Chapter 7, Problem 7.8P

a

To determine

Whether the given function exhibit constant returns to scale or not? It’s cost function and its nature of economies of scale and its importance.

Answer to Problem 7.8P

Function exhibits constant returns to scale.

Explanation of Solution

Given Information:

  $TC=B{q}^{1/(a+b)}{v}^{a/(a+b)}{w}^{b/(a+b)}$

Cobb Douglas production function is: $q=K^a{L}^b$

Total Cost function derived from this production function is:

  $TC=B{q}^{1/(a+b)}{v}^{a/(a+b)}{w}^{b/(a+b)}$

According to the situation given in the question

  $a=b=0.5$

  $TC=B{q}^{1/(0.5+0.5)}{v}^{0.5/(0.5+0.5)}{w}^{0.5/(0.5+0.5)}$

  $TC=Bq{v}^{0.5}{w}^{0.5}$

Yes, this function exhibits constant returns to scale. This is because the average Cost $(AC)$ function is constant for all $q(AC=B{v}^{0.5}{w}^{0.5})$ and we know that if AC is constant, the firm faces CRS. As far as returns to scale are concerned, the input prices do not matter. If there is a change in input prices, then also in this case, the firm will face constant returns to scale.

Economics Concept Introduction

Introduction:

Constant returns to scale means when input and outputs are increased in same proportion. It means when inputs to a given function are increased in certain proportion, with the same proportion, output is also increased.

b)

To determine

Role of q and its relation with production function and effect on shape of total cost curve.

Answer to Problem 7.8P

‘q’ determines the nature of returns to scale.

Explanation of Solution

Given Information:

  $TC=B{q}^{1/(a+b)}{v}^{a/(a+b)}{w}^{b/(a+b)}$

Cobb Douglas production function is: $q=K^a{L}^b$

Here, the exponent of $\text{'}q\text{'}$ plays a very vital role.

If the exponent $(1/(a+b))=1$ , then there will be constant returns to scale. This is because, AC is constant.

If $(1/(a+b))>1$ , then the firm will experiences the increasing returns to scale. This is because, AC is increasing.

If $(1/(a+b))<1$ , then the firm experiences decreasing returns to scale because $AC$ is decreasing (rectangular hyperbola).

Economics Concept Introduction

Introduction:

Total cost curve is a graphical representation showing cost incurred for producing output. It is sum of fixed cost and variable cost.

The cost function in its general form, i.e. $TC=B{q}^{1/(a+b)}{v}^{a/(a+b)}{w}^{b/(a+b)}$ .

c)

To determine

Value of s and its relation with returns to scale.

Answer to Problem 7.8P

It determines the nature of returns to scale

Explanation of Solution

Given Information:

  $TC=B{q}^{1/(a+b)}{v}^{a/(a+b)}{w}^{b/(a+b)}$

Cobb Douglas production function is: $q=K^a{L}^b$

Here, the exponent of ‘s’ plays a very vital role. It determines the nature of returns to scale.

If the exponent $(1/(a+b))=1$ , then there will be constant returns to scale. This is because, AC is constant.

If $(1/(a+b))>1$ , then the firm will experiences the increasing returns to scale. This is because, AC is increasing.

If $(1/(a+b))<1$ , then the firm experiences decreasing returns to scale because $AC$ is decreasing (rectangular hyperbola).

Economics Concept Introduction

Introduction:

Returns to scale is a function of long run which determines impact of change in input to overall production process. It has three parts: Increasing returns to scale, Constant returns to scaleand Diminishing returns to scale.

d)

To determine

Impact of sizes of a and b on cost function

Answer to Problem 7.8P

It determines the nature of returns to scale

Explanation of Solution

Given Information:

  $TC=B{q}^{1/(a+b)}{v}^{a/(a+b)}{w}^{b/(a+b)}$

Cobb Douglas production function is: $q=K^a{L}^b$

Total cost curve is a graphical representation showing cost incurred for producing output. It is sum of fixed cost and variable cost.

Relative sizes of $\text{'}a\text{'}$ and $\text{'}b\text{'}$ plays an important role in determining cost functions. $1/(a+b)$ Moreover, the relative sizes of a and b will affect the wage rate and the rental rate. The point to note is that $a/(a+b)$ and $b/(a+b)$ will always be less than one. If the case $(1/(a+b))<1$ is true, then any increase in the rental rate will lead to small increase/shift in the cost function/curve. It can be seen this with the help of derivatives also.

Suppose $a=0.5$ and $b=0.5$ , then

  $TC=q{v}^{0.5}{w}^{0.5}$

  $d(TC)/dv=0.5q{v}^{-0.5}{w}^{0.5}$

  $d^2(TC)/d{v}^2=-0.25q{v}^{-1.5}{w}^{0.5}<0$

See that the slope of $TC$ with respect to $v$ is decreasing. This means that $TC$ with respect to $v$ is increasing at a decreasing rate.

Economics Concept Introduction

Introduction:

Total cost curve is a graphical representation showing cost incurred for producing output. It is sum of fixed cost and variable cost.

e)

To determine

Reason for usefulness of given function and implication of coefficients.

Answer to Problem 7.8P

Given function shows elasticity.

Explanation of Solution

Given Information:

  $TC=B{q}^{1/(a+b)}{v}^{a/(a+b)}{w}^{b/(a+b)}$

Cobb Douglas production function is: $q=K^a{L}^b$

According to the situation given in the question,

  $InTC=InB+1/(a+b)Inq+a/(a+b)Inv+b/(a+b)Inw$

This equation is very useful. Equations in this form help to find elasticity. For example,

  $d(InTC)/dInq=1/(a+b)$

This is nothing but the percentage change in $TC$ due to percentage change in q. Similar is the case for $d(InTC)/dInv=a/(a+b)$ . This is elasticity of $TC$ due to v, that is, percentage change in $TC$ due to percentage change in the rental rate $(v)$ .

Economics Concept Introduction

Introduction:

Elasticity is the numerical expression showing percentage change in a variable due to percentage change in another variable. It can be positive or negative.

f)

To determine

Findings of Translog Cost Function.

Explanation of Solution

According to the situation given in the question

  $InTC=InB+1/(a+b)Inq+a/(a+b)Inv+b/(a+b)Inw$

This type of function is called Trans log Cost Function. It is a very useful function and is mostly used in empirical research. See in Christenson and Greene paper on electric generation power references in table given in the text. Note that the entries in the table are an estimate of Cobb-Douglas production function.

Economics Concept Introduction

Introduction:

Translog cost function is a system of equation in which firm’s cost behavior, output, revenue is estimated.