Basic Mathematics for Economists - Rosser - Chapter 10 potx

10.1 Partial differentiation and the marginal product For the production function Q = fK, L with the two independent variables L and K the value of the function will change if one indepe

10 Partial differentiation

Learning objectives

After completing this chapter students should be able to:

• Derive the first-order partial derivatives of multi-variable functions

• Apply the concept of partial differentiation to production functions, utilityfunctions and the Keynesian macroeconomic model

• Derive second-order partial derivatives and interpret their meaning

• Check the second-order conditions for maximization and minimization of afunction with two independent variables using second-order partial derivatives

• Derive the total differential and total derivative of a multi-variable function

• Use Euler’s theorem to check if the total product is exhausted for a Cobb–Douglasproduction function

10.1 Partial differentiation and the marginal product

For the production function Q = f(K, L) with the two independent variables L and K the

value of the function will change if one independent variable is increased whilst the other is

held constant If K is held constant and L is increased then we will trace out the total product

of labour (TPL) schedule (TPLis the same thing as output Q) This will typically take a shape

similar to that shown inFigure 10.1

In your introductory microeconomics course the marginal product of L(MPL)was probablydefined as the increase in TPLcaused by a one-unit increment in L, assuming K to be fixed

at some given level A more precise definition, however, is that MPLis the rate of change of

TPLwith respect to L For any given value of L this is the slope of the TPLfunction (Referback to Section 8.3 if you do not understand why.) Thus the MPLschedule in Figure 10.1 is

at its maximum when the TPLschedule is at its steepest, at M, and is zero when TPLis at itsmaximum, at N

Partial differentiation is a technique for deriving the rate of change of a function with

respect to increases in one independent variable when all other independent variables in the

function are held constant Therefore, if the production function Q = f(K, L) is differentiated with respect to L, with K held constant, we get the rate of change of total product with respect

to L, in other words MPL

The basic rule for partial differentiation is that all independent variables, other than the one

that the function is being differentiated with respect to, are treated as constants Apart fromthis, partial differentiation follows the standard differentiation rules explained inChapter 8

A curved ∂ is used in a partial derivative to distinguish it from the derivative of a single

variable function where a normal letter ‘d’ is used For example, the partial derivative of the

production function above with respect to L is written ∂Q/∂L.

(The 14x is treated as a constant and disappears One then just differentiates the term 3z with respect to z.)

For the production function Q = 20K 0.5 L 0.5

(i) derive a function for MPL, and

(ii) show that MPLdecreases as one moves along an isoquant by using more L.

We can now see that for any Cobb–Douglas production function in the format Q = AK α L β the law of diminishing marginal productivity holds for each input as long as 0 < α, β < 1.

If K is fixed and L is variable, the marginal product of L is found in the usual way by partial

differentiation Thus, when

which falls as K increases in value.

When there are more than two inputs in a production function, the same principles stillapply For example, if

which decreases as X3increases, ceteris paribus.

We can also see that for a production function in the usual Cobb–Douglas format themarginal product functions will continuously decline towards zero and will never ‘bottom

out’ for finite values of L; i.e they will never reach a minimum point where the slope is zero.

If, for example,

then MPLwill first rise and then fall since

Although there are now three independent variables instead of two, the same rules still apply,

this time with two variables treated as constants Therefore, holding y and z constant

To avoid making mistakes when partially differentiating a function with several variables,

it may help if you write in the variables that do not change first and then differentiate In

the above example, when differentiating with respect to x for instance, this would mean first writing in y 0.2 z 0.3 as y and z are held constant.

When a function has a large number of variables, a shorthand notation for the partialderivative is usually used For example, for the function f = f(x1, x2, , xn)one can write

This function is a summation of several terms Only one term, the j th, will contain x j If one

is differentiating with respect to x jthen all other terms are treated as constants and disappear

Therefore, one only has to differentiate the term 6x j 0.5 with respect to x j, giving

Uses of second-order partial derivatives will be explained in Section 10.3

Test Yourself, Exercise 10.1

1 Find ∂y/∂x and ∂y/∂z when

(a) y = 6 + 3x + 16z + 4x2+ 2z2

(b) y = 14x3z2

(c) y = 9 + 4xz − 3x−2z3

2 Show that the law of diminishing marginal productivity holds for the

produc-tion funcproduc-tion Q = 12K 0.4 L 0.4 Will the MPLschedule take the shape shown in

does not obey the law of diminishing marginal productivity

5 If Q = 18K 0.3 L 0.2 R 0.5 , will the marginal products of any of the three inputs K, L and R become negative?

6 Derive a formula for the partial derivative Q j , where j is an input number, for the

10.2 Further applications of partial differentiation

Partial differentiation is basically a mathematical application of the assumption of ceteris paribus (i.e other things being held equal) which is frequently used in economic analysis.

Because the economy is a complex system to understand, economists often look at the effect

of changes in one variable assuming all other influencing factors remain unchanged Whenthe relationship between the different economic variables can be expressed in a mathematicalformat, then the analysis of the effect of changes in one variable can be discovered via partialdifferentiation We have already seen how partial differentiation can be applied to productionfunctions and here we shall examine a few other applications

In introductory economics courses, price elasticity of demand is usually defined as

e = (−1)percentage change in quantity demanded

percentage change in price

This definition implicitly assumes ceteris paribus, even though there may be no mention

of other factors that influence demand The same implicit assumption is made in the moreprecise measure of point elasticity of demand with respect to price:

e = (1) p

q

1

dp/dq

Recognizing that quantity demanded depends on factors other than price, then point elasticity

of demand with respect to price can be more accurately redefined as

We know the value of p and we can easily derive the partial derivative ∂q/∂p = −0.4.

Substituting these values into the elasticity formula

The actual value of elasticity cannot be calculated until specific values for m, pc, psand n

are given Thus this example shows that the value of point elasticity of demand with respect

to price will depend on the values of other factors that affect demand and thus determine theposition on the demand schedule

Other measures of elasticity will also depend on the values of the different variables in thedemand function For example, the basic definition of income elasticity of demand is

e m= percentage change in quantity demanded

percentage change in income

If we assume an infinitesimally small change in income and recognize that all other factorsinfluencing demand are being held constant then income elasticity of demand can be defined as

Consumer utility functions

The general form of a consumer’s utility function is

U = U(x1, x2, , x n )

where x1, x2, , x nrepresent the amounts of the different goods consumed

Unlike output in a production function, one cannot actually measure utility and this oretical concept is only of use in making general predictions about the behaviour of largenumbers of consumers, as you should learn in your economics course Modern economictheory assumes that utility is an ‘ordinal concept’, meaning that different combinations ofgoods can be ranked in order of preference but utility itself cannot be quantified in any way.However, economists also work with the concept of ‘cardinal’ utility where it is assumedthat, hypothetically at least, each individual can quantify and compare different levels of theirown utility It is this cardinal utility concept which is used here.

the-If we assume that only the two goods A and B are consumed, then the utility function willtake the form

U = U(A, B)

Marginal utility is defined as the rate of change of total utility with respect to the increase

in consumption of one good Therefore the marginal utility functions for goods A and B,respectively, will be

MUA =∂U

∂B

Three important principles of utility theory are:

(i) The law of diminishing marginal utility says that if, ceteris paribus, the quantity

consumed of any one good is increased, then eventually its marginal utility will decline.(ii) A consumer will consume a good up to the point where its marginal utility is zero if

it is a free good, or if a fixed payment is made regardless of the quantity consumed,e.g water rates

(iii) A consumer maximizes satisfaction when each good is consumed up to the point where

an extra pound spent on one good will derive the same utility as an extra pound spent

on any other good

Some applications of the first two principles are given in the following examples We shallreturn to principle (iii) inChapter 11, when we study constrained optimization

Example 10.7

Find out whether the law of diminishing marginal utility holds for both goods A and B in thefollowing utility functions:

(i) U = A 0.6 B 0.8

(ii) U = 85AB − 1.6A2B2

(iii) U = 0.2A−1B−1+ 5AB

Thus MUAfalls as A increases (when B is held constant) and MUBfalls as B increases (when A is held constant) As both marginal utility functions decline, the law of

diminishing marginal utility holds

(ii) For the utility function U = 85AB − 1.6A2B2the marginal utility functions will be

As A increases, the term 0.2A−2B−1gets smaller As this term is subtracted from 5B, which

will be constant as B remains unchanged, this means that MUArises Similarly, MUB will

rise as B increases Therefore the law of diminishing marginal utility does not hold for this

function

Example 10.8

Given the following utility functions, how much of A will be consumed if it is a free good?

If necessary give answers in terms of the fixed amount of B

(i) U = 96A + 35B − 0.8A2− 0.3B2

(ii) U = 72AB − 0.6A2B2

(iii) U = A 0.3 B 0.4

Solutions

In each case we need to try to find the value of A where MUAis zero (The law of diminishingmarginal utility holds for all three functions.) Consumers will not consume extra units of Awhich have negative marginal utility and hence decrease total utility

(i) For utility function U = 96A + 35B − 0.8A2− 0.3B2marginal utility of A is zerowhen

(ii) When U = 72AB − 0.6A B then MUAis zero when

This marginal utility function will decline continuously but, for any non-zero value of

B,MUAwill not equal zero unless the amount of A consumed becomes infinitely large.Therefore no finite solution can be found

The Keynesian multiplier

If a government sector and foreign trade are introduced then the basic Keynesian nomic model becomes the accounting identity

where Ydis disposable income and t is the tax rate.

Investment I , government expenditure G and exports X are exogenously determined and

c, m and t are given parameters Substituting (2), (3) and (4) into (1) we get

to derive the investment multiplier Thus the investment multiplier is found by partially

differentiating (5) with respect to I , which gives

What is the equilibrium level of Y ? What increase in G would be necessary to increase Y to

2,500? If this increased expenditure takes place, what will happen to

(i) the government’s budget surplus/deficit, and

(ii) the balance of payments?

Solution

First we derive the relationship between C and Y Thus

Next we substitute (1) and the other functional relationships and given values into the

accounting identity to find the equilibrium Y Thus

Y = C + I + G + X − M

= 0.8(1 − 0.2)Y + 300 + 400 + 288 − 0.2(1 − 0.2)Y

= 0.64Y + 988 − 0.16Y (1− 0.48)Y = 988

The amount spent on imports will be

This is the increase in G required to raise Y to 2,500.

At the new level of national income, the amount of tax raised will be

i.e there is an increase of 192 in the deficit

The new level of imports will be

Cost and revenue functions

Some firms produce several different products When common production facilities are usedthe costs of the individual products will be related and this will be reflected in the total costschedules The marginal cost schedules of the individual products can then be derived bypartial differentiation

These marginal cost schedules show that the level of marginal cost for one good will depend

on the amount of the other good that is produced

Some firms may produce different goods which compete with each other in the marketplace, or are complements This means that the price of one good will influence the quantitydemanded of the other goods sold by the same firm Marginal revenue for one good willtherefore be the partial derivative of total revenue with respect to the output level of thatparticular good, assuming that the price of the other goods are fixed

as a function of quantity Thus, for good A

Test Yourself, Exercise 10.2

1 The demand function for a good is

q = 56.6 − 0.25p − 0.03m + 0.45ps+ 0.6n

where q is the quantity demanded per week, p is the price per unit, m is the average weekly income, ps is the price of a competing good and n is the population in millions Given values are p = 65, m = 350, ps= 60 and n = 24.

(a) Calculate the price elasticity of demand

(b) Find out what would happen to (a) if n rose to 26.

(c) Explain why this is an inferior good

(d) If producers of the competing product and the manufacturer of this goodboth increased their prices by the same percentage, what would happen to thequantity demanded (of the original good), assuming that the proportional pricechange is small and relevant elasticity measures do not alter significantly

2 Do the following utility functions obey the law of diminishing marginal utility?

(a) U = 5A + 8B + 2.2A2B2− 0.3A3B3

(b) U = 24A 0.8 B 1.2

(c) U = 6A 0.7 B 0.8

3 An individual consumes two goods and has the utility function U = 2A 0.4 B 0.4,

where A and B represent the quantities of the two goods consumed Will she ever

consume either good up to the point where its marginal utility is zero?

4 In a Keynesian macroeconomic model of an economy, using the usual terminology,

Y = C + I + G + X − M Yd= (1 − t)Y C = 0.75Yd

M = 0.25Yd I = 820 G = 960 t = 0.3 X = 650

What will be the equilibrium value of Y ? Use the export multiplier to find out what

will happen to the balance of payments if exports exogenously increase by 100

5 A multiplant firm faces the total cost schedule

TC= 850 + 18q1+ 25q2+ 0.6q2

1q2+ 1.2q1q22where q1and q2 are output levels in its two plants What marginal cost sched-ule does it face if output in plant 2 is expanded while output in plant 1 is keptunchanged?

6 In a closed economy (i.e one with no foreign trade) the following relationshipshold:

C = 0.6Yd Yd= (1 − t)Y Y = C + I + G

where C is consumer expenditure, Ydis 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

These represent the marginal product functions for K and L Differentiating these functions

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

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