ME Exercise 3 - isoquants and isocosts PDF

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ME EXERCISE 3 Exercise 3.1: TP, AP and MP Cocktails and bartenders – we are running a cocktail bar: L 0 1 2 3 4 5 6 7 8 9 10 11 12

Q 0 17 116 279 488 725 972 1211 1424 1593 1700 1727 1656

AP (L) -30 17 58 93 122 145 162 173 178 177 170 157 138

MP (L) -30 61 134 189 226 245 246 229 194 141 70 -19 -126

a) Fill in the table for average production and marginal production (Hint for MP: how much Q changes with one more unit of labour?) 3

2

2

TP=−3 L +50 L −30 L

AP =−3 L +50 L−30

2

MP=−9 L +100 L−30

b) Plot graphically TP, AP and MP as a function of Labor (use Excel)

LABOR BARTENDERS

TP 2000 1800 1600 1400 1200 1000 800 600 400 200 0

f(x) = − 3 x³ + 50 x² − 30 x Q Polynomial (Q)

0

2

4

6

8

10

QUANTITY COCKTAILS

12

14

PRODUCTION FUNCTIONS 2000

QUANTITY OF COCKTAILS

1800 f(x) = − 3 x³ + 50 x² − 30 x

1600 1400

Q Polynomial (Q) AP (L) Polynomial (AP (L)) MP (L) Polynomial (MP (L))

1200 1000 800 600 400 200 0

0

f(x) = − 9 x² + 100 x − 30 f(x) = − 3 x² + 50 x − 30 2 4 6 8

10

12

14

LABOR - BARTENDERS

c) Indicate and explain the approximate point of inflection on the TP graph. What does it mean? At about L = 6 we have the point of inflection where the TP graph changes its curvature and graphically where MP is at its highest. It signifies the beginning of Law of Diminishing Return meaning MP will decrease from now on. Where MP is at its maximum is the second derivative of MP equal to 0 -> MP’=0 '

M P =−18 L+100

−18 L+100=0

L=5.55

d) How many bartenders do we want to employ? Let us say we can sell as many cocktails as we would like… There are increasing returns from L=0 to L=6, so at least 6 bartenders. We want to be somewhere from the intersection of AP and MP to MP=0, because at MP=0 TP is at its maximum an from there it will start to decrease. So approx.. from L=6 to L=8.33≈ 8 Here is where M P L and M P K are both positive and the only segment where a sensible merger should be hiring and producing.

Exercise 3.2: The Isocost line. We often use another notation for the isocost/budgetline, then what is used in H&B (p. 255). The two inputs are often divided into capital (K) and labour (L), which is why the following is often used: C = wL + rK Where w (wage) is the price of Labour, r (rent) is the price of capital and C (Cost) is the cost or budget. Traditionally L is on the x-axis. Please draw and explain what happens to the isocost if…

a) Wage decreases?

Because they are going to substitute capital with more labor b) Rental price decreases?

c) How do you think a company’s isocost line in one of the BRIC countries (developing), say China, would look like relative to an isocost line of a similar company in a developed country? Most likely they would substitute capital (K) with Labor in China

An isocost can be written as: C = wL + rK, where w could be the monthly paycheck to the employee (the wage), and r is the cost of capital (the rent). Assume that w = 1200 $ and r = 200 $. d) If the monthly budget for wages and rent is 13,200 $, how will the isocost line look? (Show both mathematically and graphically) 13,200=1200 L+200 K L=11 K=66 L

K 0 11

66 0

200 K=13,200−1,200 L

K=66 −6 L

0=66−6 L

ISOCOST 70 60

f(x) = − 6 x + 66

50

K

40

K Linear (K)

30 20 10 0

0

2

4

6

8

10

12

L

e) What happens if the wage decreases by 50%? (Show both mathematically and graphically) w = 1200-600= 600 13,200=600 L+ 200 K L

200 K=13200−600 L

K=66−3 L

L=22

K=66

K 0 22

66 0

50% W DECREASE 70

K

60 50 40

f(x) = − 3 x + 66 K Linear (K)

30 20 10 0

0

5

10

15

20

25

L

f) Now our monthly budget is doubled. Show the new isocost line mathematically and graphically 26,400=1200 L+200 K

L

K 0

132

K=132−3 L

44

0

Exercise 3.3: Isoquants Table 1 below shows the relationship between labor and capital for two long term quantities of output. The average wage is 260 Indian Rupee per hour and the price of capital can be set to 1 (the cost is the price of used equipment). C=260 L+ K K=B−260 L

a) Draw the two isoquants from the table above (Hint: the isoquants have a l-shape, power function).

ISOQUANTS 180000 160000 f(x) = 390000 x^-1 140000 120000 K (500) Power (K (500)) K (250) Power (K (250))

CAPITAL

100000 80000 60000 f(x) = 136500 x^-1

40000 20000 0

0

20

40

60

80

100

120

LABOR

c) Determine an expression for the isoquants (by Excel) 390000 L 136500 −1 K ( 250)=136500 L → L

K ( 500 )=390000 L−1 →

d) How many workers do we need to hire to produce an output of 250 Q? (Hint: tangency point between isocost line from Exercise 2 and isoquants you calculate in point b) You need to equal the slopes of both by taking their derivatives C=260 L+K 2

K=C−260 L

−260=−136500 L

−2

−260=

−136500 L2

2

L=22.9 −260 L =−136500 L =525 Now substitute the value into K(250) to find K, when L is 22.9 then K is K=

136500 22.9

K=5960.7

e) EXTRA: How much does it cost to produce 250 Q?

Now we can find C C=260 L+ K

C=260∗22.9+ 5960.7

C=11914.7

f) EXTRA: What is the cheapest way to produce 500 Q? Repeat the process of equaling the slopes of both and then calculate the costs −390000 L2 K=10069.76 −260=

−260 L2=−390000

C=260∗38.73 + 10069.76

C=20139.56

L2=1500

L=38.73

K=

390000 38.73...