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THE CONSUMPTION FUNCTION

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- Looking at Aggregate Demand (closed economy)
- Ep = C + Ip + G
- Assuming G is exogenous, this leads to enquiring into determinants of Consumption and Investment
- Consumption is of particular interest (multipliers, etc)
- Previously we have:
- C = (1 - s)Y (0 s < 1)
- or, C = C(Y - T)

- We need to model the behaviour of C

- Keynes (1936) made three main assertions:
- C = C(Y), (not r)
- 0 MPC 1, (where MPC is dC/dY)
- APC falls as Y increases (APC is C/Y)
- Taken together these imply a Consumption Function of the form: C = A + bY
- where A and b are positive constants
- APC = A/Y + b
- MPC = b
- and A/Y must fall as Y increases

- As Y increases, C/Y falls: also dC/dY C/Y

C

45O

C = A + bY

dC/dY = b

A

0

Y

- Keynes hadn’t have much statistical evidence on consumption
- Early estimates in the 1940s for the USA and elsewhere were conflicting.
- Short-medium term annual data (1929-45)
- C = A + bY; A 0; b 0.7

- Long-term data (1869-1945)
- C = bY: A 0, b 0.9

- Which is “right”?
- We need a proper model to answer this.

- 1929-45: C = A + bY
- 1869-45; C = b*Y

C

45O

b* 0.9

C = b* Y

C = A + bY

b 0.7

0

Y

- Basic Intertemporal Choice model (Fisher)
- The Life-Cycle theory of Consumption (Modigliani, etc)
- The Permanent Income theory of Consumption (Friedman)

- Generally we require: PV(C) or PV(Y)
- i.e. C1 + C2 (1+r) or Y1 + Y2 (1+r)
- or Ci (1+r)i or Yi (1+r)i
- Households maximize Utility over expected lifetime
- i.e. Max: U = U (C1, ..., Ci , ... , Cn)
- s.t. Ci (1+r)i or Yi (1+r)i(i : 1 n)

Indifference Curves represent U = U(C1 , C2 )

C2

C1

0

Endowment at E: OB = PV(Y) = y1 + y2 (1 + r)

Slope of AB is (1 + r)

Y2

A

.

E

y2

y1

Y1

0

B

Why is slope AB = - (1 + r) ?

Suppose (present) savings increase by €100

i.e. C1 = - 100

This allows an increase in C2 of 100(1 + r)

i.e. C2 = +100 (1 + r)

Slope AB = C2 C1 = 100 (1 + r)/ - 100

= - (1 + r)

An increase in r: AB pivots at E CD

Y2

C

A

.

E

y2

y1

Y1

0

D

B

Saving is (oy1- oc*1) : future dis-saving is (oc*2 - oy2)

Y2

A

c*2

c*

.

y2

E

0

c*1

y1

B

Y1

Y2 increases: E’ E”, AB CD, c’1 c”1

Y2

C

A

.

E”

.

E’

0

c’1

c”1

B

D

Y1

Income effect 1 3; Substitution effect 3 2

Y2

C

F

A

2

3

1

.

y2

E

0

y1

Y1

c31

c21

D

B

G

c11

Inc. effect 1 2; Sub. effect 2 3

Y2

C

A

.

F

E

3

1

2

0

Y1

y1

c31

c11

c21

D

G

B

Borrowing rate (EB) > lending rate (AE)

C2

A

.

Y2

E

0

Y1

B

C1

.

C2

I”

Constraint: ADB

I’

A

Consumer cannot

borrow more than Y1B

E

Y2

D

0

Y1

B

C1

- Income shows a marked life-cycle variation
- It is low in the early years, reaches a peak in late middle age and declines, especially on retirement
- Smoothing consumption over a lifetime is a rational strategy (diminishing MUy)
- This implies C/Y will vary during the lifetime of an individual

.

C2

E’: low Y1/Y2 high C1/Y1

E”: high Y1/Y2 low C1/Y1

A

E’

.

C2*

.

E”

C1

B

0

Y1’

C1*

Y1”

Y, C and W over the life-cycle

Y, C

Ct

Yt

Age

18

65

+W

Wt

Age

W

- Let retirement age = 65; life expectancy = 75
- Years to retirement = R (= 65 – present age)
- Expected life = T (= 75 – present age)
- Assuming no pension, no discounting:
- CT = W + RY is the lifetime constraint
- i.e. C = (W + RY)/T
- and C = (1/T)W + (R/T)Y
- or C = W + Y ( = 1/T; = R/T)

- C = W + Y
- MPC = C Y =
- APC = C Y = (W Y) +
- clearly MPC < APC
- for a “typical” individual, age 35
- R=30, T = 40
- = 1/T 0.03; (MPC) = RT 0.75
- APC = [0.03 (W Y) + 0.75] > MPC

- Saving and Consumption behaviour may depend on population age-structure
- Does Social Security displace personal savings?
- What is the effect of Medicare (USA) or Medical Cards for over 70s (IRL) on Savings?
- Savings and Uncertainty:
- “rational” behaviour: run down wealth to zero
- individual circumstances unpredictable (care needs)
- individual life expectancy unpredictable
- on average even selfish people will die with W > 0

- Cp = kYp (0 k 1 )
- Y = Yp+ Ytr
- C = Cp + Ctr
- Permanent income is the return to all wealth, human and non-human:
- Yp = rW
- which implies: Cp = rkW
- NB: C is not related to Ytr i.e. dC dYtr = 0

- Are Cpand Yp observable?
- E(Ytr ) = 0
- E(Ctr ) = 0
- which imply that E(Y) = E(Yp ), etc.
- However this is ex ante: ex post, actual measures may reveal more
- (a) in a recession: Y < Yp : Ytr < 0
- (b) in a boom: Y > Yp : Ytr > 0

- Cross-section measurements of C and Y

C

45o

Ci, Yi.

.

.

.

.

Ci = A + bYi

.

.

Cm

.

0

Y

Ym

- Where Yj > Ym, Ytr > 0 and Yj > Ypj

C

45o

Cp =kYp

Cj

Ci = A + bYi

Cm

Ytrj

0

Y

Yj

Ym

Ypj

- Aggregate: Ytr > 0 in boom, < 0 in recession
- Measured C/Y should be < in boom than in recession (Recent experience?)
- Aggregate Ctr = 0: individual Ctr is > or < 0
- Average Ctr = 0 for all income groups
- Measuring Yp:
- Adaptive expectations: Yp = f(Yt, Y t - 1, ...Y t-n)
- Rational expectations: only new information (shocks) change Yp
- Consumption V Consumption Expenditure, which highlights the role of durables (Investment and saving rather than consumption

- Also we may express the PYH as an error-correction model:
- Ypt = Ypt-1 + j(Yt – Ypt-1) 0 < j < 1
- which with: Ct = Cpt = kYpt
- gives: Ct = kYpt = kYpt-1 + kj(Yt – Ypt-1)
- Re-arranging: Ct = (k – kj)Ypt-1 + kjYt
- j 0 implies slow adaptation, j 1 implies rapid adaptation
- assume k = 0.9, j = 0.3, so kj = 0.27
- then: Ct = (0.9 – 0.27)Ypt-1 + 0.27Yt or 0.63Ypt-1 + 0.27Yt
- However this is not an explicitly forward-looking model.
- Now suppose C = Cp = kYp, then Yp = 1/k(Cp)
- Thus Ct = (0.63/k)Ct – 1 + 0.27Yt = 0.7Ct – 1 + 0.27Yt

- Y < Yp in short-run (mild) recession
- Suppose there is a shock to the system (financial crisis)
- Pwople expect a severe long-drawn-out recession: i.e. Yp falls, ie. E(Y) falls
- It is possible that initiallyY > Yp
- C (and Cp) will fall
- If people anticipate a fall in Yp, then C/Y may fall
- Current (mid-2009) situation: big fall in W, both the Permanent and Life-cycle theories predict that this will hit C (independently of current measured Y)