Basic Mathematics for Economists phần 7 potx

In this example, the cross partial derivative ∂2Q/∂L∂Ktells us that the rate of change of MPLwith respect to changes in K will be positive and will fall in value as K increases.. with th

where C is consumer expenditure, Yd is disposable income, Y is national income,

I is investment, t is the tax rate and G is government expenditure What is the marginal propensity to consume out of Y ? What is the value of the govern-

ment expenditure multiplier? How much does government expenditure need to

be increased to achieve a national income of 700?

10.3 Second-order partial derivatives

Second-order partial derivatives are found by differentiating the first-order partial derivatives

a second time we get

be negative (assuming positive values of K and L) and as L increases, ceteris paribus, the

absolute value of this slope diminishes

We can also find the rate of change of ∂Q/∂K with respect to changes in L and the rate

of change of ∂Q/∂L with respect to K These will be

∂2Q

∂K∂L = 3K −0.6 L −0.7 ∂2Q

∂L∂K = 3K −0.6 L −0.7 and are known as ‘cross partial derivatives’ They show how the rate of change of Q with

respect to one input alters when the other input changes In this example, the cross partial

derivative ∂2Q/∂L∂Ktells us that the rate of change of MPLwith respect to changes in K will be positive and will fall in value as K increases.

You will also have noted in this example that

∂2Q

∂K∂L = ∂2Q

∂L∂K

In fact, matched pairs of cross partial derivatives will always be equal to each other

Thus, for any continuous two-variable function y = f(x, z), there will be four second-order

with the cross partial derivatives (iii) and (iv) always being equal, i.e.

and these represent the marginal product functions MPKand MPL

The four second-order partial derivatives are as follows:

(i) ∂

2Q

∂K2 = 0.6L

This represents the slope of the MPKfunction It tells us that the MPKfunction will have a

constant slope along its length (i.e it is linear) for any given value of L, but an increase in L

will cause an increase in this slope

(ii) ∂

2Q

∂L2 = 2.4

This represents the slope of the MPLfunction and tells us that MPLis a straight line with

slope 2.4 This slope does not depend on the value of K.

from (ii) above, its actual position will depend on the amount of K used.

Some other applications of second-order partial derivatives are given below

To find the total revenue function for good 1 (TR1) in terms of q1we first need to derive the

inverse demand function p1 = f(q1 ) Thus, given

MR1= ∂TR1

∂q1 = 150 + 0.625p2 − 2.5q1

This marginal revenue function will have a constant slope of−2.5 regardless of the value of

p2or the amount of q1sold

The effect of a change in p2on MR1is shown by the cross partial derivative

∂TR1

∂q1∂p2 = 0.625

Thus an increase in p2of one unit will cause an increase in the marginal revenue from good

1 of 0.625, i.e although the slope of the MR1schedule remains constant at−2.5, its position shifts upward if p2rises (Note that in order to answer this question, we have formulated thetotal revenue for good 1 as a function of one price and one quantity, i.e TR1= f(q1 , p2).)

Example 10.14

A firm operates with the production function Q = 820K 0.3 L 0.2 and can buy inputs K and L

at £65 and £40 respectively per unit If it can sell its output at a fixed price of £12 per unit,

what is the relationship between increases in L and total profit? Will a change in K affect the extra profit derived from marginal increases in L?

To determine the effect of a change in K on the marginal profit function with respect to L,

we need to differentiate (1) with respect to K, giving

Second-order and cross partial derivatives can also be derived for functions with three

or more independent variables For a function with three independent variables, such as

y = f(w, x, z) there will be the three second-order partial derivatives

Example 10.15

For the production function Q = 32K 0.5 L 0.25 R 0.4 derive all the second-order and crosspartial derivatives and show that the cross partial derivatives with respect to each possiblepair of independent variables will be equal to each other

Test Yourself, Exercise 10.3

1 For the production function Q = 8K 0.6 L 0.5derive a function for the slope of the

marginal product of L What effect will a marginal increase in K have upon this

MPLfunction?

2 Derive all the second-order and cross partial derivatives for the production function

Q = 35KL + 1.4LK2+ 3.2L2and interpret their meaning

3 A firm operates three plants with the joint total cost function

TC= 58 + 18q1 + 9q2 q3+ 0.004q2

1q32+ 1.2q1 q2q3

Find all the second-order partial derivatives for TC and demonstrate that the crosspartial derivatives can be arranged in three equal pairs

10.4 Unconstrained optimization: functions with two variables

For the two variable function y = f(x, z) to be at a maximum or at a minimum, the first-order

conditions which must be met are

∂y

∂x = 0 and ∂y

∂z = 0These are similar to the first-order conditions for optimization of a single variable functionthat were explained inChapter 9 To be at a maximum or minimum, the function must be at

a stationary point with respect to changes in both variables

The second-order conditions and the reasons for them were relatively easy to explain inthe case of a function of one independent variable However, when two or more indepen-dent variables are involved the rationale for all the second-order conditions is not quite sostraightforward We shall therefore just state these second-order conditions here and give abrief intuitive explanation for the two-variable case before looking at some applications Thesecond-order conditions for the optimization of multi-variable functions with more than twovariables are explained inChapter 15using matrix algebra

For the optimization of two variable functions there are two sets of second-order conditions

For any function y = f(x, z).

(2) The other second-order condition is

This must hold at both maximum and minimum stationary points.

To get an idea of the reason for this condition, imagine a three-dimensional model with x and z being measured on the two axes of a graph and y being measured by the height above the flat surface on which the x and z axes are drawn For a point to be the peak of the y

‘hill’ then, as well as the slope being zero at this point, one needs to ensure that, whichever

direction one moves, the height will fall and the slope will become steeper Similarly, for

a point to be the minimum of a y ‘trough’ then, as well as the slope being zero, one needs

to ensure that the height will rise and the slope will become steeper whichever direction onemoves in As moves can be made in directions other than those parallel to the two axes, itcan be mathematically proved that the condition

First-order conditions for maximization of this profit function are

Substituting this value for q2into (1)

598.8 − 0.6q1 − 0.2(896.7) = 0

598.8 − 179.34 = 0.6q1

419.46 = 0.6q1 699.1 = q1

Checking second-order conditions by differentiating (1) and (2) again:

∂2π

∂q12 = −0.6 < 0 ∂2π

∂q22 = −0.4 < 0

This satisfies one set of second-order conditions for a maximum

The cross partial derivative will be

and so the remaining second-order condition for a maximum is satisfied

The actual profit is found by substituting the optimum values q1 = 699.1 and q2 = 896.7.

into the profit function Thus

517− 7(110) + 2.2p2= 0

2.2p2= 253

p2= 115Checking second-order conditions:

∂2TR

∂p12 = −7 < 0 ∂2TR

∂p22 = −8.8 < 0 ∂2TR

∂p1∂p2 = 2.2

Substituting this value of q1into (1)

Note that this method could also be used to solve the multiplant monopoly problems in

Chapter 5that only involved linear functions The unconstrained optimization method usedhere is, however, a more general method that can be used for both linear and non-linearfunctions

Example 10.19

A firm sells its output in a perfectly competitive market at a fixed price of £200 per unit It

buys the two inputs K and L at prices of £42 per unit and £5 per unit respectively, and faces

the production function

q = 3.1K 0.3 L 0.25

What combination of K and L should it use to maximize profit?

Checking second-order conditions:

to scale and the average and marginal cost schedules continued to fall, a firm facing a fixedprice would wish to expand output indefinitely and so no profit-maximizing solution would

be found by this method

If its total output is sold in a market where the demand schedule is p = 320 − 0.1q, where

q = q1 + q2, how much should it produce in each plant to maximize total profits?



q12= 0.6q2

1

= −0.35492 ±

√

11.64598 0.018

to determine optimum values From the way these problems are constructed it will be obviouswhether or not a maximum or a minimum value is being sought, and it will be assumed thatsecond-order conditions are satisfied for the values that meet the first-order conditions

Example 10.21

A firm operates with the production function

Q = 95K 0.3 L 0.2 R 0.25

and buys the three inputs K, L and R at prices of £30, £16 and £12 respectively per unit If

it can sell its output at a fixed price of £4 a unit, what is the maximum profit it can make?(Assume that second-order conditions for a maximum are met at stationary points.)

Taking each side of (6) to the power of 3

R 0.75=27,000(0.8) 2.1 L 1.5

1143Inverting

Test Yourself, Exercise 10.4

(Ensure that you check that second-order conditions are satisfied for these strained optimization problems.)

uncon-1 A firm produces two products which are sold in separate markets with the demandschedules

How much should the firm sell in each market in order to maximize total profits?

2 A company produces two competing products whose demand schedules are

q1= 219 − 1.8p1 + 0.5p2 q2= 303 − 2.1p2 + 0.8p1

What price should it charge in the two markets to maximize total sales revenue?

3 A price-discriminating monopoly sells in two separable markets with demandschedules

p1= 215 − 0.012q1 p2= 324 − 0.023q2

and faces the total cost schedule TC = 4,200 + 0.3q2, where q = q1 + q2 What should it sell in each market to maximize total profit? (Note that negative

quantities are not allowed, as was explained in Chapter 5.)

4 A monopoly sells its output in two separable markets with the demand schedules

p1= 20 −q1

6 p2= 13.75 − q2

8

If it faces the total cost schedule TC= 74 + 2.26q + 0.01q2where q = q1 + q2,

what is the maximum profit it can make?

5 A multiplant monopoly operates two plants whose cost schedules are

TC1= 2.4 + 0.015q2

1 TC2= 3.5 + 0.012q2

2

and sells its total output in a market where p = 32 − 0.02q.

How much should it produce in each plant to maximize total profits?

6 A firm operates two plants with the total cost schedules

TC1= 62 + 0.00018q3

1 TC2= 48 + 0.00014q3

2

and faces the demand schedule p = 2,360 − 0.15q.

To maximize profits, how much should it produce in each plant?

7 A firm faces the production function Q = 0.8K 0.4 L 0.3 It sells its output at a fixed

price of £450 a unit and can buy the inputs K and L at £15 per unit and £8 per

unit respectively What input mix will maximize profit?

8 A firm selling in a perfectly competitive market where the ruling price is £40 can

buy inputs K and L at prices per unit of £20 and £6 respectively If it operates with the production function Q = 21K 0.4 L 0.2, what is the maximum profit it canmake?

9 A firm faces the production function Q = 2.4K 0.6 L 0.2 , where K costs £25 per unit and L costs £9 per unit, and sells Q at a fixed price of £82 per unit Explain why it cannot make a profit of more than £20,000, no matter how efficiently it

plans its input mix

10 A firm can buy inputs K and L at £32 per unit and £20 per unit respectively andsell its output at a fixed price of £5 per unit How should it organize production to

ensure maximum profit if it faces the production function Q = 82K 0.5 L 0.3?

10.5Total differentials and total derivatives

(Note that the mathematical methods developed in this section are mainly used for the proofs

of different economic theories rather than for direct numerical applications These proofs may be omitted if your course does not include these areas of economics as they are not essential in order to understand the following chapters.)

In Chapter 8, when the concept of differentiation was introduced, you learned that the

derivative dy/dx measured the rate of change of y with respect to x for infinitesimally small changes in x and y For any non-linear function y = f(x), the value of dy/dx will alter if x and y alter It is therefore not possible to predict the effect of a given increase in x on y with complete accuracy However, for a very small change (5x) in x, we can say that it will be approximately true that the resulting change in y will be

5y= dy

dx 5x

The closer the function y = f(x) is to a straight line, the more accurate will be the prediction,

as the following example demonstrates

Example 10.22

For the functions below assume that the value of x increases from 10 to 11 Predict the effect

on y using the derivative dy/dx evaluated at the first value of x and check the answer against

the new value of the function

(i) y = 2x (ii)y = 2x2 ( iii)y = 2x3

The actual values are y = 2(10) = 20 when x = 10

y = 2(11) = 22 when x = 11

Thus actual change is 22− 20 = 2 (accuracy of prediction 100%)

(ii) y = 2x2 dy

dx = 4x = 4(10) = 40 Therefore, predicted change in y is

The above method of predicting approximate actual changes in a variable can itself beuseful for practical purposes However, in economic theory this mathematical method istaken a stage further and helps yield some important results

It is usual to write dy, dx, dz etc to represent infinitesimally small changes instead of

5y, 5x, 5z, which usually represent small, but finite, changes Thus

dy= ∂y

∂x dx+∂y

∂z dz

In introductory economics texts the MRTS of K for L (usually written as MRTSKL) is

usually defined as the amount of K that would be needed to compensate for the loss of one unit

of L so that the production level remains unchanged This is only an approximate measurethough and more accuracy can be obtained when the MRTSKLis defined at a point on anisoquant For infinitesimally small changes in K and L the MRTSKL measures the rate atwhich K needs to be substituted for L to keep output unchanged, i.e it is equal to the negative

of the slope of the isoquant at the point corresponding to the given values of K and L, when

K is measured on the vertical axis and L on the horizontal axis

For any given output level, K is effectively a function of L (and vice versa) and so, movingalong an isoquant,

If we are looking at a movement along the same isoquant then output is unchanged and so

dQ is zero and thus

We already know that ∂Q/∂L and ∂Q/∂K represent the marginal products of K and L.

Therefore, from (1) and (2) above,

MRTSKL= MPL

MPK

Euler’s theorem

Another use of the total differential is to prove Euler’s theorem and demonstrate the conditions

for the ‘exhaustion of the total product’ This relates to the marginal productivity theory offactor pricing and the normative idea of what might be considered a ‘fair wage’, which wasdebated for many years by political economists

Consider a firm that uses several different inputs Each will contribute a different amount

to total production One suggestion for what might be considered a ‘fair wage’ was that eachinput, including labour, should be paid the ‘value of its marginal product’ (VMP) This is

defined, for any input i, as marginal product (MP i) multiplied by the price that the finished

good is sold at (PQ), i.e.

VMPi = PQMP i

Any such suggestion is, of course, a normative concept and the value judgements on which it

is based can be questioned However, what we are concerned with here is whether it is even

possible to pay each input the value of its marginal product If it is not possible, then it would

not be a practical idea to set this as an objective even if it seemed a ‘fair’ principle

Before looking at Euler’s theorem we can illustrate how the conditions for product tion can be derived for a Cobb–Douglas production function with two inputs This examplealso shows how the product price is irrelevant to the product exhaustion question and it is theproperties of the production function that matter

exhaus-Assume that a firm sells its output Q at a given price PQ and that Q = AK α L β where

A, α and β are constants If each input was paid a price equal to the value of its marginal product then the prices of the two inputs K and L would be

TR= P Q Q

Total expenditure on inputs (which are paid the value of their marginal product) will equaltotal revenue when TR= TC Therefore

Thus the conditions of product exhaustion are based on the physical properties of the

production function If (2) holds then the product is exhausted If it does not hold then therewill be either not enough revenue or a surplus

For the Cobb–Douglas production function Q = AK α L β we know that

∂Q

∂K = αAK α−1L β ∂Q

∂L = βAK α L β−1Substituting these values into (2), this gives

Q = K(αAK α−1L β ) + L(βAK α L β−1)

= αAK α L β + βAK α L β

= αQ + βQ

The condition required for (3) to hold is that α + β = 1 This means that product exhaustion

occurs for a Cobb–Douglas production function when there are constant returns to scale

We can also see from (3) and (1) that:

(i) when there are decreasing returns to scale and α + β < 1, then

TC= P Q (α + β)Q < P Q Q= TR

and so there will be a surplus left over if all inputs are paid their VMP, and

(ii) when there are increasing returns to scale and α + β > 1, then

TC= P Q (α + β)Q > P Q Q= TR

and so there will not be enough revenue to pay each input its VMP

Euler’s theorem also applies to the case of a general production function

the total differential of this production function will be

i.e there are constant returns to scale

If there are decreasing returns to scale, output increases by a smaller proportion than theinputs Therefore,

(i) equal total revenue if there are constant returns to scale;

(ii) be less than total revenue if there are decreasing returns to scale;

(iii) be greater than total revenue if there are increasing returns to scale

Example 10.24

Is it possible for a firm to pay each input the value of its marginal product if it operates with

the production function Q = 14K 0.6 L 0.8?

Total derivatives

In partial differentiation it is assumed that one variable changes while all other independentvariables are held constant However, in some instances there may be a connection between

the independent variables and so this ceteris paribus assumption will not apply For example,

in a production function the amount of one input used may affect the amount of anotherinput that can be used with it From the total differential of a function we can derive a totalderivative which can cope with this additional effect

Assume y = f(x, z) and also that x = g(z).

Thus any change in z will affect y:

(a) directly via the function f(x, z), and

(b) indirectly by changing x via the function g(z), which in turn will affect y via the function f(x, z).

The total differential of y = f(x, z) is

dy= ∂y

∂x dx+∂y

∂z dz Dividing through by dz gives

The first term shows the indirect effect of z, via its impact on x, and the second term shows

the direct effect

Example 10.25

If Q = 25K 0.4 L 0.5 and K = 0.8L2what is the total effect of a change in L on Q? Identify

the direct and indirect effects