Chapter 10

Choice of consumption over time.

New things in the model:

Instead of two goods, there is one good (“composite good”) with p=1

2 time periods (t=1, 2)

Endowment: (m1, m2) (inc. at t=1, 2)

a) Without saving or borrowing

Budget constraints are:

c1=m1 and c2=m2

b). With saving, the budget constraints become:

c1 m1 (1)

c1+c2=m1+m2 (2)

c) Borrowing/lending with interest r (r*100 %)

Budget constraint changes from:

c1 m1

to

c2 = m2 + (1+r)(m1 – c1) (3)

(1+r)c1+c2=(1+r)m1+m2 (4)

122

111

mr)(

m

r)(

c c

(5)

(4) shows the future value of the income stream (m1, m2), (5) shows the present value.

(4) and (5) look very similar to

p1x1+p2x2= p11+p22

the budget constraint from chapter 9

c2

c1

A consumption plan (c1, c2) is AFFORDABLE if its present value does not exceed the present

value of income.

Endowment (m1, m2) dominates (m1´, m2´)

present value of (m1, m2) > present value of (m1´,m2´)

r)(

mm

r)(

mm

11

'2'

12

1

(m1, m2) permits more consumption in each period

Endowment

m1

m2

slope= -(1+r)

”Borrower” m1 < c1 (fig. 10.3 A)

c2

c1

”Lender” c1<m1 (fig. 10.3 B)

c2

c1

Endowment

Endowment

m1

m2

slope of budget line= - (1+r)

m1

m2

slope of budget line = -(1+r)

Choice

Choice

Decrease in interest rate, r “decrease in price of present consumption”

Slutsky equation revisited:

Assume r which is price of c1

m

ccm

p

c

p

c mst

111

1

1

1

1 )(

p1<0

c1s>0

if ”consumption in period one” is a normal good

For a lender (saver) m1>c1 so the first term is negative and the second positive. The sign of

the total depends on their size.

For a borrower m1<c1 so both terms are negative and 0

1

p

ct

Since p1<0, Δct1 and the quotient c

t1 /p1 have opposite signs. Since c

t1 /p<0, we must

have ct1 > 0.

Endowment

01

m

cm

Three applications:

1. Inflation

* 100 %

p2=(1+)

)1)(()(1)(1 1122 rcmmc

(m2 measured in units of consumption)

Divide both sides by 1+

)(1

11122 cm

rmc

))(1( 1122 cmmc (6)

where 11

1

r is the real rate of interest

For small r,

r -

2. Maximising utility over several periods of time

Future value (period t) of 1 SEK now: (1+r)t-1

Present value of having 1 SEK in period t:

1)1(

1

tt

rp

Budget constraint:

t

ii

it

ii

i

tt

tt

r

m

r

c

r

m

r

m

r

mm

r

c

r

c

r

cc

11

11

12

321

12

321

)1()1(

)1(...

)1()1(

)1(...

)1()1(

This assumes that the interest rate r is constant. If there are different interest rates r1, r2 and so

on at times t1, t2 etc. the formula is modified to:

)1)...(1)(1(...

)1)(1()1(

)1)...(1)(1(...

)1)(1()1(

12121

321

12121

3

1

21

t

t

t

t

rrr

m

rr

m

r

mm

rrr

c

rr

c

r

cc

An investment is worth making if:

of present values of expenses < of present value of income stream

Best of alternative investments:

The one with biggest NET PRESENT VALUE (difference between expenses and income at

present value.)

3. Present value of security maturing at date T

Nominal value: F

yearly pay-off: x

TT r

F

r

x

r

x

r

x

r

x

)1()1(....

)1()1()1(PV

122

Om T = ∞ (if the bond is a perpetuity).

r

x

r

x

r

x

r

x

r

xi

.......)1(

....)1()1()1(

PV22

r PV

Exercise 10.3 (All sums in 100s of SEK)

m1 = 200 r=0.1 π=0

m2 = 110 p1= p2=1 U(c1,c2)= c1c2

a) Present value of endowment: 3001002001.1

110200 PV

(Rule: r

mmPV

1

21 )

b) Future value: 3301102201101.1200 FV

(Rule: 21 )1( mrmFV ))

c)

1

2

2

1

c

c

cU

cU

MRS

d) Budget constraint:

122

1 1.13301.1

100200 ccc

c

slope = -1.1 and slope = MRS = c2/ c1 c2/ c1 = -1.1 c2 = 1.1 c1

Insert into budget constraint to get

c1=150 and c2=165

e) i) A lender is better off with higher r

She could still choose (150, 165) but will increase her savings. According to the

principle of revealed preferences, she is not worse off.

Chapter 12 Uncertainty Ex 1

Outcome Insured Not insured

Car stolen

(p=0.01)

100 000 – 10 900=89 100 0

Not stolen

(p=0.99)

99 100 100 000

”Excepted

value”

0,01*89100+0,99*99100=

= 99 000

0+0,99*100000=99000

premium=900, own risk=10 000

The decision depends on probabilities, outcomes, preferences

Assume that the consumer has one contingent consumption plan for each state of nature

Ex: Insurance: K=100 000

Premium γ*K= 0,01*K

Insured Not ins.

Cg 99 000 100 000

Cb 99 000 0

Line in (Cb, Cg)-graph through (0, 100 000) & (99000, 99000) and with slope 1000/(-99 000)

= -0,01/(1-0,01) =

1b

g

C

C

Please note: γ is not necessarily = the probability of outcome b

Modified version (from Varian):

Insured Not ins.

Cg 349 000 350 000

Cb 349 000 250 000

More generally, a new model:

Two outcomes: c1, c2

with probabilities 1, 2, respectively. 1= 1- 2

Utility: U = U(c1, c2, 1, 2)

The expected value of consumption:

1c1+2c2=E[c]

Examples of utility functions:

1. U= 1c1+ 2c2

2. 22

11

ccU

3. U=1 lnc1+ 2lnc2

Expected utility function (Neumann-Morgenstern) is a utility function which can be written in

the form

U(c1, c2, 1, 2) = 1v(c1)+ 2v(c2)

where v(c) is the utility from consumption of c

V is a positive affine transformation of U V=aU+b där a>0

If U is an expected utility function, every positive affine transformation of U is an expected

utility function.

*Optional:

Independence assumption For contingent consumption U is a sum of utilities of different

contingent consumption bundles.

U(c1, c2, …., cn) = i v(ci)

MRSj, k= MUj/MUk= j*v’(cj)/ k*v’(ck)

independent of ci for i ≠ j, k

Risk Preferences:

Relation between U(1*c1+2*c2) and 1U(c1)+2U(c2)

Utility of expected value Expected utility

If 0<π1 < 1 and π1+π2 = 1 then 1*c1+2*c2 = 1*c1+(1-1)*c2 is a point between c1 and c2 on

the (horizontal) c-axis and every point between c1 and c2 can be written as *c1+(1-)*c2 for

some number π, such that 0<π<1.

In the same way, 1U(c1)+2U(c2) is a point between 1U(c1) and U(c2) on the (vertical) U-

axis.

Assume U’>0

U’’<0 (U concave), Risk averse

1. U(1*c1+2*c2) >1U(c1)+2U(c2)

U’’ > 0 (U convex), Risk lover

2.

U(1*c1+2*c2) < 1U(c1)+2U(c2)

Ex. 1 If you play: Tails you get 5 kr, heads you get 7kr

If you don’t play: You get 6 kr.

Ex 2: Throw a die, get 10 kr/eye. Is it worth 70 kr to play?

Ex 3: Throw die, get 600 kr if you get a six. Is this better or worse than getting 100 kr with

certainty?

c

U

Ex 4. U(c) = 1 + 6c – c2

Concave utility function

0

2

4

6

8

10

12

0 0,25 0,5 0,75 1 1,25 1,5 1,75 2 2,25 2,5 2,75 3 3,25 3,5 3,75 4 4,25 4,5 4,75 5 5,25

Consumption

Uti

lity U(c)

Serie2

Exercise: Show that

points that can be written t*1 + (1-t)*4, where 0≤ t ≤ 1are in the interval [1, 4] on the c-axis

points (t+(1-t)*4; U(t+(1-t)*4)) are on the blue curve above [1,4}

points (t+(1-t)*4; t*U(1)+(1-t)*U(4) are on the red line.

Ex 5: M=350 000 Possible loss: 100 000 Probability of loss:

Amount insured: K Premium γ*K Outcomes: G, B

Consumption: G = 350 000 - γ*K and B = 250 000 + K - γ*K

U = *v(B) + (1-)*v(G) (1)

slope of budget line:

1B

G (2)

Tangency point of budget line and level curve of utility function::

)(')1(

)('

1 Gv

Bv

MU

MUMRS

G

B

(3)

Assume γ= (4)

Insert (4) in (3):

)('

)('

)1()1( Gv

Bv

(5)

1)('

)('

Gv

Bv (6)

(6) + risk aversion B=G

350 000 - γ*K= 250 000 + K - γ*K K = 100 000

Example 12.1 22112121 ),,,( ccccU

He has M = 100 000

J. wins J loses J doesn’t play

Game outcome+10000 -10000 0

Probability ½ ½ 1

Total wealth 110000 90000 100000

a) Expected utility if J doesn’t play: E[U]=100000≈316.2

Expected utility if he plays: E[U]= ½ 110000 + ½ 90000 ≈ 315.8

He will not take the bet. (Is J risk neutral?)

b) E[U] if he plays when c1=200 000 and c2=0 (the probabilities are the same) is

½ 200000 + ½ 0 ≈ 223.6<316.2

He won’t take the bet this time either.

c) If c1=600 000 and c2=0,

E{U] = ½ 600000 + ½0 ≈ 387.3

He will take the bet.

d) The problem is to find a number x such that ½ 90 000 + ½ x+100000 = 316.2

x+100 000 =332.4

x+100 000 = 332.42 ≈ 110 526.7

x ≈ 10527

Chapter 14 Consumer Surplus Model:

2 goods in quantities x, y

Utility function U=v(x) + y (quasilinear)

x discrete (=exists in indivisible units only)

rn reservation price for x=n

U(0, m)=v(0)+m = v(1)+m –r1 = U(1, m-r1)

r1 = v(1) - v(0)

nn nrm v(n))rm-(n-)v(n- 11

rn = v(n) – v(n-1) (1)

If v(0) = 0 then v(1) = r1

r2 = v(2) – v(1) r2 = v(2) - r1 v(2) = r2+ r1

Analogously: v(3) = r3 + r2+ r1

v(n) = rn + rn-1+...+ r2+ r1

r1 < p x=0

rn+1 < p < rn x=n (p. 248) (2)

v(n) = Gross utility from consuming n units of good 1.

NOTE: Total utility U = v(n) + m – pn

Consumer surplus: CS = v(n) – pn = Gross utility minus cost.

The most that the consumer would pay minus what she does pay.

”The smallest sum that would make her refrain from consuming any of good 1”

v(0) + m + CS = v(n) + m –pn

CS = v(n) – p*n

Demand curve for x, discrete good when preferences are quasilinear.

Striped area+ area with squares = total utlity from 4 units of x

Area with squares = consumer surplus from cons. of 4 units of x when the price is p

Striped area = 4p = cost of buying 4 units of x

If we do the same for smaller and smaller units of x, the columns get more and more narrow.

When the units get very small, we can hardly see the “step-shape” on a graph and if they are

very small indeed we can approximate the step-shaped (piece-wise linear) curve by a straight

line.

p

With a quasilinear utility function we can identify

the area under the curve with the consumer’s utility

from consuming this good and the level of the curve

at a each value of q with the marginal utility.

If the utility function is not quasilinear these would

be approximations.

q

Demand function q(p): quantity demanded at price p

Inverse demand function p(q): price at which the consumer demands quantity q

Gross consumer utility

x

dqqpxv0

)()(

(“the area under the inverse demand curve”)

p

Consumer surplus

xpxxvCS ()( )

Demand curve for good 1 (x)

*

x’’ x’

p from p’ to p’’

x from x’ to x’’

+ + + = CS at p=p’

= CS at p=p’’

p’

p’’

Compensating variation (CV) - m needed after price change to get same utility as before

the price change

.

U(old price, m)=U(new price, m + CV)

Blue line – original budget constraint

Red broken line – budget constraint after price increase

Orange dotted line – a budget constraint with the new price and an income that would

allow the same utility after the price change

CV

Equivalent variation (EV) - -m that without price change gives same change of utility as

price change would.

U(new price, m)=U(old price, m-EV)

Blue line – original budget constraint

Red broken line – budget constraint after price increase

Orange dotted line – a budget constraint with an income that at the old prices would

allow the same utility as the new price with unchanged income does.

Producer (supplier) surplus

EV

p

x

p=p’

Example 14.2

9,01,0),( yxyxU

m0 = 100, p0 = q0 = 1 where p is the price of x, q the price y

Caclulate CV and EV when p increases to 2

i) Initially U is a Cobb-Douglas function so we know that

9,01,0 9010

901

1009,0

101

1001,0

U

q

bmy

p

amx

ii) If p = 2 then x = 0,1 *100*1/2 = 5

y is independent of p, and therefore unchanged. 9,01,0 905 U

iii) CV: Let m’ = 100 + CV and assume that p = 2

x = 0,05m’ y = 0,9m’

The definition of CV

18,7

177,1071002100210021,0

10'

9,01,0

9,0100210)'(

9010)'(9,0)'(2

1,0

9010)90,10()1

'9,0,

2

'1,0(

1,09,01,01,09,01,01,0

9,01,0

9,09,01,01,09,01,0

9,01,09,09,01,0

1,0

1,0

9,01,0

CV

m

m

mm

Umm

U

iv) EV: Find the income m’= 100 –EV such that the maximal utility when p=2 and m=100 is

the same as when p=1 and m = m’

If p =2 and m = 100, 9,01,0 905U according to ii)

If p = 1, m = m’ then x = 0,1m’ och y = 0,9m’

')9(1,0)9,0()1,0(')'()9,0()'()1,0( 9,09,01,09,09,01,01,0 mmmmU

should be equal to 9,01,0 905

After simplification, this implies that m’ = 100*2-0,1

≈93,30

EV ≈ 6,7

Chapter 15 Aggregate demand

Goods x, y and n consumers

Aggregate demand for good x:

n

iiyxnyx mppxmmmppX

121 ),,(),....,,,,(

where ),,( iyx mppx =demand of consumer i

Aggregate demand - X(P)

Aggregate inverse demand - P(X)

Example of aggregate demand with two consumers:

Consumer 1 demands x1(p) units

Consumer 2 demands x2(p) units

x1 = 20 – p if p < 20

x1 = 0 if p 20

x2 = 10 – 2p om p < 5

x2 = 0 om p 5

X = 30 – 3p om p < 5

X = 20 – p om 5 p < 20

X = 0 om p 20

ELASTICITIES

q – demand

p – price

Price elasticity of demand:

q

p

p

q

pp

qq

(arc elasticity – two points on the demand curve are compared).

q

p

dp

dq

pq

dp

dq

(1)

(point elasticity)

= -1 unit elastic demand

< 1 inelastic demand

> 1 elastic demand

R – aggregate revenue

R = pq

)1()1(1 qq

p

dp

dqqp

dp

dqqp

dp

dqq

dp

dp

dp

dR (2)

1)(' qpR (3)

p

q

= qp

=pq

= qp (very small)

qdq

dppq

dq

dpp

dq

dq

dq

dR

(4)

111 p

p

q

dq

dpp

dq

dR (5)

dR/dq and dR/dp are both zero when ε = -1. Revenue is maximised when a 1% price increase

induces a 1 % decrease in the quantity sold.

qppqR

p

qpq

p

R

Demand functions with constant elasticity:

Let Appq )( where A and 1 constants

1)(' pApq

Ap

ppA

q

ppq

1

)('

Special case: = -1

p

AAppq 1)(

Exercise 15.1

Demand function D(p) = q = 10 -2p

a) What is the price elasticity of demand when p = 3

2dp

dq q(3) = 10 – 2*3 = 4

ε= -2 * ¾ = -3/2

b) At what price and quantity is ε = -1?

ε = -1

5.2

52

210

2102

1210

2

1210

p

pp

p

pp

p

p

p

p

dp

dq

c) If the demand function is D(p)=a-bp, what is the elasticity of demand?

bpa

pb

q

p

dp

dq

15.2 q(p) = (p+1)-2

a) The price elasticity of demand

1

2)1()1(2

)1()1(2 23

2

3

p

pppp

p

pp

q

p

dp

dq

b) If p≠-1 11211

2

ppp

p

p

c) Total revenue R = pq = p(p+1)-2

d) Answers to a)-c) when q(p)=(p+a)b, a>o and b<-1

ap

bp

= -1 when p = -a/(b+1) R(p) = p(a+p)

b