### Essay preview

0956201

0953949

0892377

Midterm paper

Hand-in date:

04.11.2013

Campus:

BI Oslo

Examination code and name:

GRA 60312 Microeconomics

Programme:

Master of Science in Business, major in Finance

Midterm paper in GRA 60312

04.11.2013

Content

Content

............................................................................................................................ i

Figures

............................................................................................................................ ii

Tables

.............................................................................................................................. ii

Question

1.

Production

function

............................................................................ 1

Question

2.

Dynamic

Hotelling

model

................................................................. 4

Question

3.

Cheese

market

in

a

convenience

store

.........................................

9

Question

4.

Juvenile

crime

..................................................................................... 13

References

................................................................................................................... 20

Page i

Midterm paper in GRA 60312

04.11.2013

Figures

Figure 1. 1. Increasing Return to Scale ................................................................... 1

Figure 1. 2. Marginal cost and average cost when β = 1 ........................................ 3

Figure 2. 1. Best response function of Firm 2 in the Dynamic Hotelling game ...... 5

Figure 2. 2. Best response functions for the two firms ............................................ 6

Figure 2. 3. Best response functions for the two firms ............................................ 7

Tables

Table 2. 1. Comparison between Simultaneous game and Dynamic game ............. 8

Table 3. 1. Summary information of Brands and Products ..................................... 9

Table 3. 2. Some examples of special and traditional products............................. 10

Page ii

Midterm paper in GRA 60312

04.11.2013

Question 1. Production function

(a) Characterize the returns to scale of the technology, using the formal definition, and draw it in a graph.

Return to scale is the term that describe the change in output when we scale all of inputs proportionally. So an increasing return to scale means that production changes in a greater rate than the one which is scaled in each input. A decreasing return to scale means an inverse thing, and constant return to scale describes the situation when production changes in the same rate with the scale rate of input. Following these arguments, with a production function 𝑄(𝐾, 𝐿) of Capital (𝐾), and Labour (𝐿), with 𝑎 > 0 we have:

Constant returns to scale: 𝑄(𝑎𝐾, 𝑎𝐿) = 𝑎𝑄(𝐾, 𝐿) Increasing returns to scale: 𝑄 𝑎𝐾, 𝑎𝐿 > 𝑎𝑄(𝐾, 𝐿) Decreasing returns to scale: 𝑄 𝑎𝐾, 𝑎𝐿 < 𝑎𝑄(𝐾, 𝐿) The production function is given as follows:

𝑦 = 𝑄(𝐾, 𝐿) = 𝐴𝐾 !.! 𝐿!.!

When we scale both Capital and Labor in (for example) two times we have: 𝑄 2𝐾, 2𝐿 = 𝐴 2𝐾

!.!

(2𝐿)!.! = 𝐴. 2!.! 𝐾 !.! 𝐿!.! 2!.! 𝑄 𝐾, 𝐿 > 2𝑄(𝐾, 𝐿) (1)

The above inequality proves that our technology is Increasing Return to Scale. Figure 1. 1. Increasing Return to Scale

6

Output rate

of change

4

2

Input scale rate

0

0

1

2

3

Page 1

4

5

Midterm paper in GRA 60312

04.11.2013

Following the equation (1) we can trace a graph which represents the relationship between the scale rate of input and the rate of change in output. This function is a convex function:

𝑌 = 𝑥 !.!

Where 𝑌 is the rate of change in output; and 𝑥 is the scale rate. (b) Should you be worried about the necessary size of production in order to break-even? [State the assumptions you make in order to arrive at an answer]

Concerning size of production, every firm always concerns the break-even point when his marginal cost (denoted 𝑀𝐶) equal to average cost (denoted 𝐴𝐶). After this point the production is usually inefficient when the marginal cost is greater than average cost.

With an assumption that the cost of each unit of Capital (denoted 𝑅), and of Labor (denoted 𝑊) is constant in all of the process, we can conclude that the firm should not be worried about production size neither break-even point. The explanation will be given below.

Assume that 𝑊 and 𝑅 is constant and greater than 0, we have: 𝐶 = 𝑊. 𝐿 + 𝑅. 𝐾 (2)

And

𝑊 𝑀𝑃𝐿

=

(3)

𝑅 𝑀𝑃𝐾

Where 𝑀𝑃𝐿 is marginal product of Labor, and 𝑀𝑃𝐾 is marginal product of Capital. The formula (3) gives us:

𝑊

𝐾

= 2 (4)

𝑅

𝐿

Plug (4) into (2) we have:

𝐶 = 3𝑅𝐾 𝑜𝑟 𝐾 =

𝐶

(5)

3𝑅

3

2𝐶

𝐶 = 𝑊𝐿 𝑜𝑟 𝐿 =

(6)

2

3𝑊

Plug (5) and (6) into 𝑄(𝐾, 𝐿) = 𝐴𝐾 !.! 𝐿!.! we have: Page 2

Midterm paper in GRA 60312

04.11.2013

𝑄=

2!.! . 𝐴. 𝐶 !.!

(3𝑅)!.! (3𝑊)!.!

𝐶 !.! = 𝑄.

3!.! 𝑅!.! 𝑊 !.!

2!.! . 𝐴

Finally we have:

3!.! 𝑅!.! 𝑊 !.!

𝐶=

𝐴. 2!.!

!

!.!

!

. 𝑄!.! (7)

!

Let us denote 𝛽 =

!!.! ! !.! ! !.! !.!

!.!!.!

which is a constant and greater than 0, given

by the parameters 𝐴, 𝑊, 𝑅 that we choose with an assumption that 𝐴 > 0. The !

formula (7) can be rewritten as: 𝐶 = 𝛽. 𝑄!.! (8)

!

!

!

From (8) we have: 𝑀𝐶 = !.! . 𝛽. 𝑄!! (9) and 𝐴𝐶 = 𝛽. 𝑄!! (10) Where 𝑀𝐶 is marginal cost and 𝐴𝐶 is average cost. From (9) and (10) we can conclude that whatever the size of production is, our marginal cost is always smaller than average cost of production.

Therefore, with an assumption that 𝐴 > 0, 𝑊, 𝑅 which are respectively cost of 01 unit of Labor and cost of 01 unit of Capital, are constant and there is neither sunk cost nor fixed cost. Hence, we can conclude that the firm should not be worried about the size of production or break-even point , 𝑀𝐶 = 𝐴𝐶, because 𝑀𝐶 is smaller than 𝐴𝐶 whatever the size of production.

Figure 1.2 shows us an illustration when 𝛽 = 1 of 𝑀𝐶 and 𝐴𝐶. Figure 1. 2. Marginal cost and average cost when 𝛃 = 𝟏 2

MC/AC

1.5

1

0.5

0

0

2

4

6

8

MC

Page 3

10

AC

12

14

16

Quantity

Midterm paper in GRA 60312

04.11.2013

Question 2. Dynamic Hotelling model

Two firms are competin...

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*𝐶*

*𝐷*

*𝐺*

*𝐻*

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*𝐿*

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*𝑀𝑃𝐿*

*𝑀𝑎𝑥*

*𝑂*

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*𝑆*

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*𝑋*

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*𝑎*

*𝑎𝐾*

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*𝜀*

*𝜋*

*𝜖*

*𝟏*