Solution 6 - Yioryos Makedonis PDF

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Yioryos Makedonis...

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Queen Mary, University of London BSc. (Econ), ECN-124 MMEB2 Exercises L06 Solutions

1. Which of the following functions are homogeneous, and if so, of what degree?

Where the function is homogeneous, verify Euler's theorem. (a)

z  3 x2 y 2  2 xy 3 10

x5 y

The new value of z is

z1  3 ( x )2 ( y )2  2( x )( y )3  10

  4 x5   4 x 2y 2   4 2xy 3  10  y 

( x )5 y

1

3   



3  4 x2 y 2

4 3   2xy  10

 4x 5 y

1

 

4 3

 2 2 x5  3 3  x y  2xy  10   y  

Therefore the function is homogeneous of degree

4 3

4

  3 z0

(the power to which  is raised.)

(b)

z  3x 2 y  4 xy  xy 2 The new value of z is

z1  3(x )2 (y )  4(x )(y )  (x )(y )2



 33x 2y   2 4xy   3xy 2

As r, for any r, is not a common factor of every term this expression, it is not homogeneous of any degree.

2.

(a) For each of the following production functions, assess the degree of homogeneity. (i)

Q  25K 0.3 L0.5 Homogeneous of degree 0.8

(ii)

Q  AK L1  Homogeneous of degree  + (1 – )  1

(b) Verify Euler's theorem in each function in (a). (As stated it in section 17.6 of the book, this theorem says that if a function Q  f( K , L) is homogeneous of degree , then L

Q Q K  θQ L K

(i) Q  25K 0.3 L0.5 . In this case: L

Q Q K  25K 0.3 (0.5)L0.5 25(0.3)K 0.3L0.5 = 0.8Q L K

(ii) Q  AK L1 

L

Q Q K  AK  (1   )L1   AK L1   Q L K

(c) What is the significance for the theory of competitive equilibrium of this extension of Euler's theorem? If the production function Q  f (K , L ) is homogeneous of degree , then Euler’s theorem tells us that L

Q Q K  Q . L K

Under perfect competition, the real reward of each factor is equal to its marginal productivity, so we have

Q w Q r and (where as usual w is the money   L P K P

wage, r is the rental rate of capital, and P is the product price.

Combining these, we have L

w r  K  Q P P

 wL  rK   PQ

Since wL  rK is total cost, and PQ is total revenue, then if  total cost > total revenue, while if  total cost < total revenue. We also know that if , the production function has increasing returns to scale, while if , the production function has decreasing returns to scale. Therefore, if there are increasing returns to scale, total cost > total revenue; while if there are decreasing returns to scale, total cost < total revenue. Only if there are constant returns to scale will total cost = total revenue. (Note this assumes perfect competition and that the real reward of each factor is equal to its marginal productivity.)

3.

Given the production function Q  120K 0.5L0.6 (a)

Derive the total, average and marginal products of labour when the capital input is fixed at K = 100.

(b)

Sketch their graphs, showing how they are related.

(c)

Comment on the suitability of this function for use as a production function.

(a)

Derive the total, average and marginal products of labour when the capital input is fixed at K = 100.

0.6 0.5 When K = 100, K  10 , so Q  1200L . This is the total product. The marginal

product of labour is

dQ 0.4 (you may think it more correct to write this as  720L dL

Q 1200L0.6 Q  1200 L0.4 ). The average product of labour is  L L L

(b)

Sketch their graphs, showing how they are related. Q Total product, Q

4777.29

1200

1

10

L

Q 1200 Values are approximate

720 477.73

APL

286.34

MPL

1

(c)

10

L

Comment on the suitability of this function for use as a production function.

In general the Cobb-Douglas production function, Q  AK L , where A,  and  are parameters, is the simplest function giving plausible shapes to the total, average and marginal product curves, and to the isoquants. In this case we have Q  120K 0.5L0.6 , which is homogeneous of degree 1.1. As this is greater than 1, we have increasing returns to scale, which many economists believe found in many industries in the real world....