Lecture 03 differentiation w5

Application Il: Optimisation Cu tri - Stationary points diém dừng ¢ Second-order derivatives dao ham cap hai ° Applications eMarket Equilibrium and Taxation eRevenue Costs and Prof

VU TUAN ANH, MSA, CMA INTERNATIONAL UNIVERSITY, VNU-HCMC, VIETNAM

SEMESTER 2 2019-2020

Application Il: Optimisation (Cu tri)

- Stationary points (diém dừng)

¢ Second-order derivatives (dao ham cap hai)

° Applications

eMarket Equilibrium and Taxation

eRevenue Costs and Profits

Stationary Points

e Stationary points are the turning points or critical points of a function

A

Note: Slope of tangent to curve its zero at stationary points

Inflection Point

Stationary Points

e Classifying stationary points

e Sign of first derivative around a turning point:

Before At After Maximum plus zero minus Minimum minus zero plus

f'(x)=0

Second Order Derivatives

e Derivative of the first derivative —- Rate of Change

dy d“y_ oy

ao Se) dr 2 ƒ (x)

e Classifying stationary points:

: function decreases beyond this point

2

qd*y <0 so point is a local maximum

dx 7

° a2? : function increases beyond this point

—>0 so point is a local minimum

dx

° d*y : class of stationary point

he? =0 jndeterminate

Second-order derivatives

Classification of Turning Points

Application Il: Optimisation (Cu’c tri)

- Maximisation and Minimisation

Practice Problem

1 Find and classify the stationary points of the following functions Hence sketch their

graphs

(a) y= 3x“ + 12x - 35 (b) y =—2x° + 15x* — 36x + 27

Application Il: Optimisation (Cu’c tri)

- Maximisation and Minimisation

Practice Problem

2 A firm’s short-run production function is given by

where L denotes the number of workers Find the size of the workforce that maximises

the average product of labour and verify that at this value of L

MP, = AP,

Application Il: Optimisation (Cu’c tri)

- Maximisation and Minimisation

Practice Problem

3 The demand equation of a good is given by

P+20=20

and the total cost function is

(a) Find the level of output that maximises total revenue

Application Il: Optimisation (Cu’c tri)

- Maximisation and Minimisation

Practice Problem

4 The total cost function of a good is given by

TC =0°+30+ 36 Calculate the level of output that minimises average cost Find AC and MC at this value of O What do you observe?

Practice Problem

5 The supply and demand equations of a good are given by

P='hO, + 25

and

P=-2Q0, + 50

respectively

The government decides to impose a tax, t, per unit Find the value of t which maximises the government’s total tax revenue on the assumption that equilibrium con-

ditions prevail in the market

Application Il: Optimisation (Cu’c tri)

- Maximisation and Minimisation

Practice Problem

1 A monopolist’s demand function is

P=25-0.5Q

The fixed costs of production are 7 and the variable costs are Q + 1 per unit

(a) Show that

TR = 250 — 0.507 and TC=Q°+Q0+7 and deduce the corresponding expressions for MR and MC

(b) Sketch the graphs of MR and MC on the same diagram and hence find the value of

O which maximises profit

Practice Problem

2 A firm has the possibility of charging different prices in its domestic and foreign

markets The corresponding demand equations are given by

Ó, =300 - P,

QO, = 400 — 2P,

The total cost function is

TC = 5000 + 1000

where O=QO,+Q)

Determine the prices (in dollars) that the firm should charge to maximise profits (a) with price discrimination

(b) without price discrimination

Compare the profits obtained in parts (a) and (b)

Application Il: Optimisation (Cu’c tri)

- Maximisation and Minimisation

Practice Problem

3 Calculate the price elasticity of demand at the point of maximum profit for each of the

demand functions given in Practice Problem 2 with price discrimination Verify that the firm charges the higher price in the market with the lower value of LE |