Managerial exercises

Utility function exercise

Cobb Douglas

Question )( , = 1 2 1 2 = = 1

Old Price: 1 21′ = 2

New price: = 10

Total income:

Solution

This is a Cobb Douglas utility function, corresponding to this is the square root of the utility function 1 1( ) ( )2 2)( ; = 1 2 1 2( , , ) =2 1 2 22( , , ) = 2 1 2 2

110̂1∗ = 5; = = 2.5 1 2∗

210̂2∗ = 5 ; = =5 2 2∗

115̃ = = 3.75 1 2∗

215̃ = = 7.5 2 2∗1

Now we can compute the price effect and the subs effect.

̂ ∗ (2.5− = − 5)(5 − 5) = (−2.5; 0)

Price effect on good 2 is 0.

̃ ∗ (− = 3.75 − 5 ; 7.5 − 5) = (−1.25; 2.5)

We now can proceed to calculate the income effect.

̂ ̃ (2.5− = − 3.75; 5 − 7.5) = (−1.75; −2.5)

Leontev/Leontief function

)( , = min( , )1 2 1 2 = = 1

Old Price: 1 21′ = 2

New price: = 10

Total income:

Solution

This is continuous but not differentiable. In this case you use = 1 2 ( (→ , , ) = , , ) =1 1 2 2 1 2 + 1 2

101∗ 2∗ = = =5 1+1 10

10̂ ̂= = =1 2 2+1 3 10 10 5 5

̂ ∗− = ( − 5; − 5) = (− ; − )3 3 3 3

15̃ ̃= = 5 =1 2 2+1

̃ ∗ (5 (0;− = − 5; 5 − 5) = 0)

Subs effect is 0 and all the price effect is income effect.

10 10 5 5̂ ̃− = ( − 5; − 5) = (− ; − )3 3 3 3

Endowment (only Leontief)

)( , = min( , )1 2 1 2

To solve it you set up = 1 2

Substitute this to the budget constraints + = + 1 1 2 2 1 1 2 2

Then we have + 1 1 2 2

1∗ 2∗( ) , = =1 2 + 1 2

Suppose we are initially endowed with good 1 or leisure of 15 then we have our demand of leisure or consumption goods will be 15.

1( ) ( ) , = = , 1 1 2 2 1 2 + 1 2

This is only possible because our Leontief function is the simplest possible one without m.

P1 is the price of leisure, so what is the reaction of demand of leisure to this price effect. It is positive. You can say that by computing the derivative of leisure demand.

( ) 5 ∗ + − 1 ∗ 15 1 1 2 1= 2( ) + 1 1 2

15 + 15 − 15 15 1 2 2 2= >0 2 2( ) ( )+ + 1 2 1 2

So, the demand of leisure increase when price increase. So, leisure is behaving as a Giffen good. Labour supply is now decreasing in wage. How can we write labour supply from here? Labour supply is what we don’t consume from the initial endowment.

15 15 + 15 − 15 15 1 1 2 2 2 = 15 − = = + + + 1 2 1 2 1 2

Elasticity - Markup

1. Chapman cosmetic and perfume product

Chapman cosmetic and perfume product, the manager raise the price of a line of mascara product from 9 to 12 following increases in the cost of labour and material.

= 9 1

= 12 2

Unfortunately, sales dropped.

(9) = 16200

(12) = 9000

To regain sale, Chapman ran a coupon promotion 5$ off the regular price. The coupon printing and distribution cost: = 500/ℎ

Represented a substantial increase over the normal advertising cost of 3250/month.

Despite this added cost, the promotion was judged to be a success because it stimulates the consumer, 40% of purchase used coupon with a total sale of 15000.

Calculate the arc price elasticity.

9000 − 16200
9000 + 16200
2

12 − 9
12 + 9
2

= −2

If you increase the price by 1 %, sales decrease by 2%.

Advertising cost:

15000 − 9000
15000 + 9000

3750 − 3250
3750 + 3250

= 3.5

This is the arc elasticity with sale respect to advertising.

If you have an increase of 1% in ad cost you have 3.5 % increase in sale.

Interpretation: the price elasticity changes along the range.

2. Last prob of chap 4: Optimal pricing

Ford offer 1% discount in SUV, the consumer increase sale by 10%.

Calculate point price elasticity.

The elasticity 10% = −10

−1%

= 23500

= 350

3. Markup

In order to use excess end-of-the-year inventory, Henry Ford offered 1% discount of the avg price of the car sold. Customers responds were enthusiastic with unit sales rising by 10% over the period. Calculate the point-price elasticity demand (we did last time).

The increase of sale is the elasticity of demand.

(10)10 = 10

−1

The elasticity due to percentage changes. Then we have to calculate the maximizing price per unit.

Original Costs 23500$ and incurred marginal selling cost of 350/unit -> New marginal producing cost C = 23850.

Calculating the mark-up.

1 1̅ = = 1.1

1− 10

You are going toward each other regime. You are going toward perfect competition.

Why elasticity so high? Because there are many substitutes, so this is a monopolistic competition rather than a pure monopoly. And you have many types of substitutable, which mean the elasticity may rise to a very high level. You have a very low mark-up, which mean you can increase your marginal cost and increase them by 11% to get your optimal price.

10∗ = ∗ 23850 = 26500

94

BBline

BBline is a retailer of a wide variety of sporting good products. Although the customer respond to the spring catalogue was actually good, sales of BBline declines from 10000 units to 4800 units. During this period, a competitor offered a 52 dollars of the regular 157 price of the deluxe garment bag.

First point is to compute the arc cross price elasticity for this bag. We have to compute the ratio between the decrease in sale and the percentage increase in the other bag price then have to calculate the mark-up.

4800 − 10000
4800 + 10000

= 1.5

52− 15 + 137

The cross-price arc elasticity is equal to 1.5.

If the price of competitor go up by 1 % our sale go up by 1.5 % and vice versa, which is what happen in average so that’s why were talking about arc elasticity and not point elasticity.

2nd point

BBline recover sale.

Price reduction 140 -> 130

Sales recovery 4800 -> 6000

Calculate BBline arc price elasticity for its product.

6000 − 4800
6000 + 4800

= −3

130 − 140
130 + 140

3rd point

Assuming the same arc price elasticity of b, calculating the price reduction BBline has to make to regain lost sale = −3 (6000 → 10000).

We need to compute the percentage increase in sale that we need, in order to obtain the same arc price elasticity, we know percentage change in sale will be:

%∆ = ∗ %∆

We must compute the increase of percentage in sale that we want to obtain.

∆ 10000 − 6000 6̅

= = 0. 6000

If we use the same trick we used for the arc price elasticity, then we should arrive here, not 6000 but 8000 which will give us:

The midpoint of sale:

4000 1

= = 0.5

8000 2

So it’s either 66% if you consider the initial sale value or 50% if you consider the midpoint, and then we should fill this number to the equation.

2∆% = −3⏟ %∆ → ∆% = − 9

So we should decrease price by 2/9 (22%).

Optimal consumption under risk

1. Calculating expected payoff

1 1 = 4 = 16 = =1 2 1 2

2 2

1 1(̃) = ∗ 1 + ∗ 16 = 10

2 2

1 1̃ = + = 3

( ) √16√4 1 2 2

1 = 9 = 9 = =1 2 1 2

2 1 1̃ = + = 3

( ) √9 √9 2 2

2 = 0 = 25 = = 1/2 1 2 1 2

1 1̃ √25

( ) = + = 2.5

√10 3 2 2

() = √ = 0 ; = 9 1 2

1 = =1 2

2 1 1(̃) = + = 1.5

√0 √9 2 2

√4.5 =̃

We find this individual prefer to get the mean -> risk aversion.

2= 1.5 → = 1.5

√̃ ̃

2 = 4.5 − 1.5

̃3 () = log 10 = 1

1 = 100

2 1 = =1 2

2 1 1̃ = 1 + + 100 + = 50.5

2 2

1 1(̃) = log 1 + log 100 = 1

10 10 2 2

→ log = 1 → = 10

10

The individual is risk adverse.

= 50.5 − 10 = 40.5

̃This individual is willing to give up 40.5 of mean to eliminate risk.

Given the certainty equivalent a