Problem Set 1

1 Growth Accounting

1.1. Labor and capital share of national income

Labor share of national income :labor income

national income=

wNd

Y

Capital share of national income :capital income

national income=

r K

Y

Production function : Y = zF IK, NdM = zK1-INdM

Profit-maximizing and price-taking behaviour of firms:

MaxNdHL =MaxNdHY -VCL =MaxNdIzK1-INdM-wN

d- r KM

NdHL = zK1-INdM-1 -w U 0

zK1-INdM-1 =w

zK1-INdM-H1-L = w

z J KN

dN1-

=w

Labor share of national income :wN

d

Y=

zK1-INdM-1Nd

zK1-INdM

=INdMNd

INdMNd=

MaxKHL =MaxKHY -VCL =MaxKIzK1-INdM-wN

d- r KM

KHL = H1 - L zK-INdM - r U 0

H 1 - L zK-INdM = r

H 1 - L z J Nd

KN

= r

Capital share of national income :r K

Y=

H1-L zK-INdMK

zK1-INdM

=H1-L K- K K

K= 1 -

1.2 Firms demand for labour

MaxNdHL =MaxNdHY -VCL =MaxNdIzK1-INdM-wN

d- r KM

NdHL = zK1-INdM-1 -w U 0

zK1-INdM-1 =w

INdM-1 = wzK

1-

1

-1

-1

INdM = J wzK

1-N

1

-1

1.3 Labor Tax

U(c,l) = log c + log l, where (0,1)

FINdM = zNd

l +Ns= 100 | N

s= h - l = 100 - l

Constraint:

C w Ns+ H - TL

C w Ns

C w(100-l)

Optimization:

m ac,l

x{ log c + log l } s.t. c = w (100-l)

0 = c-w(100-l)

L(c,l,) = log c + log l + [c-w(100-l)]

FOC:

I.

cLHc, l, L = 1

c+ = 0

II.

lLHc, l, L =

l+ w = 0

III.

LHc, l, L = c - 100 w + l = 0

I. = -1

c

II.

l+ I-1

cM w = 0

l=

w

c l =

c

w in c = w (100-l)

c=w I100 - cw

M = 100 w1-

l = 100

1+

Problem Set 1

1 Growth Accounting

1.1. Labor and capital share of national income

Labor share of national income :labor income

national income=

wNd

Y

Capital share of national income :capital income

national income=

r K

Y

Production function : Y = zF IK, NdM = zK1-INdM

Profit-maximizing and price-taking behaviour of firms:

MaxNdHL =MaxNdHY -VCL =MaxNdIzK1-INdM-wN

d- r KM

NdHL = zK1-INdM-1 -w U 0

zK1-INdM-1 =w

zK1-INdM-H1-L = w

z J KN

dN1-

=w

Labor share of national income :wN

d

Y=

zK1-INdM-1Nd

zK1-INdM

=INdMNd

INdMNd=

MaxKHL =MaxKHY -VCL =MaxKIzK1-INdM-wN

d- r KM

KHL = H1 - L zK-INdM - r U 0

H 1 - L zK-INdM = r

H 1 - L z J Nd

KN

= r

Capital share of national income :r K

Y=

H1-L zK-INdMK

zK1-INdM

=H1-L K- K K

K= 1 -

1.2 Firms demand for labour

MaxNdHL =MaxNdHY -VCL =MaxNdIzK1-INdM-wN

d- r KM

NdHL = zK1-INdM-1 -w U 0

zK1-INdM-1 =w

INdM-1 = wzK

1-

1

-1

-1

INdM = J wzK

1-N

1

-1

1.3 Labor Tax

U(c,l) = log c + log l, where (0,1)

FINdM = zNd

l +Ns= 100 | N

s= h - l = 100 - l

Constraint:

C w Ns+ H - TL

C w Ns

C w(100-l)

Optimization:

m ac,l

x{ log c + log l } s.t. c = w (100-l)

0 = c-w(100-l)

L(c,l,) = log c + log l + [c-w(100-l)]

FOC:

I.

cLHc, l, L = 1

c+ = 0

II.

lLHc, l, L =

l+ w = 0

III.

LHc, l, L = c - 100 w + l = 0

I. = -1

c

II.

l+ I-1

cM w = 0

l=

w

c l =

c

w in c = w (100-l)

c=w I100 - cw

M = 100 w1-

l = 100

1+

2. Sales Tax

2.1 Consumers nominal budget constraint

(a) ad valoram tax on consumption of c1:

Y = c1 p1H1 - t1L + c2 p2

(b) per-unit tax on consumption of c1:

Y = c1Hp1 - t1L + c2 p2

2.2 Change in the optimal choice of goods c1, c2 through imposition of sales tax on c1

A per-unit sales tax on c1 has a similar effect as a price increase on good 1 for a consumer who has

a budget constraint that is subject to goods 1 and 2.

As a consequence, the price of c1has risen by which the budget constraint line rotates inwards

along the c1 axis. This means a lowering of real income for the consumer where he or she has less

of c1.

We extract the pure substitution effect from an increase in price of c1 by hypothetically imagining,

how the consumers optimal choice of goods would change, if the opportunity costs between c1 and

c2 would change for him or her (without a change in real income).

Thus we hypothetically compensate the initial real income loss by shifting the inwardly rotated

budget constraint line to the right until it tangentially intersects the consumers initial indifference

curve.

The distance on the c1 axis from the consumers initial optimal choice of goods to his or her new

optimal bundle of goods on his initial indifference curve constitutes the substitution effect.

The distance on the c1 from the consumers optimal choice of goods on his inwardly rotated budget

constraint line to the new optimal bundle of goods on his initial indifference curve would be the

income effect.

(Hicks Substitution)

2.3 Optimization

u Hc1, c2L = log c1 + log c2

m ac1,c2

x{ log c1 + log c2 } s.t. Y = c1 p1 + c2 p2

0 =Y - Hc1 p1 + c2 p2L

LHc1, c2, ) = log c1 + log c2 + [Y - Hc1 p1 + c2 p2L] = log c1 + log c2 + Y - c1 p1 - c2 p2

FOC:

I.

c1LHc1, c2, L = 1

c1- p1 = 0

II.

c2LHc1, c2, L = 1

c2- p2 = 0

III.

LHc1, c2, L = Y - c1 p1 - c2 p2 = 0

I. p1= 1

c1 =

1

c1 p1; c1 =

1

p1

II. 1

c2= p2 =

1

c2 p2; c2 =

1

p2

1

c1 p1=

1

c2 p2

c1 =c2 p2

p1

c2 =c1 p1

p2

2 | Problem Set 1.nb

2. Sales Tax

2.1 Consumers nominal budget constraint

(a) ad valoram tax on consumption of c1:

Y = c1 p1H1 - t1L + c2 p2

(b) per-unit tax on consumption of c1:

Y = c1Hp1 - t1L + c2 p2

2.2 Change in the optimal choice of goods c1, c2 through imposition of sales tax on c1

A per-unit sales tax on c1 has a similar effect as a price increase on good 1 for a consumer who has

a budget constraint that is subject to goods 1 and 2.

As a consequence, the price of c1has risen by which the budget constraint line rotates inwards

along the c1 axis. This means a lowering of real income for the consumer where he or she has less

of c1.

We extract the pure substitution effect from an increase in price of c1 by hypothetically imagining,

how the consumers optimal choice of goods would change, if the opportunity costs between c1 and

c2 would change for him or her (without a change in real income).

Thus we hypothetically compensate the initial real income loss by shifting the inwardly rotated

budget constraint line to the right until it tangentially intersects the consumers initial indifference

curve.

The distance on the c1 axis from the consumers initial optimal choice of goods to his or her new

optimal bundle of goods on his initial indifference curve constitutes the substitution effect.

The distance on the c1 from the consumers optimal choice of goods on his inwardly rotated budget

constraint line to the new optimal bundle of goods on his initial indifference curve would be the

income effect.

(Hicks Substitution)

2.3 Optimization

u Hc1, c2L = log c1 + log c2

m ac1,c2

x{ log c1 + log c2 } s.t. Y = c1 p1 + c2 p2

0 =Y - Hc1 p1 + c2 p2L

LHc1, c2, ) = log c1 + log c2 + [Y - Hc1 p1 + c2 p2L] = log c1 + log c2 + Y - c1 p1 - c2 p2

FOC:

I.

c1LHc1, c2, L = 1

c1- p1 = 0

II.

c2LHc1, c2, L = 1

c2- p2 = 0

III.

LHc1, c2, L = Y - c1 p1 - c2 p2 = 0

I. p1= 1

c1 =

1

c1 p1; c1 =

1

p1

II. 1

c2= p2 =

1

c2 p2; c2 =

1

p2

1

c1 p1=

1

c2 p2

c1 =c2 p2

p1

c2 =c1 p1

p2

3|Problem Set 1.nb 3 | Problem Set 1.nb