Economics of technology and management - Esercizi

Economics of Technology and Management

Master Degree in Mechanical Engineering – Sapienza University of Rome

Settore scientifico-disciplinare: ING-IND/35 Ingegneria Economico-Gestionale Teacher: Tiziana D’Alfonso Day: 06/03/2017

Problem 1

Consider the production function (CES production function) whose equation is given by the formula:

Q = (Lp^β + Kp^β)^1/β

1. (a) What is the elasticity of substitution between capital and labour?
2. (b) Is it a constant number or not?
3. (c) What if β → 0? Indicate which production function describes the CES in this situation.

Problem 2

Determine returns to scale in the following production functions:

1. (a) Q = 2√K^1/2 + L²
2. (b) Q = K^1/4L^3/4
3. (c) Q = ⅗ (K + L)

Problem 3

Suppose that the production function for microchips is given by Q = KL² – L³, where Q is the number of microchips produced per year, K is machine-hours of capital, and L is man-hours of labor.

1. a) Suppose K = 600. Find the total productivity function and graph it over the range L = 0 to L = 500. Then sketch the graphs of the average and marginal product functions. At what level of labor L does the average product curve appear to reach its maximum? At what level does the marginal product curve appear to reach its maximum?
2. b) Replicate the analysis in (a) for the case in which K = 1200.
3. c) When either K = 600 or K = 1200, does the total product function have a region of increasing marginal productivity?

Problem 4

Are the following statements correct or incorrect?

1. a) If average product is increasing, marginal product must be less than average product.
2. b) If marginal product is negative, average product must be negative.
3. c) If average product is positive, total product must be rising.
4. d) If total product is increasing, marginal product must also be increasing.

Problem 5

Suppose the production function is given by the equation Q = L^1/2R^1/2. Graph the isoquants corresponding to Q = 10, Q = 20, and Q = 50. Do these isoquants exhibit diminishing marginal rate of technical substitution?

Problem 6

Let Q be the number of bicycles produced from F bicycle frames and T tires. Every bicycle needs exactly two tires and one frame.

1. Draw the isoquants for bicycle production.
2. Write a mathematical expression for the production function for bicycles.

Problem 7

A firm produces a quantity Q of semiconductors using labor L and material M with the production function Q = 50/ML + M + L.

1. Determine the marginal product functions for this production function.
2. Are the returns to scale increasing, constant, or decreasing for this production function?
3. Is the marginal product of labor ever diminishing for this production function? If so, when? Is it ever negative, and if so, when?

Problem 8

Consider a CES production function given by Q = (L^0.5 + K^0.5)².

1. What is the elasticity of substitution for this production function?
2. Does this production function exhibit increasing, decreasing, or constant returns to scale?
3. Suppose that the production function took the form Q = (100 + L^0.5 + K^0.5)². Does this production function exhibit increasing, decreasing, or constant returns to scale?

Problem 9

Suppose a firm’s production function initially took the form Q = 500( L + 3K). However, as a result of a manufacturing innovation, its production function is now Q = 1000(0.5L + 10K).

1. Show that the innovation has resulted in technological progress. Is the technological progress neutral, labor saving, or capital saving?

Problem 10

Suppose that in the the production of semiconductors requires two inputs: capital (denoted by K) and labor (denoted by L). The production function takes the form Q = ^KL. However, in 100 years it is forecasted that the production function for semiconductors will take the form Q = √K. In other words, it will be possible to produce semiconductors entirely with capital (perhaps because of robots).

1. Does this change in the production function change the returns to scale?
2. Is this change in the production function an illustration of technological progress?

Problem 4

a) FALSE: MUST BE MORE THAN...

b) FALSE: NO

c) FALSE: IF MP IS POSITIVE...

d) FALSE: INVERTIRE MP & TOTAL P...

Problem 5

Q = LK

K=^Q₂/_L2

K =

• ¹⁰⁰ - 1^a ISOQUANT.
• ⁶⁰⁰ - 2^a ISOQUANT.
• ^2'500 - 3^a ISOQUANT.

0 ∞ 1^a

5 4

10 1

15 0,44

20 0,25

L K

0 ∞ 2^a

10 6

20 1

30 0,44

40 0,25 L K

0 ∞ 3^a

25 4

50 1

75 0,44

100 0,25

INCREASING _L THERE IS A DIMINSHING

OF MRTS_L,K

MP_L = √K

MP_k = ^L / _2√K

MRTS_L,K=¹/_2^L/ 1^K

MRTS_L,k⬇ IF L⬆

Problem 5

A paint manufacturing company has a production function Q = K + L^1/2. The firm faces a price of labor w that equals $1 per unit and a price of capital services r that equals $50 per unit.

1. Verify that the firm's cost-minimizing input combination to produce Q=10 involves no use of capital.
2. What must the price of capital fall to in order for the firm to use a positive amount of capital, keeping Q=10 and w=1?
3. What must Q increase to for the firm to use a positive amount of capital, keeping w=1 and r=50?

Problem 6

A plant's production function is Q=2KL+K. The price of labor services w is $4 and of capital services r is $5 per unit.

1. In the short run, the plant's capital is fixed at K=9. Find the amount of labor it must employ to produce Q=45 units of output.
2. How much money is the firm sacrificing by not having the ability to choose its level of capital optimally.

Problem 7

A firm operates with the production function Q = K²L. Q is the number of units of output per day when the firm rents K units of capital and employs L workers each day. The manager has been given a production target: Produce 8,000 units per day. She knows that the daily rental price of capital is $400 per unit. The wage rate paid to each worker is $200 day.

1. Currently, the firm employs 80 workers per day. What is the firm's daily total cost if it rents just enough capital to produce at its target?
2. Compare the marginal product per dollar spent on K and on L when the firm operates at the input choice in part (a). What does this suggest about the way the firm might change its choice of K and L if it wants to reduce the total cost in meeting its target?
3. In the long run, how much K and L should the firm choose if it wants to minimize the cost of producing 8,000 units of output day? What will the total daily cost of production be?

Total cost in long run:

_Cl = w_lL* + w_kK* ≃ 40,14

So the firm is sacrificing:

_Cs - _Cl = 53-66.6 ≃ 12.6

Problem (4)

Q = k²L

w_k = 100

w_l = 200

@ if L = 80and L must produce 8000 = Q

k = _Q√

⁡L

k = _√8000√

√80

= √100 = 10

Total cost in this case:

C = (w_kk + w_ll = 20,000 per day

↓₅↓

_mp k²= 100

mp_k = 2kl - l⁶⁰⁰⁰

But

C = w_kK* +w_IL*-12,000

MP_I = ¹⁄₁₆ But w_I = ¹⁄₂

We must MP↓ ↑

Up to have tangency condition

↓\ ₅K²_!

@ 1↓

w_l*_x𝑝

Tangency point of isocost is on the left