/4) 200L^{(}^{1/4 - 4/4)}

MP_{L}= 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 MP_{L} and MP_{k}

A) linear (bc there is a plus sign)

B) Q= 4(2) +(3) = 17

C) MP_{L}= 3, MP_{k}= 4

300

An economist estimated that the cost function of a single-product firm is:

C(Q) = 100 + 20Q +15Q^{2} + 10Q^{3}

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= 400L^{1/4}K^{1/2}

Determine optimal amount of labor the firm should use

(1) Find VMP_{L}

VMP_{L} = MPL_{ * }P

= 500L^{(}^{-}^{ 3/4) }* ($5) = 2500L^{(- 3/4)}

VMP_{L}= 2500L^{(- 3/4)}

(2) Set VMP_{L} = Wage

2500L^{(-3/4)}=20 or 2500/(L^{3/4})

2500 = 20(L^{3/4} )

2500/20 = L^{3/4}

(125)^{4/3} = (L^{3/4})^{4/3}

L = 625

400

Given a production function Q= (10K^{0.5})(L^{.5}), K=9 L=16

a) what kind of Production Function is this. How can you tell?

b) What is Q?

C) Find MP_{L} and MP_{K}

A)Cobb Douglas, exponents

b) Q= (10(9)^{0.5})((16)^{0.5}) --> Q=10(3)(4) --> Q=120

c)

MP_{L}= (0.5)(10)K^{(0.50)}L^{(0.50 -1.00)}= 5K^{0.5}L^{(-0.5)}

MP_{k}=(0.5)(10)K^{(0.5-1.00)}L^{0.5}= 5K^{-0.5}L^{0.5}

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.

MP_{L}/w ? MP_{K}_{/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 +15Q^{2} + 10Q^{3}

Based on this, determine:

g) The marginal cost when Q= 10

TC = 100 +20Q + 15Q^{2}+ 10Q^{3}

MC (Q) = 20 + 30Q + 30Q^{2}

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^{.5}L^{.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) AP_{L }= 67.11/6 = 11.18