Valuation: capital & discount rate

(Cost Benefit Analysis DEC 51304)

R. Jongeneel

Z&D Ch.3 and 13

Lecture Plan Interest rate in 1st best worldThe present value criterionEthical basis for discountingEconomic growth rateDiscount rate in 2nd best worldShadow price-approach

Interest rate in 1st best world

1)( 00 MRTP

P

PMRSslope

ii

C0

C2

Cons in P0

Cons in P1

Opportunity cost of cons - today is loss of available cons next year (MRT)

Interest rate in 1st best world

Transformation curve (PPC)

MRT = P0/P1

= MOCR + 1

Interest rate in 1st best world

Transformation curve (PPC)

MRT = MRS

MRTP = MOCR

C0

C1

Indifference

curve

Interest in 1st best world MRT is equal to marginal opportunity costs

of capital (MOCR) plus 1

Efficiency: MRTPs for all consumers equal social rate of time preference (SRTP)

SRTP=SOCR (social opp.cst of capital)

MOCRMRTPMRTMRS 010101 11

Interest rate in a 1st best world

Relevant interest rate = called discount rate

In 1st best world economy the correct discount rate for B/C-analysis is just the market rate

The Present Value-criterion

Thus:

Discounting: calculate the equivalent of C1 in period 0 (C0) (calculated along indifference curve: i.e. consumer is indifferent)

SRTPMRSC

C 101

0

1

SRTP

CC

11

0

The Present Value-criterion

cons.

project rule

0 (1 )

Ttt

t

CPV

r

1

0(1 )

Tt t

tt

B CNPV

r

Ethical basis for discounting

“Veil of ignorance”-principle

choose a discount rate before knowing to

what generation you will belong to

positive discount rate implies placing more

value on 1 euro in the present generation

than 1 euro in a future generation

Economic Growth Rate

Discount rate effects investment level Investment level determines economic

growth rate What proportion of GNP should be saved? Optimal growth path (max cons): marginal

return to capital should equal the growth rate of the economy due to technical change

Discount rate in 2nd best world Economy is economy with taxes Assume:

corporate profit tax 36%marginal income tax 27%MRTP 8%

Before-tax rate should be 10,9% to end up with 8% after (8 / (1 – 0.27) = 10.9)

Corporation must earn 17% before taxes

%9.1064.017100

)36100(17

Discount rate in a 2nd best world In a 2nd best world (tax) the choice for a discount

rate is no longer obvious

Wedge between rate of return and MRTP causes

biases (underinvestment, favors short-term

projects)

Debate:

MRTP-rate: trade present cons. for future cons.

OCR-rate: public investment displaces private

investment

Discount rate in 2nd best world

But….

SRTP (or MRTP)-approach not yet takes into

account displacement of private investment

Solution: adjust for OCR-effect

NEW: shadow price approach

Shadow Price of Capital

Trick: express both benefits and costs of an

investment in consumption terms

Total capital: K(t)=K(0)esrt

sr: growth factor

s: fraction of proceeds reinvested

r: private rate of return (OCR)

i: social rage of time preference (SRTP)

Shadow Price of Capital

srtsrt

resK

eKrstC )1(

)0(

)0()1()(

Consumption: (1-s)* net capital income

Present value (general rule)

Thus:

itPV Peitsrt eerstCPV )1())((

Shadow Price of Capital

Total cons. stream

with s . r < i (necessary cond.)

Simplifies to

T

t

itsrtresCPV0

)()1()(

sri

rsVSPC

sri

rsCPV

t

)1(

)1()(

Present value of the

consumption from

€1 of private

investment

Shadow price of

private capital

Shadow Price of Capital

Shadow price (SPC):

In 1st best world: i=r and Vt=1

sri

rsVt

)1(

Rationale: convert all benefits and costs to consumption equivalents

and discount by the social rate of time preference (shadow price of private

capital SPC)

Shadow Price of Capital (optional)

: fraction of public spending that displaces

private spending

costs

benefits

c

*)1()( ttctct CCVCPV

*)1()( ttbtbt BBVBPV

Shadow Price of Capital (optional)

Fundamental present value equation for B/C-A in a 2nd best world

with FB and FC conversion terms

T

tt

tctB

T

tttt

i

CFBF

i

CBNPV

0

0

**

)1(

)1(

Shadow Price of Capital (optional)

T

tt

tctB

i

CFBFNPV

0 )1(

Case 1: (multiplier affects size but not sign)

Case 2: (all benefits go to private capital)

Case 3: (all costs from private capital)

FFF cBcb ,

0,1 cb

1,0 cb

Shadow Price of Capital

SRTP still good choice for case 1 and when relying on global capital market

Empirical estimates SPC or Vt [ 2.5 - 3.5 ] SRTP [ 2.0 - 5.5 ]

Reasonable approx: rate on government bonds of the same length as project