Extra Session-Numerical Problems With Calculus

Economic Analysis Economic Analysis for Businessfor Business

Numerical Problems for Numerical Problems for practicepracticeInstructorInstructorSandeep BasnyatSandeep [email protected][email protected]

Solved ProblemsSolved ProblemsConsider a market with Demand curve q = 16 − 10p and Supply curve q = −8 + 20p. (Here q is in millions of pounds per day and p is in dollars per pound.)

(a)Determine the market equilibrium price and quantity and the total revenue in this market.

(b) Calculate the price elasticity of demand and the price elasticity of supply at the market equilibrium.

Solved ProblemsSolved ProblemsConsider a market with Demand curve q = 16 − 10p and Supply curve q = −8 + 20p. (Here q is in millions of pounds per day and p is in dollars per pound.)

(a) Determine the market equilibrium price and quantity and the total revenue in this market.Simultaneously solving the demand and supply equations:

p = $0.80 per pound and q = 8 million pounds per day.

Total revenue is: p x q = 0.8 x 8 = $6.4 million per day.

(b) Calculate the price elasticity of demand and the price elasticity of supply at the market equilibrium.

The slope of the demand curve is −10, so the price elasticity of demand at the market equilibrium is −10.(0.8/8) = −1.

Similarly, the slope of the supply curve is 20, so the price elasticity of supply at the market equilibrium is 20.(0.8/8)= 2.

(c) Imagine that the government imposes a $0.60 per-unit tax on the buyers. Write down the new market supply and demand curves, and find the new market equilibrium price and quantity. How much of the tax burden is borne by the buyers, and how much by the sellers?

Calculate the ratio of these tax burdens, and compare with the ratio of the elasticities calculated above.

Solved ProblemsSolved Problems

(c) Imagine that the government imposes a $0.60 per-unit tax on the buyers. Write down the new market supply and demand curves, and find the new market equilibrium price and quantity. How much of the tax burden is borne by the buyers, and how much by the sellers?

Calculate the ratio of these tax burdens, and compare with the ratio of the elasticities calculated above.

A tax of $0.60 on the buyers will change the demand curve to

q = 16 − 10(p + 0.6), i.e., q = 10 − 10p.

But, the supply curve is still q = −8 + 20p, so the new market equilibrium is at

p = $0.60 per pound and q = 4 million pounds per day.

Therefore, the buyers end up paying (P + T) =0.60 + 0.60 = $1.20 per pound.

The sellers get $0.60 per pound (Amount buyers pay).

Note: The original equilibrium was at a price of $0.80 per pound, so the buyers pay $0.40 more and the sellers end up getting $0.20 less.

The ratio of these tax burdens is (.40 /.20) = 2.


(d) Now imagine that the government instead decides to impose a $0.60 per-unit tax on the sellers. How will this change things? Write down the new market supply and demand curves, find the new market equilibrium price and quantity, and compare with your answer from above (where the tax is on the buyer)


Some variations-Solved ProblemSome variations-Solved ProblemMarket Demand and Supply Case:

Given:

500 consumers with individual demand curves of qi = 15−p

300 consumers with individual demand curves of qi = 30−2p

Total demand from the market is: qM = 16500 − 1100p.

Find:

a) The total quantity buyers want to buy at price of $10:

b) The quantity and price each of the 500 and 300 buyers want to pay.

Some variations-Solved ProblemSome variations-Solved ProblemMarket Demand and Supply Case:

Given:

500 consumers with individual demand curves of qi = 15−p

300 consumers with individual demand curves of qi = 30−2p

Total demand from the market is: qM = 16500 − 1100p.

Find:

a) The total quantity buyers want to buy at price of $10:

At a price of $10, the buyers want to buy 16500− 1100 ・ 10 = 5500 units.

b) The quantity and price each of the 500 and 300 buyers want to pay.

Each of the 500 buyers with an individual demand curves of

qi = 15 − p wants to buy 15 − 10 = 5 units, for a total of 2500.

And each of the 300 buyers with individual demand curves of qi = 30 − 2p wants to buy 30 − 2 ・ 10 = 10 units, for a total of 3000.

Sales Tax on Buyers and Sellers case: Consider a world with 300 consumers, each with demand curve

q = 25−2p, and 500 suppliers, each with supply curve q = 5 + 3p. The government decides to impose a 50% sales tax on the sellers. Find the new market equilibrium price and quantity.

Some variationsSome variations

Sales Tax on Buyers and Sellers case: Consider a world with 300 consumers, each with demand curve

q = 25−2p, and 500 suppliers, each with supply curve q = 5 + 3p. The government decides to impose a 50% sales tax on the sellers. Find the new market equilibrium price and quantity.

Solution:With a 50% sales tax on the sellers, the market

supply curve becomes q = 2500+1500(.5p),i.e., q = 2500 + 750p.

The demand curve is, as originally, q = 7500 − 600p.

So the new equilibrium is at: p = 500 / 135 ≈ $3.70q ≈ 5278.

Some variationsSome variations

Worked out ProblemWorked out ProblemTC = 1000 + 10Q - 0.9Q2 + 0.04Q3

Find:

1)MC, TVC, AVC and Minimum AVC

Worked out ProblemWorked out ProblemTC = 1000 + 10Q - 0.9Q2 + 0.04Q3

1) MC = ΔTC / ΔQ = d(TC) / dQ

= 10-1.8Q+ 0.12Q2

2) TVC = TC –TFC

= 1000 + 10Q - 0.9Q2 + 0.04Q3 – 1000

= 10Q - 0.9Q2 + 0.04Q3

3) AVC = TVC / Q =(10Q - 0.9Q2 + 0.04Q3 )/Q

= 10 - 0.9Q + 0.04Q2

4) Minimum AVC occurs at the intersection of AVC and MC.

So, AVC = MC

10 - 0.9Q + 0.04Q2 = 10-1.8Q+ 0.12Q2

Or, - 0.08Q2 + 0.9Q = 0

Or, Q(- 0.08Q+ 0.9) = 0

Or, Q =0 and - 0.08Q+ 0.9 = 0 i.e, Q = 11.25 (Minimum AVC)

Market Structure ProblemsMarket Structure Problems

1) Assume the cost function: TC = 1000 + 2Q + 0.01Q2 and Price is $10 per unit.

Calculate the profit maximizing output and economic profit.


1) Assume the cost function: TC = 1000 + 2Q + 0.01Q2 and Price is $10 per unit.

Calculate the profit maximizing output and economic profit.

Solution:

MC = dTC /dQ = 2+0.02Q

In a perfectly competitive market, profit maximizing output is at where P = MC

10 = 2+0.02Q

Therefore, Q = 400

Economic Profit = TR –TC = 10(400) – (1000 + 2(400) + 0.01(4002)) =$600

2) Suppose the Total Cost for the Monopoly(TC) = 500 + 20Q2

Demand Equation (P) = 400 – 20Q

Total Revenue (TR) = 400Q – 20Q2

What is the profit maximizing price and quantity?


2) Suppose the Total Cost for the Monopoly(TC) = 500 + 20Q2

Demand Equation (P) = 400 – 20Q

Total Revenue (TR) = 400Q – 20Q2

What is the profit maximizing price and quantity?

Solution:

MR = dTR / dQ = 400 -40Q

MC = dTC / dQ = 40Q

Profit Maximizing price is achieved when MR =MC

Or, 400 – 40Q = 40Q

Therefore, Q =5 (Profit maximizing output)

Putting the value of Q in demand equation

Profit Maximizing Price P = $300.


3) Consider a firm which has a horizontal demand curve. The firms Total Cost is given by the function:

TVC = 150Q – 20Q2 +Q3. Below what price should the firm shut down operation?


3) Consider a firm which has a horizontal demand curve. The firms Total Cost is given by the function:

TVC = 150Q – 20Q2 +Q3. Below what price should the firm shut down operation?

Solution:

In the competitive market, the shut down condition is when Price (P) = Minimum Average Variable Cost

But Profit maximization theory require P =MC

MC =dTVC / dQ = 150 -40Q +3Q2

AVC = TVC /Q = (150Q – 20Q2 +Q3) / Q = 150 -20Q +Q2

Equating, both equations:

MC = AVC or 150 -40Q +3Q2 = 150 -20Q +Q2

Or, 2Q2 – 20Q = 0 or 2Q (Q – 10) = 0

Or, Q = 0 and Q = 10

Substituting Q = 10 into marginal cost, P = MC = 150 – 40(10) + 3 (100) = $50

Similarly, substituting Q = 0 in the marginal cost, P = $150

Therefore, if the price falls below $50, the firm shuts down.


4) Consider a monopolist sells in two markets and has constant marginal cost equal to $2 per unit. The demand and marginal revenue equations for two markets are:

PI = 14 -2QI : MRI = 14 -4QI

PII= 10 –QII : MRIi = 10 -2QI

Using third degree discrimination, find profit maximizing prices and quantities.

Note:

1st degree: Charging maximum price for each unit sold.

2nd degree: Different prices depending upon quantities of goods bought by consumers.

3rd Degree: Separating consumer market and charge separate prices.


Solution:

Marginal Cost =$2 per unit. The demand and marginal revenue equations for two markets are:

PI = 14 -2QI : MRI = 14 -4QI

PII= 10 –QII : MRIi = 10 -2QI

Profit maximizing is possible when MRI = MRII =MC

So, for Market I: 14 -4QI = 2. Therefore, QI = 3

For Market II: 10 –2QII = 2. Therefore, QII = 4

Substituting values of QI and QII in PI and PII, we have

PI = 8 and PII = 6


More Practice questions:1) Studies indicate that the price elasticity of demand for cigarettes is 0.4. If a pack of cigarettes currently costs $2 and the government wants to reduce smoking by 20 percent:a) By how much (percentage) should it increase the price?b) What would be the price of a pack of cigarettes after the increase in price (use midpoint method to calculate the price)

Answer :1) Studies indicate that the price elasticity of demand for cigarettes is 0.4. If a pack of cigarettes currently costs $2 and the government wants to reduce smoking by 20 percent:a) By how much (percentage) should it increase the price? (Ans: 50%)b) What would be the price of a pack of cigarettes after the increase in price (use midpoint method to calculate the price) (Ans: $3.33)

Price

($)

Demand

(millions)

Supply

(millions) 60 22 14 80 20 16100 18 18120 16 20

(2)

• Calculate the price elasticity of demand when the price is $80

• Calculate the price elasticity of supply when the price is $80

• What is the equation for the demand curve? • What is the equation for the supply curve?

Price

($)

Demand

(millions)

Supply

(millions) 60 22 14 80 20 16100 18 18120 16 20

(2)

• Calculate the price elasticity of demand when the price is $80 (Ans. : - 0.4 )

• Calculate the price elasticity of supply when the price is $80 (Ans. : 0.5)• What is the equation for the demand curve?(Ans.: Q = 28 - 0.1p)• What is the equation for the supply curve? (Ans.: Q = 8+0.1p)

For the following equations determine whether the demand is elastic, inelastic or unitary elastic at the given prices. (5+5)◦Q = 100 – 4P and P = $20◦Q = 1500 - 20P and P = $5

(3)

For the following equations determine whether the demand is elastic, inelastic or unitary elastic at the given prices. (5+5)◦Q = 100 – 4P and P = $20 (Ans.: Elastic)

◦Q = 1500 - 20P and P = $5 (Ans. : Inelastic)

(3)

Assume that in the short run, a firm is operating in a competitive market for computer keyboard at a price level of Rs. 22 each. The firm’s marginal cost curve is given by MC=2+4Q where Q is the firm’s output. Average fixed costs are given by 20/Q and average variable costs by 2+2Q.

a) what level of output does the firm maximize profits?

b) At this profit maximizing output level, calculate average total costs and the value of maximum profit.

c) Below what price would the firm shut down in the short run and in the long run?

(4)

Assume that in the short run, a firm is operating in a competitive market for computer keyboard at a price level of Rs. 22 each. The firm’s marginal cost curve is given by MC=2+4Q where Q is the firm’s output. Average fixed costs are given by 20/Q and average variable costs by 2+2Q.

a) what level of output does the firm maximize profits?

(Ans. Q = 5)

b) At this profit maximizing output level, calculate average total costs and the value of maximum profit.

(Ans. ATC = ATC + AFC = Rs. 16)

c) Below what price would the firm shut down in the short run and in the long run?

(Ans. Short run (AVC) = Rs. 12; Long run (ATC) = Rs. 16)

(4)

Consider the following table of long-run total cost for three different firms:

QUANTITY 1 2 3 4 5 6 7

Firm A $60 $70 $80 $90 $100 $110 $120

Firm B 11 24 39 56 75 96 119

Firm C 21 34 49 66 85 106 129

Does each of these firms experience economies of scale or diseconomies of scale?

(5)

QUANTITY 1 2 3 4 5 6 7

Firm A $60 $70 $80 $90 $100 $110 $120

Firm B 11 24 39 56 75 96 119

Firm C 21 34 49 66 85 106 129

Does each of these firms experience economies of scale or diseconomies of scale?

Firm A: Economies of Scale – ATC fallingFirm B: Diseconomies of Scale – ATC risingFirm C: Economies and then Diseconomies of Scale

(5)

Utility FunctionUtility FunctionTwo ways to represent consumer

preferences:◦ Indifference Curve◦ Utility

Utility is an abstract measure of the satisfaction that a consumer receives from a bundle of goods.

Indifference Curve and Utility are closely related.◦ Because the consumer prefers points on higher

indifference curves, bundles of goods on higher indifference curves provide higher utility.

If the individual’s utility function is U(L,K) = L1/2K1/2 at the utility level of 2 then indifference curve corresponding to it is the set of all consumption bundles that provide the individual with a utility of 2.Therefore, the equation for this indifference curve is L1/2K1/2 = 2Squaring both sides by 2 and rewriting the equation:

LK = 4 or K = 4L-1

The slope of this indifference curve,

Utility Function and Indifference Utility Function and Indifference CurveCurve

dK

dL= - 4L-2 (Marginal rate of substitution (MRS)

between two goods L and K)

The slope of this indifference curve,


dK

dL= - 4L-2 (Marginal rate of substitution (MRS)

between two goods L and K)

An individual with a utility level of 2 who currently has L and K goods would be willing to trade up to 4L−2 pieces of K in order to gain an extra L. Such a substitution would leave the individual on the same indifference curve, and therefore with the same utility.

ULUK

MRS

Marginal utility of L (MUL =∂U / ∂L), the extra utility the individual would get from an additional L

Marginal utility of cake (MUK =∂U / ∂K), the extra utility the individual wouldget from an additional K

=


ULUK

MRS=

Since the slope of this indifference curve is MRS,

= Slope of this indifference curve

Production functions and Production functions and IsoquantsIsoquantsGiven inputs of labor (L) and capital (K), the production function f(L,K) describes the quantity of output that can be produced from these inputs.

If the firm’s production function is Y = L1/2K1/2,

then the Isoquant corresponding to an output level of, say, 2 is the set of all input bundles that the firm can use to produce 2 units of output.

Therefore, the equation for this Isoquant is L1/2K1/2 = 2 Squaring both sides by 2 and rewriting the equation:

LK = 4 or K = 4L-1

The slope of this Isoquant,

dK

dL= - 4L-2 (Marginal rate of technical substitution

(MRTS) between two Labour L and Capital K)

A firm with an output target of 2 which currently has L units of labor and K units of capital would be willing to trade up to - 4L-2 units of capital in order to gain an extra unit of labor. Such a substitution would leave the firm on the same isoquant, and therefore with the same output.

Production functions and Production functions and IsoquantsIsoquants

fLfK

MRTS

Marginal product of Labour (MPL = ∂f / ∂L), the extra output the firm would get from an additional unit of labor

Marginal product of capital (MPK = ∂f / ∂K) the extra output the firm would get from an additional unit of capital.

=

fLfK

MRTS = = Slope of this isoquant

Some formulae relating to Production Some formulae relating to Production FunctionFunctionTotal Product of Labour function (TPL) = f (K, L);

(keeping K constant)

Total Product of Capital Function (TPK)= f (K, L); (keeping L constant)

Average Product of Labour (APL) = TPL / L

Average Product of Capital (APK) = TPL / K If the Production Function is Q = AKαLβ then,

Marginal Product of Labour (MPL) = dQ / dL = βAKαLβ-

1

Marginal Product of Capital (MPK) = dQ / dK = αAKα-1Lβ

Value of Marginal Product of Labour (VMPL) = P x MPL

Worked out examplesWorked out examplesFor the Production function: Q = 10K0.5L0.5 whose

labour input (L) is 4, selling price per unit is Rs. 5, capital (K) is fixed 9 units calculate: TPL, APL, MPL, VMPL

Solution: Here, (TPL) = Q = f (K, L) = 10K0.5L0.5

= 10 x 90.5 x 40.5

= 10 x 3 x 2 = 60APL = TPL / L = 60 / 4 = 15

MPL = dQ / dL = βAKαLβ-1

= 10 (1/2)K0.5L0.5-1 =5(K0.5 / L0.5)= 5 x (3/2) = 7.5

VMPL = P * MPL = 5 x 7.5 = Rs. 37.5

More FormulaeMore Formulae Marginal Revenue Product (MRPL): Value of the extra unit of

labour hired. It determines how much labour is hired. Thus the labour is hired until MRPL equals the wage rate (w). That is,

MRPL = w

Similarly, the in case of capital, capital could be employed until MRPK equals the price of the capital (r).

That is, MRPK = r

Similarly, MRPL = MR . MPL

If Price P is constant, then P = MR. In that case, MRPL = P. MPL

We know, MR = ΔTR /ΔQ and MPL = ΔQ / ΔL

Therefore, MRPL = MR . MPL

= ( ΔTR /ΔQ).( ΔQ / ΔL)

= ΔTR / ΔL

MRPL = d (TR) / dL

Worked out exampleWorked out exampleFor the Production function: Q = 10K0.5L0.5 whose, capital (K) is fixed 9 units, selling price per unit is Rs. 5 and wage rate is Rs. 3, calculate how many labour should be hired or find out he profit maximizing rate of labour hired.

Solution:MRPL = P. MPL = P. dQ / dL = P. βAKαLβ-1

= 5.10 (1/2)K0.5L0.5-1 =25(K0.5 / L0.5)= 25 x (3/L) = 75/ L0.5

Total labour hired until:MRPL = w

75/ L0.5 = 3L = 252 = 625Therefore total no. of labour hired is 625.

Worked out examplesWorked out examplesJane receives utility from days spent traveling on vacation domestically (D) and days spent traveling on vacation in a foreign country (F), as given by the utility function U(D,F) = 10DF. In addition, the price of a day spent traveling domestically is $100, the price of a day spent traveling in a foreign country is $400, and Jane’s annual travel budget is $4,000.Find Jane’s utility maximizing choice of days spent traveling domestically and days spent in a foreign country.

Solution:

The optimal bundle is where the slope of the indifference curve is equal to the slope of the budget line, and Jane is spending her entire income. The slope of the budget line is:

-PD / PF = - ¼ and the slope of the indifference curve is

MRS = - MUD / MUF = - 10F/10D = - F/D

Setting the two equal we get: F / D = ¼ or, 4F = D

From the above question: 100 D + 400 F = 4000

Solving the above two equations gives D=20 and F=5. Utility is 1000.

Thank youThank you

Extra Session-Numerical Problems With Calculus

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