301_psetsa.pdf

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Unit 1.1

1. Solving the demand curve for P: 500 50

50 500 1 10

50

Q P

P Q

= −

So the demand curve has a y-intercept of 10 and a slope of -1/50 = -0.02 Similarly for the supply curve:

50 25 25 50

1 2

25

Q P

P Q

= +

= − +

So the supply curve has a y-intercept of -2 and a slope of 1/25 = 0.04. Here is a sketch:

2. Equilibrium occurs when quantity demanded equals quantity supplied: 500 50 50 25

450 75 6 D S

Q Q

P P

=

− = +

= =

Plug this back into either the supply or the demand curve to get Q=200 3. With a regulated price of $8:

a. Quantity demanded is QD =500 50(8) 100− = Quantity supplied is QS =50 25(8)+ =250

The quantity actually exchanged is Q=100 units – the lower of the two.

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4. With a regulated price of $0:

a. Quantity demanded is QD =500 50(0)− =500 Quantity supplied is QS =50 25(0)+ =50

The quantity actually exchanged is Q=50 units – the lower of the two.

b. Since 500 units are demanded, but only 50 units are supplied, there is a shortage of 450 units.

5. With a quantity restriction of 100 units:

a. A price floor is a price above the equilibrium price, when the quantity demanded is lower than the quantity supplied. Therefore, we need to find a price so high that quantity demanded drops to 100 units:

500 50 100 500 50

8 D

Q P

P P

= −

=

Thus, a price floor of $8 causes demand to fall to 100 units, so this will be the quantity exchanged (notice that you actually did this already in problem 3).

b. A price ceiling is a price below the equilibrium price, when the quantity supplied is lower than the quantity demanded. Therefore, we need to find a price so low that quantity supplied drops to 100 units:

50 25 100 50 25

2 S

Q P

P P

= +

=

Thus, a price ceiling of $2 causes supply to drop to 100 units, so this will be the quantity exchanged.

6. With a tax of $3 imposed on sellers, the price actually received by sellers is lowered from P to P−3, so the supply curve becomes QS =50 25+

(

P−3

)

a. Equilibrium occurs where quantity demanded equals quantity supplied (using the modified after-tax supply curve):

(

)

500 50 50 25 3 500 50 50 25 75 525 75

7 D S

Q Q

P P

=

− = + −

= =

Plug this back into either the demand or the post-tax supply curve to get Q=150 b. Buyers pay P=7.

c. Sellers receive a net price of 7 3− =4 after deducting the tax.

d. Buyers pay $7 instead of the original price of $6, meaning that they pay $1 of the $3 tax (33.3%).

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7. With a tax of $3 imposed on buyers, the price actually paid by buyers is raised from Pto 3

P+ , so the demand curve becomes QD =500 50−

(

P+3

)

a. Equilibrium occurs where quantity demanded equals quantity supplied (using the modified after-tax demand curve):

(

)

500 50 3 50 25 500 50 150 50 25 300 75

4 D S

Q Q

P P

=

− + = +

− − = +

= =

Plug this back into either the supply or post-tax demand curve to get Q=150 b. Buyers pay a net price of 4 3+ =7after including the tax.

c. Sellers receive P=4

d. Buyers pay $7 instead of the original price of $6, meaning that they pay $1 of the $3 tax (33.3%).

Sellers receive $4 instead of the original price of $6, meaning that they pay $2 of the $3 tax (66.7%)

8. With a subsidy of $6 paid to sellers, the price actually received by sellers is raised from Pto 6

P+ , so the supply curve becomes QS =50 25+

(

P+6

)

a. Equilibrium occurs where quantity demanded equals quantity supplied (using the modified after-subsidy supply curve):

(

)

500 50 50 25 6 500 50 50 25 150 300 75

4 D S

Q Q

P P

=

− = + +

= =

Plug this back into either the demand or the post-subsidy supply curve to get Q=300 b. Buyers pay P=4.

c. Sellers receive a net price of 4 6 10+ = after including the subsidy.

d. Buyers pay $4 instead of the original price of $6, meaning that they receive $2 of the $6 subsidy (33.3%).

Sellers receive $10 instead of the original price of $6, meaning that they receive $4 of the $6 subsidy (66.7%).

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9. With a 40% VAT tax imposed on sellers, the price actually received by sellers is lowered from Pto

(

1 0.4−

)

P=0.6P, so the supply curve becomes QS =50 25 0.6+

(

P

)

a. Equilibrium occurs where quantity demanded equals quantity supplied (using the modified after-tax supply curve):

(

)

500 50 50 25 0.6 500 50 50 15 450 65

6.92 D S

Q Q

P P

=

− = +

= =

Plug this back into either the demand or the post-tax supply curve to get Q=153.85 b. Buyers pay P=6.92.

c. Sellers receive a net price of 0.6 6.92

(

)

=4.15 after deducting the tax. d. Notice that the tax wedge (dollar-tax paid per unit) is 6.92 4.15− =2.77

Buyers pay $6.92 instead of the original price of $6, meaning that they pay $0.92 of the $2.77 tax wedge (33%).

Sellers receive $4.15 instead of the original price of $6, meaning that they pay $1.85 of the $2.77 tax wedge (67%).

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Unit 1.2 1. Derivatives

f ∂ ∂f / x

2 2

ax +bxy+cy 2ax by+

40 xy 1 40 1/ 2 1/ 2 20 1/ 2 1/2

2 x y x y

− −

⎛ ⎞ ₌

⎜ ⎟ ⎝ ⎠

a b

Ax y aAxa−1yb

1 a a

)

ln 3 ln 2

a x− +b y− 1

3 3

a a

x x

⎛ _{⎞ =}

⎜ ₋ ⎟ ₋

⎝ ⎠

(

)

(

)

ln 3 ln 2

a −x +b −y 1

3 3

a a

x x

−

⎛ ⎞

− _⎜ _⎟=

− −

3

x +y

(

)

2

1/3 1/3 1 2/3 3

3 x +y ⎛_⎜ x− ⎞_⎟

⎝ ⎠

1/ 2 2 2 1

40x y

− − −

⎛ ₊ ⎞

⎜ ⎟

⎝ ⎠

( )

3/ 2

2 2 3

1 1 1

2

2 40x y 40x

−

− − −

⎛ ⎞ ⎛ ⎞

− _⎜ + _⎟ _⎜ − _⎟

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2. The maximum occurs when the first derivative is equal to zero. Calculating the first derivative of the function:

( )

1 1

5

200 5 df

dx x x

⎛ ⎞

= + − ⎜_⎝ ₋ ⎟_⎠

We can now find the value of x where the first derivative is equal to zero:

1 5

0 200 5

1 5

200 5 5 200 5

10 200 20

x x

− =

−

= −

= ⇒ =

So the function reaches its maximum value when x=20.

3. The maximum occurs when the first partial derivatives are set equal to zero. Calculating the partial derivatives with respect to x and y:

2 2 f

x x

∂ _{= − +}

∂

2 4 f

y y

∂ _{= −} ₊

∂

The maximum value of the function occurs when both partial derivatives are equal to zero: 2x 2 0 x 1

− + = ⇒ =

2y 4 0 y 2

− + = ⇒ =

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Unit 1.3

1. Tax incidence

a. 1.266 0.634 0.366

0.732 1.266 buyer seller η σ σ ε η = = = ⇒ = + +

b. 0.455 1 0

0 0.455 buyer seller η σ σ ε η = = = ⇒ = + +

c. 0 0 1

1.528 0 buyer seller η σ σ ε η = = = ⇒ = + +

d. 1.266 0 1

1.266 buyer seller η σ σ ε η = = = ⇒ = + ∞ +

2. Linear supply and demand a. Equilibrium occurs where

2500 50 150 500

3000 200 15, 1750 D S

Q Q

P P

P P Q

=

− = −

= ⇒ = =

b.

(

50

)

15 0.429 1750

D D

Q P

P Q

ε = ∂ = − ⎛_⎜ ⎞_⎟= −

∂ ⎝ ⎠

( )

15

150 1.286 1750 S S Q P P Q

η =∂ = ⎛_⎜ ⎞_⎟=

∂ ⎝ ⎠

c. In slope-intercept form, the demand curve is 50 1 50

P= − Q and the supply curve is

10 1 1

3.33

3 150 150

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d. 2 D D Q P P Q ∂ − = ∂

(

)

2 50 2500 50 5000 100 50

150 5000 33.33

P P P P P P ⎛ ⎞ − = − _⎜ _⎟ − ⎝ ⎠ − = = ⇒ =

3. Elasticity and price changes a. %

% Q P

ε = Δ Δ

25%

2.58 % 9.69

% P P

− = ⇒ Δ = −

Δ

Price should be cut 9.69% in order to raise quantity 25%. b. %

% Q P

ε = Δ Δ

%

2.58 % 64.5

25% Q

Q

Δ

− = ⇒ Δ = −

Output should be cut 64.5% in order to raise price 25%. 4. Immigration and wages

a. Immigration causes a shift of the labor supply curve, which maps out various points on the demand curve. Thus, this information helps us estimate the elasticity of demand for labor.

b. % 11% 3.44

% 3.2%

Q P

ε = Δ = = −

Δ −

In words, each 1% increase in wage causes a 3.44% reduction in quantity of labor demanded.

5. Demand for beef

a. Notice that the demand for beef rises when the price of rice rises, which is the definition of a substitute.

b. ,

(

1 1/3 1/3 1/ 2

)

1 1/3 2/3 1/ 2

0.2 2 0.2

0.3 0.3 3

B R R

B R B L R

R B B L R

Q P P

P P P Y

P Q P P P Y

ε − −

−

⎛ ⎞

∂

= = _⎜ _⎟= =

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c. The cross-price elasticity is constant at ε_{B R}_, =2 / 3, regardless of the values of the prices or income, so it wouldn’t change. To see why, notice that the demand function has the constant elasticity form discussed in class.

d. No – You would need the demand function for rice as it depends on the price of beef. 6. Income and comparative statics

a. Setting supply equal to demand:

1 1 1

1000

20 2 40

3 1

1000

2 40

2000 1 *

3 60

Q Y Q Y

Q Y

− + = +

= +

Substitute back to either curve to find equilibrium price: 1

* 1000

20

2000 1 1 1000

3 60 20

1000 1 3 30

P Q Y

Y Y

Y

= − +

⎛ ⎞

= −_⎜ + _⎟+

⎝ ⎠

= +

b. The derivative is * 1

30 dP

dY =

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Unit 2.1

1. Budget Sets

a. The budget constraint is below. She can consume 20 waters, 10 salads or on the linear segment connecting them.

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c. T th sh w w

2. Since her eq lesson tuba l down more

The budget co he budget co

he wants to j water. Therea with 16 bottle

e she dislikes qually well o ns, she likes lessons. The nward slopin ballet lesson

onstraint is b onstraint sinc

join the club after, Petra o es of water a

s her first fiv off if we giv s her tuba les e indifferenc ng after 5 les

ns.

below. The b ce she could b, she immed only gives up and no salads

ve tuba lesso ve her additio

ssons and so e curve is up ssons. Clara p

bundle conta consume 10 diately gives p half a salad

s is the other

ons, we have onal tuba les she’s willin pward slopin prefers indif

aining no wa 0 salads if sh s up 6 salads

d for each bo r endpoint.

e to give her ssons. Howe ng to give up ng for fewer fference curv

ater and 10 s he does not jo

, but receive ottle of wate

more ballet ver, after the p ballet lesso

than 5 tuba ves with bun

alads is still oin the club. es 4 bottles o r. The bundl

lessons to m e first five ons to get mo

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3. Using for “prefers” and ∼ for indifferent, the information in the problem is: C A,

A W, P∼D, D A.

a. Since D A and A W, transitivity gives us that D W . But since P∼D, this means that by extension P W.

b. Uncertain – we know that C A and that D A, but nothing about preference ordering over dates and cashews.

4. The initial price ratio is 120 1.5 80 X Y P

P = = . The new price ratio is

138 1.5 92 X Y P

P = = . So the two budget lines are parallel, but the new budget line is shifted inwards.

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Unit 2.2

1. The relevant tool for decision-making is marginal utility per dollar, MU/P. Assembling a table:

Quantity MU/P (apple) MU/P (banana) MU/P (cherry)

1 5 10 8

2 5 10 7

3 4 6 5

4 3 4 2

5 2 2 1

6 1 1 1

The consumer starts with $50. At any point where he has money left, he additional units of whatever commodity gives him the most marginal utility per dollar. In order:

• Purchase the first banana (total expenditure = 4)

• Purchase a second banana (total expenditure = 8)

• Purchase the first cherry (total expenditure = 18)

• Purchase the second cherry (total expenditure = 28)

• Purchase the third banana (total expenditure = 32)

• Purchase the first apple, second apple and third cherry (total expenditure = 44) (note that all give marginal utility per dollar of 5)

• Purchase the third apple and the fourth banana (total expenditure = 49) (note that both give marginal utility per dollar of 4)

• Purchase the fourth cherry (total expenditure = 50)

At this point, the consumer has used up his income. So the optimal bundle is 4 apples, 4 bananas and 3 cherries.

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Unit 2.3

1. Price decrease for a normal good.

a. $200 – For example, one bundle on budget line B₀ is 100 carrots and no coconuts, which costs $200. Any other bundle on the budget line also costs $200.

b. The price decrease, pivoting the budget line from B₀ to B₁ caused the consumer to increase his coconut purchases from 14 units to 30 units. The total effect is 16 coconuts. c. The part of the increase from B' to B1 represents a pure increase in purchasing power

with no change in the price ratio – so the movement from 17 coconuts to 30 coconuts is the income effect – increase of 13 coconuts.

d. The slope change between B₀ to B' represents a change in the relative price ratio with no change in purchasing power – so the movement from 14 coconuts to 17 coconuts is the substitution effect – increase of 3 coconuts.

e. The increase in income represented by the parallel shift from B' to B₁ caused Lisa to increase in consumption of coconuts, so coconuts are a normal good.

f. The decline in price, pivoting the budget line from B₀ to B₁ caused the consumer to increase his coconut purchases, so coconuts are not a Giffen good. Alternatively, a Giffen good must be an inferior good, and since we determined that coconuts are normal goods, this means that they cannot be Giffen goods.

g. When the price of coconuts fell, pivoting the budget line from B₀ to B₁, demand for carrots falls from 44 to 40 carrots. This means that the two are substitutes.

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i. Using the information obtained in (h): 30 14

% ₂₂

1.09 4 8

%

6 Q

P

ε

− Δ

= = ₋ = −

Δ

j. Budget lines B₁ and B' correspond to different levels of income. On budget line B₁, where income is $200 as derived earlier, demand is 30 coconuts. Note that budget line B' corresponds to income of $128 (e.g. consider the bundle with 64 carrots and no

coconuts), where demand is 17 coconuts. 30 17

% _23.5

1.26 200 128

%

164 Q

Y

ξ

− Δ

= = =

− Δ

k. No. You would need information about what happens when carrot prices change. The price changes illustrated in this problem all refer to the price of coconuts.

l. Yes – when the price of a coconut is $8 (B₀), Lisa buys 44 carrots. When the price of a coconut is $4 (B₁), Lisa buys 40 carrots.

,

44 40

% ₄₂

0.14 8 4

%

6 carrot

carrot coconut

coconut Q P

ε

− Δ

= = ₋ =

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2. Price increase for an inferior good.

a. Inferior good – The parallel shift from B₁ to B' represents an increase in income. His demand for weapons falls from 9 to 8 as a result of this income increase.

b. No – The pivot from B₀ to B₁ represents an increase in the price of weapons. His demand for weapons falls from 12 to 9 as a result of the price increase.

c. The decline from B' to B₁ represents the decline in purchasing power due to the price increase. Notice that consumption of weapons actually rises from 8 to 9 as a result of this decline in purchasing power (inferior good), so the income effect is +1 weapon.

d. The pivot from B₀ to B' represents the change in price ratio due to the price increase. Consumption of weapons falls from 12 to 8 as a result of this pivot, so the substitution effect is -4 weapons. Notice that the substitution effect of -2 and the income effect of +1 combine for a total effect of -3 weapons.

e. When the price of weapons is $1 (on B₀), Tarzan demands 12 weapons. When the price of weapons is $2 (on B₁), Tarzan demands 9 weapons. The Marshallian demand curve contains these points.

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3. Price increase for perfect complements.

a. When the price of left shoes rises, pivoting the budget line from B₀ to B₁, demand for shoes falls from 5 shoes to 2 shoes. The total effect is a decline of 3 shoes.

b. The parallel shift from B' to B₁ represents the decline in purchasing power due to the price increase. Demand for shoes falls from 5 shoes to 2 shoes, so the income effect is a decline of 3 shoes.

c. The pivot from B0 to B' represents the change in the price ratio. Notice that demand for shoes does not change at all as a result of this change in relative prices, so the substitution effect is 0.

d. The entire decline is due to the income effect. This makes sense because the two goods in this example are perfect complements, and as a result there is no substitution between goods when relative prices change.

e. When the price of left shoes is $1 (on B₀), the consumer demands 5 left shoes. When the price of left shoes is $4 (on B₁), the consumer demands 2 left shoes. The Marshallian demand curve contains these points.

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Unit 3.1

1. Some bundles generating U =4: {(16,0), (9,1), (4,2), (1,3), (0,4)}

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2. CES utility function.

a. The constrained maximization problem is:

1 2 1 2

max x + x s t. . 10x +20x =1500 The Lagrangian is:

(

)

1 2 1500 10 1 20 2 L= x + x +λ − x − x First order conditions:

1/ 2 1 1 1 10 0 2 L x x λ − ∂ ₌ ₋ ₌ ∂ 1/ 2 2 2 1 20 0 2 L x x λ − ∂ ₌ ₋ ₌ ∂ 1 2

1500 10 20 0 L

x x

λ

∂ ₌ ₋ ₋ ₌

∂

Solving the first two FOC’s for λ: 1/ 2

1

1 1

10 0

2x λ λ 20 x

− ₋ _{= ⇒ =} 1/ 2 2 2 1 1 20 0

2x λ λ 40 x

− ₋ _{= ⇒ =}

Equating:

1 2

1 2 1 2

1 1 20 40 20 40 2 4 x x x x

x x x x

=

= ⇒ =

Substituting into the budget line:

( )

1 2 2 2

2 2

10 20 1500 10 4 20 1500

60 1500 25

x x x x x x + = + = = ⇒ = Substituting back: 1 4 2 100 x = x = b. The MRS is:

( )

(

1

)

2

1

2 2 1

1 2 ₂₅ ₁

2 100 1 2 x _x U x MRS

U x x x

∂ ∂

= = = = =

∂ ∂

The price ratio is: 1

2

10 1 20 2 P

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c. Marginal utility per dollar for good 1 is:

( )

1

1 1

1 1 1

1 2 _{1 20}

0.005 10

x

MU U x

P P P

∂ ∂

= = = =

Marginal utility per dollar for good 2 is:

(

2

)

2 2

2 2 2

1 2 _{1 10}

0.005 20

x

MU U x

P P P

∂ ∂

= = = =

d. U = x₁ + x₂ = 100+ 25=15

e. From the calculation in (a), the Lagrange multiplier is:

1

1 1

0.005 20 x 20 100

λ = = =

f. Going back to (a), but using income of Y =1501 gives the optimal bundle 1 100.0667

x = and x₂ =25.0167. This generates utility of 1 2 100.0667 25.0167 15.005

U = x + x = + =

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3. Three-good utility function.

a. The constrained maximization problem is: 1 2 3 1 1 2 2 3 3 maxx x x s t P x. . +P x +P x =Y The Lagrangian is:

(

)

1 2 3 1 1 2 2 3 3

L=x x x +λ Y−P x −P x −P x

First order conditions, and solving for λ: 2 3 2 3 1

1 1

0 x x

L

x x P

x λ λ P

∂

= − = ⇒ =

∂

1 3 1 3 2

2 2

0 x x

L

x x P

x λ λ P

∂ ₌ ₋ _{= ⇒ =}

∂

1 2 1 2 3

3 3

0 x x

L

x x P

x λ λ P

∂ ₌ ₋ _{= ⇒ =}

∂

1 1 2 2 3 3 0 L

Y P x P x P x

λ

∂

= − − − =

∂

Equating the first and second: 2 3 1 3 1 1

2

1 2 2

x x x x P x

x

P = P ⇒ = P

Equating the first and third: 2 3 1 2 1 1

3

1 3 3

x x x x P x

x

P = P ⇒ = P

Substituting into the budget line: 1 1 2 2 3 3

1 1 1 1

1 1 2 3

2 3

1 1 1

1 3

3 P x P x P x Y

P x P x

P x P P Y

P P

Y P x Y x

P + + = ⎛ ⎞ ⎛ ⎞ + _⎜ _⎟+ _⎜ _⎟= ⎝ ⎠ ⎝ ⎠ = ⇒ =

Substituting this back into 1 1 2

2 P x x

P

= and 1 1 3

3 P x x

P

= gives:

2 2 3 Y x P

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b. The constrained maximization problem is:

1 2 3 1 1 2 2 3 3 max lnx +lnx +lnx s t P x. . +P x +P x =Y The Lagrangian is:

(

)

1 2 3 1 1 2 2 3 3

ln ln ln

L= x + x + x +λ Y−P x −P x −P x

First order conditions, and solving for λ: 1

1 1 1 1

1 1

0 L

P

x x λ λ P x

∂ ₌ ₋ _{= ⇒ =}

∂

2

2 2 2 2

1 1

0 L

P

x x λ λ P x

∂ ₌ ₋ _{= ⇒ =}

∂

3

3 3 3 3

1 1

0 L

P

x x λ λ P x

∂ ₌ ₋ _{= ⇒ =}

∂

1 1 2 2 3 3 0 L

Y P x P x P x

λ

∂ _{= −} ₋ ₋ ₌

∂

Equating the first and second: 1 1 2

1 1 2 2 2

1 1 P x

x P x = P x ⇒ = P Equating the first and third:

1 1 3

1 1 3 3 3

1 1 P x

x P x = P x ⇒ = P Substituting into the budget line:

1 1 2 2 3 3

1 1 1 1

1 1 2 3

2 3

1 1 1

1 3

3 P x P x P x Y

P x P x

P x P P Y

P P

Y P x Y x

P + + = ⎛ ⎞ ⎛ ⎞ + _⎜ _⎟+ _⎜ _⎟= ⎝ ⎠ ⎝ ⎠ = ⇒ =

Substituting this back into 1 1 2

2 P x x

P

= and 1 1 3

3 P x x

P

= gives:

2 2 3 Y x P

= and ₃ 3 3 Y x P =

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4. Stone-Geary utility function

(

1

)

( )

2 1 1 2 2 max ln x − +5 ln x s t P x. . +P x =Y

The Lagrangian is:

(

1

)

( ) (

2 1 1 2 2

)

ln 5 ln

L= x − + x +λ Y −P x −P x

First order conditions, solving for λ:

(

)

1

1 1 1 1

1 1

0

5 5

L

P

x x λ λ P x

∂

= − = ⇒ =

∂ − −

2

2 2 2 2

1 1

0 L

P

x x λ λ P x

∂ ₌ ₋ _{= ⇒ =}

∂

1 1 2 2 0 L

Y P x P x

λ ∂ _{= −} ₋ ₌ ∂ Equating:

(

)

(

)

1 1 2 2

1 1 2 2 1 1 1 2

2

1 1

5

5 5

P x P x

P x

P x P x P x

P

= −

−

= − ⇒ =

Substituting into the budget line: 1 1 1 1 1 2

2

1 1 1

1 1 1 5 2 5 5 2

P x P

P x P Y

P P x Y P

Y P x P ⎛ − ⎞ + _⎜ _⎟= ⎝ ⎠ = + + = Substituting back:

(

)

1 1 1 1

2

2 2 2 2

5 ₅ ₁₀

5

5 2 2 5

2

Y P _Y _P _P

P

P x P Y P

x

P P P P

⎛ + ₋ ⎞ _⎛ ₊ ₋ _⎞

⎜ _{⎟ ⎜} _⎟

− _⎝ _{⎠ ⎝} _⎠ −

= = = =

b. The demand functions x₁ and x₂ rise as Y rises, so both are normal goods. c. The demand for x₁ does not change when P₂ rises, so good 2 is neither a

substitute nor a complement with good 1.

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5. Quasi-linear uility

( )

max ln W +F s t P W. . _W +P F_F =Y

The Lagrangian is:

( )

(

)

ln _W _F

L= W + +F λ Y−P W−P F

First order conditions, solving for λ:

1 1 0 W W L P

W W λ λ WP

∂

= − = ⇒ =

∂

1 1 _F 0

F L

P

F λ λ P

∂ = − = ⇒ = ∂ 0 W F L

Y P W P F

λ

∂

= − − =

∂

Equating:

1 1 _F

W F W

P W WP = P ⇒ = P Substituting into the budget line:

W F F W F W F F F F P W P F Y

P

P P F Y

P

Y P

P P F Y F

P + = ⎛ ⎞ + = ⎜ ⎟ ⎝ ⎠ − + = ⇒ =

b. 4 4

1 F W P W P

= = = , and 100 4 24 4 F F Y P F P − − = = =

c. W +4F=Y

d. 4 4

1 F W P W P

= = = , and 200 4 49 4 F F Y P F P − −

= = = . Notice in this case that the demand for water does not change. All the additional income is spent on food.

e. _W

( )

0 0

F W

W Y Y

Y W P P

ξ =⎛_⎜∂ ⎞⎛_⎟⎜ ⎞_⎟= ⎛_⎜ ⎞_⎟=

∂

⎝ ⎠⎝ ⎠ _⎝ _⎠

f.

(

)

1 F

F F F F

F Y Y Y

Y F P Y P P Y P

ξ =_⎜⎛∂ ⎞⎛ ⎞_{⎟⎜ ⎟}=_⎜⎛ ⎞_⎟⎛_⎜_⎜ _⎟⎞_⎟=

∂ − −

(25)

Unit 3.2

1. Engel’s Rule and the Slutsky Equation

a. According to Engel’s Rule θ ξ θ ξ_h _h+ _f _f =1. Since θ_f =0.3 and θ_h =0.7, then 0.7ξ_h+0.3ξ_f =1 and so

1 0.7 0.3

h f

ξ ξ = −

Since housing is a necessity good, the lowest value of ξ_h is 0 and the highest value of ξ_h is 1, this creates a bound 1<ξ_f <3.33.

b. Using the same calculation as in (a), with θ_f =0.5 and θ_h =0.5, then the bound is 1<ξ_f <2.

c. The Slutsky equation states that εm =εh−θξ . Since Hicksian elasticities are non-negative, the highest possible value is εh =0. Also, since 1<ξ_f <3.33, then the highest possible Marshallian elasticity will occur when ξ_f =1:

( )( )

0 0.3 1 0.3 m

ε = − = −

d. Using the same logic as in (c), the highest possible Marshallian elasticity is 0.5

m

(26)

Unit 3.3

1. Consumer theory relations a. Using Roy’s Identity:

(

)

2

1 2 1 1 1 2 1 2 1 m Y P P

V P Y

x

V Y P P

P P ⎛ ₋ ⎞ ⎜ ⎟ ⎜ ₊ ⎟ ∂ ∂ _⎝ _⎠ = − = − = ∂ ∂ ⎛ ⎞ + ⎜ ₊ ⎟ ⎝ ⎠

b. The expenditure function is the inverse of the indirect utility function:

(

)

(

)

(

)

1 2

1 2 1, 2, 1 2

Y V

P P

Y V P P E P P U U P P

= +

= + ⇒ = +

c. Using Shephard’s Lemma:

1 1 h E x U P ∂ = = ∂

2. Consumer theory relations

a. Substituting the Marshallian demands into the utility function:

(

)( )

1 1

(

1

)

2

1 2

1 2 1 2

5

5 5

5 5 5

2 2 4

Y P

Y P Y P

V x x

P P P P

−

⎛ − ⎞⎛ − ⎞

= − =_⎜ + − _⎟⎜ _⎟=

⎝ ⎠⎝ ⎠

b. The expenditure function is the inverse of the indirect utility function:

(

)

(

)

(

)

2 1 1 2 2

1 2 1

1 1 2

1 1 2 1 2 1 1 2

5 4

4 5

5 4

5 2 , , 5 2

Y P

V

P P VP P Y P

Y P VP P

Y P VP P E P P U P UP P

− =

= −

− =

(27)

c. One way is to use Shephard’s Lemma:

1/ 2 1/ 2 1/ 2 2

1 1 2

1 1

5 5

h E UP

x U P P

P P

−

∂

= = + = +

∂

)

(

)

1 1 2 2 1 5 2 L=P x +P x +λ U− x − x

(28)

Unit 3.4

1. Perfect Complements

a. For each 1-unit increase in ice cream, the consumer needs a proportionate 2-unit increase in cherries and a 10-unit increase in fudge.

(

)

1 1

, , min , , 2 10 U I C F = ⎧⎨I C F⎫⎬

⎩ ⎭

An equivalent way to write this, giving the same ratios, is:

(

, ,

)

min 10 , 5 ,

{

}

U I C F = I C F

Again, this makes intuitive sense because a 1-unit increase in I needs to be met with an equivalent 2-unit increase in C and 10-unit increase in F.

b. The optimality condition sets all the arguments equal: 10I =5C=F

Also, the budget line needs to hold: 0.5 0.1

I+ C+ F =Y

Using the optimality condition to put everything in terms of I:

( )

0.5 2 0.1 10 3

3

I I I Y

Y

I Y I

+ + =

= ⇒ =

Which implies that: 2 2

3 Y C= I =

10 10

(29)

2. Perfect substitutes

a. A one-unit increase in O is equivalent to a four-unit increase in A.

b. The solution will be a corner solution where the consumer uses all of his budget for one good or the other. If he buys all apple juice, then he will have 120 apple juice and no orange juice:

( )

15 60 15 120 60 0 1800

U = A+ O= + =

The other alternative is no apple juice and 60 orange juice:

( )

15 60 15 0 60 60 3600

U = A+ O= + =

Since the latter is higher, the optimal bundle is A=0 and O=60.

Another way to see this is to observe that marginal utility per dollar for apple juice is MU_A P_A =15 / 0.25=60. For orange juice: MU_O P_O =60 / 0.5 120= . Since marginal utility per dollar is higher, the consumer should allocate all his income to orange juice.

3. The utility function represents perfect substitutes, so the solution will be a corner solution where all income is allocated to the good for which marginal utility per dollar is highest. For good 1, MU P₁ ₁ =a₁/P₁. For good 2, MU₂ P₂ =a₂ /P₂, etc… The consumer should spend all his income on the good for which this ratio is the highest, and buy 0 units of the other goods. Writing this out formally, for the good i for which a_i /P_i is maximized, the consumer should buy _i

i Y x

P

(30)

4. Interior and corner solutions.

a. The interior solution is characterized by the Lagrangian:

( ) (

)

2

1 2 1 2

1

ln 2

2

L= x + x +λ Y− x −x

First-order conditions: 1 1 1 2 0 2 x L x

x λ λ

∂

= − = ⇒ =

∂

2 2 2

1 1

0 L

x x λ λ x

∂ = − = ⇒ = ∂ 1 2 2 0 L

Y x x

λ

∂

= − − =

∂

Equating the first two: 1

2

1 2 2

1 1 2 2 2 x x

x x x

x

=

= ⇒ =

Substituting back to the budget line: 1 2 1 1 2 2 2

x x Y

x Y

x

+ =

For 4Y = , the interior solution is characterized by:

1 1 2

2x 4

x

+ =

Which is solved by x1 =1, and so x2 =2 x1=2

1 1 2 2x 5

x

+ =

Which is solved by x₁ =2, and so x₂ =2 x₁=1

(

x x1, 2

) ( )

= 2,1

The intuition behind this problem is that ln

( )

x₂ is a concave function that initially rises fast and then slows down. However, 1 ₁2

(32)

Unit 3.5

1. Observation X: P_T =4, 6P_G = . Consumer buys T =6, G=6 Observation Y: P_T =6, 3P_G = . Consumer buys T =10, G=0 Making a table showing the cost of each bundle at each price vector:

Bundle X Bundle Y

Prices X 60 40

Prices Y 54 60

Since bundle Y was affordable when bundle X was purchased X Y Since bundle X was affordable when bundle Y was purchased Y X Since X Y and Y X , this consumer’s behavior violates WARP.

2. The table below shows the cost of each bundle at each of the price vectors. At price vector A, bundle A was purchased, but C, D and E were affordable:

A C, A D, A E

At price vector B, bundle B was purchased, but C, D and E were affordable:

B C, B D, B E

At price vector C, bundle C was purchased, but D and E were affordable:

C D, C E

At price vectors D and E, no other bundles were affordable.

Notice that there are no bundles X and Y for which X Y and Y X simultaneously. Also, all the revealed preferences are transitive. For example, A C and C D, then

A D. Thus, this consumer’s behavior satisfies WARP and SARP.

A (quantity) B (quantity) C (quantity) D (quantity) E (quantity)

A (price) 40 45 25 20 20

B (price) 75 55 40 35 30

C (price) 40 45 25 20 20

D (price) 50 115 45 30 40

(33)

3. Labeling the three observations as X, Y and Z respectively, the table below shows the cost of each bundle at each of the three price vectors.

At price vector X, bundle X was purchased, but Z was affordable: X Z At price vector Y, bundle Y was purchased, but X was affordable: Y X At price vector Z, bundle Z was purchased, but Y was affordable: Z Y

There are no bundles where A B and then B A, so the consumer’s behavior satisfies WARP.

However, the revealed preferences are intransitive: Z Y and Y X, but then X Z. So this consumer’s behavior does not satisfy SARP.

X (quantity) Y (quantity) Z (quantity)

X (price) 42 48 40

Y (price) 33 36 39

(34)

Unit 4.1

1. Labor supply and nonlabor income

240 24

L= +Y N +λ Y− −Y w wN+

First-order conditions, solved for λ:

1 0 1

L

Y λ λ

∂ _{= + = ⇒ = −}

∂

120 120

0 L

w

N N λ λ w N

∂ ₌ ₊ _{= ⇒ = −}

∂

0 24 0

L

Y Y w wN

λ

∂ _{= − −} ₊ ₌

∂

Equating:

2 120

1 120 120

w N w N N

w

− = −

=

⎛ ⎞

= ⎜ ⎟

⎝ ⎠

Substituting back into the constraint: 2 0

120 24

Y Y w

w

⎛ _⎛ _⎞ ⎞

= + ⎜_⎜ −_⎜ _⎟ ⎟_⎟

⎝ ⎠

b. The labor supply is

2 120

24 24

S

L N

w

⎛ ⎞

= − = _{− ⎜} _⎟

⎝ ⎠

Labor supply rises as w rises, so the substitution effect is stronger than the income effect.

c. An increase in Y₀ increases only consumption Y, leaving leisure time N

(35)

2. Overtime pay premium

a. The initial budget line is sketched in green, with the new budget line sketched in red. They coincide for 8 or fewer hours of work (16 or more hours of leisure). Beyond here, the new budget line has 1.5 times the slope of the initial budget line.

b. His initial optimal bundle is at 17 hours of leisure. Whether he chooses to take fewer hours of leisure with access to the red budget line depends upon the shape of his indifference curves. If he can find a higher indifference curve on the red budget line, then he will work more hours. If not, then he will continue to work only 7 hours. The diagrams below show both cases, respectively.

Intuitively, if the indifference curve is flat (MRS is low meaning that the person doesn’t value an extra hour of leisure very much), then he will choose to work more when given access to overtime pay. If the indifference curve is steep (high MRS, meaning that the person places high value on leisure), then he will continue to work only 7 hours.

(36)

(37)

4. Food preparation time.

a. Time is either spent with leisure time or preparing food F: 24N+P F_F =

b. The consumer maximizes U F N

(

,

)

subject to the constraint:

(

)

1/ 2

24 _F

L=FN +λ − −N P F

First order conditions:

1/2 0 F F L N N P

F λ λ P

∂ ₌ ₋ _{= ⇒ =} ∂ 1/2 1 0 2 2 L F FN

N λ λ N

−

∂ ₌ _{− = ⇒ =}

∂

24 _F 0

L

N P F

λ ∂ = − − = ∂ Equating: 2 2 2 F F F N F P N N FP N F P = = =

Substituting back into the constraint: 24

2

24

3 24 8

F F

F

N P F N N P P N N + = ⎛ ⎞ + _⎜ _⎟= ⎝ ⎠ = ⇒ =

And so 2 16 F F N F

P P

= =

(38)

Unit 4.2

1. Two-period intertemporal choice

a. Equating the present value of income with the present value of consumption: 2

1 1100 2000

1 0.1 1 0.1 C C

+ = +

+ +

b. Equating the future value of income with the future value of consumption:

(

)

1 2

2000 1 0.1+ +1100=C(1 0.1)+ +C

c. The consumer maximizes U C C

(

₁, ₂

)

subject to the budget constraint above:

2 2

1 2 1 1 2 1

1100

2000 3000

1 0.1 1 0.1 1.1

C C

L=C C +λ_⎜⎛ + −C − ⎞_⎟⇒ =L C C +λ⎛_⎜ −C − ⎞_⎟

+ +

⎝ ⎠ ⎝ ⎠

2 2

1

0 L

C C

C λ λ

∂ = − = ⇒ = ∂ 1 1 2 1 0 1.1 1.1 L C C

C λ λ

∂ = − = ⇒ = ∂ 2 1 3000 0 1.1 C L C λ ∂ = − − = ∂ Equating: 1 2 1.1C =C

Substituting back to the constraint: 2 1 1 1 1 1 3000 1.1 1.1 3000 1.1

2 3000 1500 C C C C C C + = + = = ⇒ =

(39)

d. Since Y1=2000, but C1=1500, this implies that Amal saves $500 of her first-period income

e. If you solve the same problem again with an interest rate of 20%, you obtain 1 1458.33

C = and C₂ =1750. Consumption in the first period falls, meaning that the consumer saves more as a result of the higher interest rate. Thus, the

substitution effect is stronger than the income effect. 2. Lifetime budget constraint

a. Perfect complements – If the utility function were U =min

{

C C C C C₁, ₂, ₃, ₄, ₅

}

, then the individual would choose to equate consumption in all 5 periods. b. Equating the present value of income with the present value of consumption:

(

) (

)

(

) (

)

1 2 3 4

3

1 2 4

0 1 2 3 4

25000 25000 25000 0 25000

1 .05 1 .05 1 .05 1 .05

1 .05 1 .05 1 .05 1 .05 C

C C C

C

+ + + +

= + + + +

+ + + +

c. If we set C₀ =C₁ =C₂ =C₃ =C₄ and plug back into the budget line (simple math gives that the left side of the budget constraint is 93081.2:

(

) (

1

) (

2

) (

3

)

4

1 1 1 1

93081.2 1

1 .05 1 .05 1 .05 1 .05 93081.2 4.54595

20475.63

C

C C

⎛ ⎞

⎜ ⎟

= + + + +

⎜ ₊ ₊ ₊ ₊ ⎟

⎝ ⎠

= ⇒ =

d. For each of his four working years, he earns $25,000 but spends only $20475.63, meaning that he saves $4524.37 each year. Now, the money saved in the first year accumulates interest for four years, the money saved in the second year

accumulates interest for three years, etc… Thus, at the beginning of retirement the value of his assets is:

(

)

4

(

)

3

(

)

2

(

)

1

(40)

Unit 4.3

1. Speeding fines

a. By driving on E-11, his wealth is $100 with probability 1/2 if he gets a ticket and $300 with probability 1/2 if he does not get a ticket. Similarly, by driving on E-311, his wealth is $0 with probability 1/4 and $300 with probability 3/4.

( )

11

1 1

100 300 200

2 2

E

EW ₋ = + =

( ) (

800 + −1 p

)

0. We need the expected fine to be at least 500:

(

)

800 0 1 500 800 500 0.625

p p

+ − ≥

)

=0.9801

If the pieces are mailed together, then the probability that both will be lost is 0.01 and the probability that both will arrive is 0.99

(41)

( )

(

)

(

)

(

)

0.0001 0 0.0099 1000 0.0099 1000 0.9801 2000 1980 separate

EW = + + + =

( )

(

)

0.01 0 0.99 2000 1980 together

EW = + =

b. Similar to (a) but using the corresponding utilities:

( )

(

)

(

)

(

)

0.0001 0 0.0099 1000

0.0099 1000 0.9801 2000 44.46 separate

EU = +

+ + =

( )

(

)

0.01 0 0.99 2000 44.27 together

EU = + =

c. The expected value of the jewelry is the same either way, but your expected utility is higher when the pieces are mailed separately. Basically, the risk is more idiosyncratic this way. The risk is more systemic if the pieces are mailed together since, if lost, both are lost.

d. In this case, you have to pay 2ω when the pieces are mailed separately but only

ω when the pieces are mailed together. Using calculations similar to (b), the corresponding utilities are:

(

)

(

)

(

)

(

)

0.0001 500 2 0.0099 1500 2 0.0099 1500 2 0.9801 2500 2 separate

EU ω ω

ω ω

= − + −

+ − + −

(

)

(

)

0.01 500 0.99 2500 together

EU = −ω + −ω

(42)

4. Car theft

a. With no garage, Michael’s wealth is W =80000 with probability 0.5 and 70000

W = with probability 0.5. With a garage, Michael’s wealth is 80000

W = −P for sure, where P is the cost of the garage. Michael pays for the garage as long as his expected utility is higher:

(

)

(

)

(

)

(

)

0.4 _0.4 _0.4

0.4

80000 0.5 80000 0.5 70000 80000 89.0825

80000 74899.89 5100.11 garage no garage

EU EU

P P

≥

− ≥ +

− ≥

− ≥ ⇒ ≤

b. Similar to (a):

(

)

(

)

(

)

(

)

0.4 _0.4 _0.4

0.4

20000 0.5 20000 0.5 10000 80000 46.1706

20000 14484.84 5515.16

P P

− ≥ +

− ≥

− ≥ ⇒ ≤

c. For this utility function, the Arrow-Pratt risk aversion is:

( )

1.6

1 0.6

'' 0.24

0.6

' 0.4

U W W

W

U W W

ρ − −

−

= − = − =

Michael’s Arrow-Pratt risk aversion is ρ=0.6 80000

(

)

−1=0.000008. Samer’s Arrow-Pratt risk aversion is ρ=0.6 10000

(

)

−1=0.00006.

(43)

5. Car insurance premiums

a. The insurance company pays out 100,000 with probability 0.02 and pays out nothing with probability 0.98. The actuarially fair premium is the company’s expected payout:

(

)

( )

0.02 100000 0.98 0 2000

EP= + =

b. With no insurance, wealth is W =100000 with probability 0.98 and is W =0 with probability 0.02. With insurance, wealth is W =100000−P for sure, where P is the cost of the insurance. Abdulla buys insurance as long as:

(

)

(

)

(

)

(

)

ln 1 100000 0.02 ln 1 0 0.98 ln 1 100000 ln 100001 11.2827

100001 79435.45 20565.55 insurance noinsurance

EU EU

P P

≥

+ − ≥ + + +

− ≥

− ≥ ⇒ ≤

c. Similar to (b):

(

)

( )

(

)

100 0.5 100000 0.02 100 0.5 0 0.98 100 0.5 100000 100 50000 0.5 49100 2000

insurance noinsurance

EU EU

P

P P

≥

⎡ ⎤ ⎡ ⎤

+ − ≥ _⎣ + _⎦+ _⎣ + _⎦

+ − ≥ ⇒ ≤

d. Similar to (b):

(

)

( )

(

)

(

)

2 ₂ ₂

2

100000 0.02 0 0.98 100000 100000 9,800, 000, 000

100000 98994.95 1005.05 insurance noinsurance

EU EU

P P

≥

− ≥ +

− ≥

− ≥ ⇒ ≤

(44)

Unit 4.4

1. Consumer welfare and perfect complements. Note that I solve the problem below intuitively. You could also solve it formally by working through expenditure functions. Recall that, for perfect complements, x₁=x₂. The budget line is P x_{1 1}+P x₂ ₂ =Y. Substituting back:

1 1 2 2

1 1 2 1 1 2

1 2 P x P x Y

Y P x P x Y x x

P P

+ =

+ = ⇒ = =

+

a. Here, ₁ ₂ 300 150 1 1 x =x = =

+ so U =min 150,150

{

}

=150

b. Here, 1 2

300 100 1 2 x =x = =

+ so U =min 100,100

{

}

=100

c. To buy x₁ =x₂ =150 (giving U =150) at the new prices, the consumer would need income of Y =450. This means $150 of additional income

d. To buy x₁ =x₂ =100 (giving U =100) at the original prices, the consumer would need income of Y =200. This means $100 of his income taken away.

e. Part (c) is exactly the definition of CV and part (d) is exactly the definition of EV, so CV =150 and EV =100.

f. Consumer surplus is the difference in area under the Marshallian demand curve as the price of good 1 rises from P₁=1 to P₁ =2:

2 2

1 1

1 2 1

1 1

300

121.64 1

Y

CS dP dP

P P P

Δ = = =

+ +

(45)

2. Quasilinear Utility

a. The constrained optimization problem is:

( )

1 2 1 1 2

max ln x +x s t P x. . +1x =Y

The Lagrangian is:

( )

1 2

(

1 1 2

)

ln

L= x +x +λ Y−P x −x

1

1 1 1 1

1 1

0 L

P

x x λ λ P x

∂ ₌ ₋ _{= ⇒ =}

∂

2

1 0 1

L

x λ λ

∂ _{= − = ⇒ =}

∂

1 1 2 0 L

Y P x x

λ

∂ _{= −} ₋ ₌

∂

Equating:

1 1

1 1 1

1 1 1 1 1 P x

P x x

P

=

= ⇒ =

Substituting back into the budget line: 1 1 2

1 2

1

2 2

1

1 1

P x x Y

P x Y

P

x Y x Y

+ = ⎛ ⎞ + = ⎜ ⎟ ⎝ ⎠ + = ⇒ = −

b. Substituting the Marshallian demands back into the utility function:

( )

(

)

( )

1 2 1 1 1 ln 1 ln 1

ln 1 ln 1 ln 1

V x x

Y P

P Y Y P

= +

⎛ ⎞

= _{⎜ ⎟}+ −

⎝ ⎠

(46)

c. Solving the indirect utility function for income gives the expenditure function:

( )

1

1 1

ln 1

ln 1 ln 1

V Y P

Y V P E U P

= − −

= + + ⇒ = + +

d. At the original prices, utility is V = −Y ln

( )

P₁ − =1 10 ln 1−

( )

( )

ln 1 8.31 ln 1 1 9.31

E= +U P + = + + =

Thus, the equivalent variation is EV =10 9.31− =0.69 f. The change in consumer surplus is:

2 1 1 1

1

0.69

CS dP

P

Δ =

_∫

=

(47)

Unit 5.1

c. Each of the first 15 workers raises output by 1000 units per worker. Beyond 15 workers, each worker does not raise output at all:

2. Grade production function

a. The marginal product of E is: 0.64 0.64 0.9

q

E P

E

−

∂ = ∂

This is the increase in grade resulting from one more hour of studying for exams. b. The MRTS is:

0.64 0.64 0.36 0.36 0.9

0.5625 1.6

q E E P P

q P E P E

− −

∂ ∂ ₌ ₌ ⎛ ⎞

⎜ ⎟

∂ ∂ ⎝ ⎠

(49)

3. Diminishing marginal returns

a. The marginal product of labor is:

10 q L

∂ ₌

∂

This is constant even as L rises, so the firm does not experience diminishing marginal returns to labor. Each worker adds 10 units of output.

b. The marginal product of labor is: 1/ 4 1/ 4 1/ 4

1/ 4

3 3

4 4

q K

L K

L L

−

∂ ₌ ₌

∂

This falls as output rises, meaning that this firm experiences diminishing marginal returns to labor. The extra output of an additional worker falls as more workers are hired.

4. Returns to scale

This production function exhibits constant returns to scale regardless of the values of the parameters a and b.

b. f

(

2 , 2L K

)

=min

{

a

( ) ( )

2L b, 2K

}

=2 min

{

aL bK,

}

=2f L K

(

,

)

This production function exhibits constant returns to scale regardless of the values of the parameters a and b.

c.

(

) ( ) ( )

(

)

(

)

(

)

(

)

1/ 1/ 1/

2 , 2 2 2 2 2 2 ,

f L K L K L K L K f L K

ρ ρ ρ

ρ ρ _ρ _ρ _ρ _ρ _ρ

= + = + = + =

This production function exhibits constant returns to scale regardless of the value of the parameter ρ.

d.

(

) ( ) ( )

(

)

( )

(

)

(

)

/ _1/

/

2 , 2 2 2 2 2 ,

f L K L K L K f L K

φ ρ _ρ

ρ ρ ρφ ρ ρ ρ φ

= + = + =

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5. Automobile production

a. The marginal product of raw materials is: 0.27 0.16 0.39

0.61 q

L K M

M

−

∂ ₌

∂

b. In this case:

(

) ( ) ( ) (

0.27 0.16

)

0.61 _1.04

(

)

2 , 2 , 2 2 2 2 2 , ,

f L K M = L K M = f L K M

This production function exhibits increasing returns to scale (just barely). 6. If we interpret his comment literally, this seems to imply that relief efforts face

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Unit 5.2

1. Extra output per dollar spent on capital is 200 0.2 1000 MPK

r = =

Extra output per dollar spent on labor is 50 0.25 200

MPL

w = =

Costs are minimized when the two are equal. In this case, since MPL MPK

w > r , the firm should increase its usage of labor with a corresponding reduction in usage of capital. 2. A table of the usual cost functions is the easiest way to start this problem. Note that the

$10 to prepare the plate is a fixed cost. After the plate is prepared, cost rises by $1 for each copy that is produced. The ATC, AVC and MC functions are graphed below. Note that the AVC and MC are constant at 1, while the ATC approaches 1 as output rises.