Prepared by finance department

7/29/2020 B03013 Chapter 1: One-variable – calculus: Applications 23 Cost function The cost function, Cq, gives the relationship between total cost and quantity produced.. SMCq=slope

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Key words

• Variable: biến

• Application: ứng dụng

• Function: hàm

• Revenue: Doanh thu

• Linear function: Hàm tuyến tính

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Key works

• Dictatorial: độc tài, độc đoán

• Treat: giải quyết, bàn về

Introduce the general cost function and a

derived cost function with one variable

Identify revenue and profit function with

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3 • Cost function and its application

4 • Revenue and profit functions in the

one variable world

5 • Applications

7/29/2020 B03013 Chapter 1: One-variable –

calculus: Applications 5

CONCEPTS

• One variable calculus: linear function

• One variable input

• Elasticity

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One-variable function

A linear function:

• y=f(x)=a+bx

• One independent variable (x)

• One dependent variable (y)

• a: constant term/intercept

• b: coefficient/slope and given rate of change

Graph: 2 points satisfied the equation and

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• Production function: labor input

• Cost function- variable cost

• Revenue/profit function: number of items sold

• Demand function: price per unit

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Production function

• Managers: not only what to produce for the market,

but also how to produce it in the most efficient or

least cost manner

• Economics offers widely accepted tools for judging

whether the production choices are least cost

• A production function relates the most that can be

produced from a given set of inputs

• Production functions allow measures of the marginal

product of each input

• A Production Function is the maximum quantity

from any amounts of inputs

• If L is labor and K is capital: the Cobb-Douglas

Production Function- a popular function:

Q = a • K b1 • L b2

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• The number of inputs: often large Economists

simplify by suggesting some, like materials or labor,

is variable, whereas plant and equipment is fairly

fixed the short run in

• A Production Function: only one variable input, labor,

is easily analyzed The one variable input is labor, L

• Q = f ( , L) K for two inputs case, where K asFixed

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•output attributable to last unit of labor applied

• Similar to profit functions, the Peak of MP occurs

before the Peak of average product

• When MP = AP, we’re at the peak of the AP curve

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• The production elasticity of labor,

- EL = MPL/APL=(∆Q/∆L)/(Q/L) =(∆Q/∆L)(L/Q)

- The production elasticity of capital has the identical

in form, except K appears in place of L

• When MP > AP , then the labor elasticity, E >1 L L L

a 1 percent increase in labor will increase output by

more than 1 percent

• When MPL<APL, then the labor elasticity, E <1 L

a 1 percent increase in labor will increase output by

less than 1 percent

• When MP > AP, then AP is RISING

• When MP < AP, then AP is FALLING

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• Avoidable costs are costs that need not be

incurred (can be avoided)

• Fixed costs do not vary with output

• Variable costs change with output

Cost function

The cost function, C(q), gives the relationship

between total cost and quantity produced

• The variable cost, C(q) function is:

C(q)=the minimum variable cost of producing q

units of output

200 1 if 000 1 200 1 72 2

314

200 1 800 if 000 1 800 128 263

800 0 if 000

1

160 135

, q ,

) , q (

, q ,

) q (

q ,

q )

q

(

C

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Cost function

• Average variable cost is variable cost per unit

of output

AV(q)=C(q)/q

Short-run marginal cost is the rate at which costs

increase in the short-run

SMC(q)=slope of C(q)

Cost function

Relation between short-run Marginal Costs and

Average Variable Costs

1 When SMC is below AVC, AVC decreases as

q increases

2 When SMC is equal to AVC, AVC is constant

(its slope is zero)

3 When SMC is above AVC, AVC increases as

q increases

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Revenue and profit function

• Most economists treat the firm as a single

decision-making unit

• The decisions are made by a single dictatorial

manager who rationally pursues some goal

• Usually profit-maximization

• A profit-maximizing firm chooses both its inputs and

its outputs with the sole goal of achieving maximum

economic profits

• seeks to maximize the difference between total

revenue and total economic costs

• If firms are strictly profit maximizers, they will make

decisions in a “marginal” way

• examine the marginal profit obtainable from producing

one more unit of hiring one additional laborer

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• Total revenue for a firm is given by

R(q ) = (pq) q

• In the production of , certain economic costs q

are incurred [C(q)]

• Economic profits ( ) are the difference

between total revenue and total costs

( q) = R(q) – C q) = p(q) q –C(q) (

• The necessary condition for choosing the level

of q that maximizes profits can be found by

setting the derivative of the function with

respect to q equal zeroto

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• To maximize economic profits, the firm should

choose the output for which marginal revenue

is equal to marginal cost

MC dq

dC dq

dR MR

• MR = MC is only a necessary condition for

profit maximization

• For sufficiency, it is also required that

• “marginal” profit must be decreasing at the

optimal level of q

7/29/2020 B03013 Chapter 1: One-variable calculus: Applications – 32

0)('

*

* 2

2

q q q

q d dq

d

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• If a firm can sell all it wishes without having any

effect on market price, marginal revenue will be

equal to price

• If a firm faces a downward-sloping demand

curve, more output can only be sold the firm if

reduces the good’s price

dq

dp q p dq

q q p d dq

dR q

) ( revenue

marginal

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• If a firm faces a downward-sloping demand

curve, marginal revenue will be a function of

output

• If price falls as a firm increases output,

marginal revenue will be less than price

• Suppose that the demand curve for a sub

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• To determine the profit-maximizing output, we

must know the firm’s costs

• If subs can be produced at a constant average

and marginal cost of $4, then

Suppose that the demand curve for a sub

sandwich is q = 300 – 20p Calculate the

revenue, marginal revenue

To determine the profit - maximizing, How many

product is produced If subs can be produced at

a constant average and marginal cost of $6

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Example 2

Suppose that the demand curve for a sub

sandwich is q = 200 – 30p Calculate the

revenue, marginal revenue

To determine the profit - maximizing, How many

product is produced If subs can be produced at

a cost function (C= 5q+300)

• The concept of marginal revenue is directly

related the elasticity the demand curve facing to of

the firm

• The price elasticity of demand is equal to the

percentage change quantity that results from a in

one percent change price in

q

p dp

dq p dp

q dq

eq p

/ /

,

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• This means that

• if the demand curve slopes downward, e < 0 and q,p

MR < p

• if the demand is elastic, eq,p < -1 and marginal

revenue will be positive

• if the demand is infinitely elastic, e q,p= - and

marginal revenue will equal price

p

e p dq

dp p

q p dq dp q p

MR

,

1 1 1

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• A firm’s profit function shows its maximal profits

as a function of the prices that the firm faces

• Homogeneity

• the profit function is homogeneous of degree one in

all prices

• with pure inflation, a firm will not change its production

plans and its level of profits will keep up with that inflation

])

,([),()

• Nonincreasing in input prices

• if the firm responded to an increase in an input price by

not changing the level of that input, its costs would rise

• profits would fall

• Convex in output prices

• the profits obtainable by averaging those from two

different output prices will be at least as large as those

obtainable from the average of the two prices

w v p p w v p w

v

p

, , 2 2

) , , ( ) ,

,

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• We can apply the envelope theorem to see

how profits respond to changes in output and

input prices

),,(),,(

w v p q p w v p

),,(),,(

w v p k v

w v p

),,(),,(

w v p w

w v

• Because the profit function is nondecreasing in

output prices, we know that if p2> p1

(p2,…) (p1,…)

• The welfare gain to the firm of this price

increase can be measured by

welfare gain = (p2,…) - (p1,…)

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• A firm’s output is determined by the amount of

inputs it chooses to employ

• the relationship between inputs and outputs is

summarized by the production function

• A firm’s economic profit can also be expressed

as a function of inputs

(k,l) = pq –C(q) = pf(k,l) – ( vk + wl)

• The first-order conditions for a maximum are

/ k = p[ f/ k] – v = 0 / l = p[ f/ l] – w = 0

• A profit-maximizing firm should hire any input

up to the point at which its marginal

contribution to revenues is equal to the

marginal cost of hiring the input

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• These first-order conditions for profit

maximization also imply cost minimization

• they imply that RTS = w/ v

• To ensure a true maximum, second-order

conditions require that

kk = f < 0 kk ll = f < 0 ll

kk ll - kl2 = f kk fll– f kl2 > 0

• capital and labor must exhibit sufficiently

diminishing marginal productivities so that marginal

costs rise as output expands

• In principle, the first-order conditions can be

solved to yield input demand functions

Capital Demand = k(p,v,w)

Labor Demand = l(p, , v w)

• These demand functions are unconditional

• they implicitly allow the firm to adjust its output to

changing prices

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• We expect l/ w 0

• diminishing marginal productivity of labor

• The first order condition for profit maximization

f p

ll

1

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Demand function

• General Form: Q = a – bP

• Why is this the general form?

• Changes in quantity demanded: movement

along a given demand curve reflecting a

change in price and quantity

• Shift in demand – Switch from one demand

curve to another following a change in a

non-price determinant of demand

• If an independent variable changes, other than

price of the good, you must draw a new

demand curve!!!

Demand function

• Demand Sensitivity Analysis: Elasticity

• Price Elasticity of Demand

• Cross Price Elasticity of Demand

• Income Elasticity of Demand

• Additional Demand Elasticity Concepts

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Demand function

• Elasticity – The percentage change in a

dependent variable resulting from a 1% change

• Elasticity = Percentage Change in Quantity

(Sales) / Percentage Change in (X)

• Percentage change = (X2-X )/X1 1

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Demand function

• Price Elasticity of Demand (Own-Price):

• Measure of the magnitude by which

consumers alter the quantity of some

product they purchase in response to a

change in the price of that product

• Responsiveness of the quantity demanded

to changes in the price of the product,

holding constant the values of all other

variables in the demand function

• Estimating from the Demand Function

• Estimating from the Demand Curve

1 Order of events dictate outcomes

2 Does not impose ceteris paribus

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• Note: If you divide by a fraction you

multiply by the reciprocal

• This is the inverse slope of the demand curve,

which we can estimate empirically (via basic

econometrics), and therefore we can impose

ceteris paribus

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How does elasticity vary along a linear demand curve?

• The upper half of a linear demand curve is elastic

• The lower half of a linear demand curve is inelastic

• BE ABLE TO EXPLAIN WHY!!!

• The search for substitutes as price increases

• Big number, small number explanation

• Calculating elasticity at the midpoint

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decreasing the price will increase TR

and marginal revenue must be positive

• If % change in Q < % change in P

increasing the price will increase TR

and marginal revenue must be negative

• be able to illustrate the relationship

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Exercise

1 Show that the function F(x) =x + x +1 has the 3

essential properties of a cost function Carefully

graph corresponding average cost function and

marginal cost function

2 What happens to a competitive firm whose cost

function exhibits decreasing marginal cost

everywhere? Construct a concrete cost

function of this type and carry out the research

for the profit- maximizing output?

3 The linear demand function x= a-b Prove that p

the point elasticity is -1 exactly at the midpoint

of the linear demand

Exercise

1) Show that the function F(x) =x + x +1 has the 3

essential properties of a cost function Carefully graph

corresponding average cost function and marginal

cost function

2) Suppose that the demand curve for a sub sandwich

is q = 200 – 15p Calculate the revenue, marginal revenue?

Suppose the cost function is C= 5q+10 Calculate the average

cost, marginal cost?

The profit - maximizing, How many product is produced If subs

can be produced at the cost function above

Tiêu đề	One-variable Calculus: Applications
Tác giả	Finance Department
Chuyên ngành	Calculus
Thể loại	Chapter
Năm xuất bản	2020

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Số trang	33
Dung lượng	3,01 MB