chapter 5: partial differentiation

Chapter 5 Partial Differentiation Chapter 5 Partial Differentiation Nguyen Thi Minh Tam ntmtam vnuagmail com December 1, 2020 1 5 1 Functions of several variables 2 5 2 Partial elasticity and marginal functions 3 5 4 Unconstrained optimisation 4 5 5 Constrained optimisation 5 5 6 Lagrange multipliers 5 1 Functions of several variables Functions of two variables A function f of two variables is a rule that assigns to each ordered pair of real numbers (x,y) a unique real number denoted by f (x,y).

Chapter 5: Partial Differentiation

Nguyen Thi Minh Tam

ntmtam.vnua@gmail.com

December 1, 2020

1 5.1 Functions of several variables

2 5.2 Partial elasticity and marginal functions

3 5.4 Unconstrained optimisation

4 5.5 Constrained optimisation

5 5.6 Lagrange multipliers

5.1 Functions of several variables

Functions of two variables

A function f of two variablesis a rule that assigns to eachordered pair of real numbers (x , y ) a unique real numberdenoted by f (x , y )

We often write z = f (x , y )

x , y are independent variables and z is the dependent variable

In general, a function of n variables can be written

u = f (x1, x2, , xn)

Example 1 Given the function

f (x , y ) = 5x + xy2− 10Evaluate f (1, 2) and f (2, 1)

First-order partial derivatives

Example 2 Find the first-order partial derivatives of the functionsa) f (x , y ) = 5x4− y2

b) f (x , y ) = x2y3− 10x

Second-order partial derivatives

Given a function z = f (x , y ) There are 4second-order partialderivatives

Example 4 Find the second-order partial derivatives of thefunctions

a) f (x , y ) = x2y3− 10x

b) z = x3+ x2y3− 2y2

Note For all functions that arise in economics fxy = fyx

Small increments formula

If x changes by a small amount ∆x and y changes by a smallamount ∆y then the change in z is

∂y at the point (2, 6).

b) Use the small increments formula to estimate the change in z

as x decreases from 2 to 1.9 and y increases from 6 to 6.1.c) Confirm your estimate of part b) by evaluating z at (2, 6) and(1.9, 6.1)

5.2 Partial elasticity and marginal functions

5.2.1 Elasticity of demand

Suppose that the demand Q for a certain good depends on itsprice P, the price of an alternative good PA, and the income ofconsumers Y This relationship can be expressed as the demandfunction

Q = f (P, PA, Y )

Price elasticity of demand

Theprice elasticity of demand is

EP = percentage change in Qpercentage change in P =

P

Q ×

∂Q

∂Pwith PA and Y held constant

Cross-price elasticity of demand

The cross-price elasticity of demandis

If the alternative good issubstitutable, then EPA > 0

If the alternative good iscomplementary, then EPA < 0

Income elasticity of demand

The income elasticity of demandis

If a good is inferior, then demand falls as income rises and

Example 7 Given the demand function

Q = 500 − 3P − 2PA+ 0.01Ywhere P = 20, PA = 30 and Y = 5000, find

a) the price elasticity of demand,

b) the cross-price elasticity of demand,

c) the income elasticity of demand If income rises by 5%,calculate the corresponding percentage change in demand.Would this good be classified as inferior, normal or superior?

Example 8 Given the utility function

U = 1000x1+ 450x2+ 5x1x2− 2x2

1 − x2 2

a) Determine the value of the marginal utilities ∂U

∂x1,

∂U

∂x2 when

x1= 138 and x2 = 500

b) Estimate the change in U if x2 increases by 15

c) Does the law of diminishing marginal utility hold for thisfunction?

Marginal rate of commodity substitution

The marginal rate of commodity substitution, MRCS, is

MRCS = −dx2

dx1 =

∂U/∂x1

∂U/∂x2The increase in x2 necessary to maintain a constant value ofutility when x1 decreases by d unit is

∆x2 ≈ MRCS × d

Example 9 Given the utility function

U = x11/4x23/4a) Find a general expression for MRCS in terms of x1 and x2.b) Calculate the value of MRCS at the point (100, 200) Henceestimate the increase in x2 required to maintain the currentlevel of utility when x1 decreases by 3 units

5.2.3 Production

Given a production function Q = f (K , L)

The marginal product of capitalis MPK = ∂Q

∂K.The marginal product of labouris MPL= ∂Q

∂L.

If capital changes by a small amount ∆K and labour changes

by a small amount ∆L, then the net change in Q is

∆Q ≈ ∂Q

∂K∆K +

∂Q

∂L∆L

An isoquant displays all combinations of two factors which give thesame level of output

Marginal rate of technical substitution

Themarginal rate of technical substitution, MRTS, is

a) Find the values of the marginal products, MPK and MPL.b) Estimate the overall effect on Q when K decreases by 1.5units and L increases by 1.3 units

c) Find the value of the marginal rate of technical substitution(MRTS)

5.4 Unconstrained optimisation

How to find and classify stationary points of a function f (x , y )

1 Solve the simultaneous equations

∂f

∂x = 0,

∂f

∂y = 0

to find the stationary points

2 Classify the stationary point (a, b)

Example 11 A firm has the possibility of charging different prices

in its domestic and foreign markets The corresponding demandequations are given by

Q1= 300 − P1

Q2= 400 − 2P2The total cost function is

TC = 5000 + 100Qwhere Q = Q1+ Q2 Determine the prices that the firm shouldcharge to maximise profit with price discrimination and calculatethe value of this profit

Exercise 5.2, 5.2*2-4, 6, 7, 9, 10 (page 416-417)

Exercise 5.4, 5.4*4-8 (page 444-445) 2, 4 , 5 (page 445-446)

5.5 Constrained optimisation

Problem

Optimize (minimize/maximize) a function

z = f (x , y )subject to

ϕ(x , y ) = M,where M is a constant

z = f (x , y ) is called theobjective function

The method of substitution:

Step 1 Use the constraint ϕ(x , y ) = M to express y in terms of x Step 2 Substitute this expression for y into the objective function

z = f (x , y ) to write z as a function of x only

Step 3 Use the theory of stationary points of functions of one variable

to optimize z

How to find and classify stationary points of a function f (x )

1 Solve the equation f0(x ) = 0 to find the stationary points

2 Classify the stationary point x = a

If f00(a) > 0, then x = a is a minimum.

If f00(a) < 0, then x = a is a maximum.

Example 12 An individual’s utility function is given by

U = x1x2where x1 and x2 denote the number of items of two goods, G1 andG2 The prices of the goods are $2 and $10, respectively

a) Assuming that the individual has $400 available to spend onthese goods, find the utility-maximising values of x1 and x2.b) Verify that the ratio of marginal utility to price is the same forboth goods at the optimum

Example 13 A firm’s production function is given by

Q = 2

√KLUnit capital and labour costs are $4 and $3, respectively Find thevalues of K and L which minimise total input costs if the firm iscontracted to provide 160 units of output

5.6 Lagrange multipliers

TheLagrange multiplier method for solving the problem ofoptimizing a function

f (x , y )subject to

ϕ(x , y ) = M

Step 1 Define a new function

(x , y ) is the optimal solution of the constrained problem The value of λ gives the approximate change in the optimal value of f due to a 1-unit increase in M.

Example 14 A consumer’s utility function is given by

U(x1, x2) = 2x1x2+ 3x1,where x1 and x2 denote the number of items of two goods G1 andG2 that are bought Each item costs $ 1 for G1 and $2 for G2.a) Use Lagrange multipliers to find the maximum value of U ifthe consumer’s income is $83

b) Estimate the new optimal utility if the consumer’s incomerises by $1

Example 15 (3 page 469) A firm’s production function is given by

Q = KLUnit capital and labour costs are $2 and $1, respectively Find themaximum level of output if the total cost of capital and labour is

$6

Exercises

Exercise 5.5, 5.5*4, 7, 8 (page 457-458), 2, 3, 5ab (page 458-459)

Exercise 5.6, 5.6*4, 5, 6 (page 469-470) 2, 6 (page 470-471)