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FINANCIAL ECONOMICS:

Amnon Levy’s Lecture Notes

Spring 2013

University of Wollongong

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Table of Content

Part 1. Investment Theory Week 1 The discrete-time discount factor: a derivation

Evaluation of investment projects: Tobin’s q criterion Net present value criterion Internal rate of return criterion

Tutorial questions The dynamics of capital stock

Tutorial questions Week 2 The Dale Jorgenson’s user cost approach to computing the desired

capital stock Net investment and the Baumol’s partial adjustment model

Tutorial questions Week 3 The role of interest rate in the optimal choice of saving and

consumption: a two-period analysis Tutorial questions

Saving with fixed income, consumption and interest rate: an analysis in continuous time The continuous-time discount factor

Tutorial questions Week 4 Intertemporal analysis of efficient investment in production capital

with costs of adjustment: The model’s building blocks The optimal control problem and the sufficient and necessary conditions The singular control and the no-arbitrage rule

Week 5 Intertemporal analysis of efficient investment in production capital

with costs of adjustment (cont.): The phase-plane diagram of capital stock and gross investment

Week 6 Intertemporal analysis of efficient investment in production capital

with costs of adjustment (cont.): The steady state capital stock and investment and comparative statics Specific Appendix: Identification of the local asymptotic stability nature of the model’s steady state General Appendix: Identification of the local asymptotic stability nature of a steady state of an infinite horizon optimal control problem

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Tutorial questions Extension: efficient investment and borrowing with adjustment and transaction cost and with tax considerations The New Theory: irreversibility, uncertainty and the delay option

Week 7 In-Session Examination Week 8 Modigliani-Miller approach to valuation of firms, leverage and cost of

capital Houses’ values and prices and the cobeweb phenomenon Tutorial questions Part 2. Uncertainty, Risk, Insurance and Portfolio Week 9 Risk, expected utility and optimal insurance The Von Neuman-Morgenstern utility function Optimal insurance (coverage) level Supply side and the actuarially fair premium rate

Moral hazard Tutorial questions

Week 10 Uncertainty, Risky Assets (Activities) and Portfolio: Basic concepts Efficient Portfolios: Mean-Variance Approaches

The Markowitz Efficient Portfolio Set Tobin’s Capital Market Line (CML) William Sharpe’s Capital Asset Pricing Model (CAPM) Tutorial: Construction of the CML

Week 11 Expected utility maximizing approach to choice of portfolio Tutorial questions

Part 3. Financial Crises Week 12 Evolution of a firm’s debt burden

Developing countries’ external sovereign debts: Structure Causes and evolution The David Howard’s model Implication for exchange rate policy Tutorial questions

Week 13 Developing countries’ external sovereign debts (cont.):

External debt and economic growth The Debt Laffer Curve and discount Repudiation and debt’s secondary market price Tutorial questions

The Global Financial Crisis, 2007-present

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Lecture 1 (Week 1)

Discrete Time Discount Factor

The discount factor transforms future nominal values to present-time ones. The

following procedure generates an agent’s discount factor in discrete time with time-

invariant (fixed) interest rate. The analogue in continuous time will be developed in

lecture 3. Suppose that the best alternative activity/asset for an agent is a saving

account with flat, time-invariant interest rate r. A sum of om is deposited in this

account and no further deposits, nor withdrawals, are made. The motion equation of

the account is therefore:

1(1 )t tm r m −= + . (1)

It can be expressed as a homogeneous, first order, linear difference equation:

1(1 ) 0t tm r m −− + = . (1’)

Let t

tm cγ= (3)

be a solution, then by substituting into (1’): 1[ (1 )] 0tc rγ γ− − + = (4)

The non-trivial solution for γ is:

1 rγ = + (5)

and consequently

(1 )ttm c r= + (6)

Substituting 0t = and recalling the initial condition, ( 0) om t m= = , then oc m= .

Hence, the solution to the initial-value problem is:

(1 )tt om m r= + . (7)

This implies that the present value of tm available in that account at some future date

t is:

11 = +

t

t om mr

. (8)

The term on the left-hand side that multiplies tm can be interpreted as the discount

factor of the said agent, 1/ (1 )+ tr . Note that if the interest rate is variable, the

discount factor of tm in discrete time can be proven to be

5

11 2 3

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

/=

= ++ + + + ∏

t

t

rr r r r

τ

τ

. (9)

Evaluation of Investment Projects: Tobin’s q, Net Present Value, Internal Rate of Return

The Tobin’s q criterion

' 1cos

market value of the investment projectTobin s qreplacement t of the investment project

>

<≡ = .

If the investment project’s Tobin’s q is smaller than 1, the project should be rejected.

The Net Present Value Criterion

0

1 1( )1 1=

= − − + + + ∑

t TT

t t t Tt

NPV R C I Sr r

where,

r is a time-invariant discount rate considered by the planner,

1,...,t T= and T the life expectancy of the project,

tI is the expected cost of the investment in the project in period t,

tR is the expected revenue generated by the project in period t,

tC is the expected operational costs of the project in period t, and

TS is the expected salvage (or resale) value of the project.

The components of investment projects’ NPV formula are tabulated below.

Year Investment Costs Revenues Operational

Costs Salvage Value

0 0I 0R 0C 1 1I 1R 1C

2 2I 2R 2C

3 3I 3R 3C

T TI TR TS

=NPV 0

11=

− + ∑

tT

tt

Ir

0

11=

+ + ∑

tT

tt

Rr

0

11=

− + ∑

tT

tt

Cr

11 + +

T

TSr

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If the project’s expected 0NPV < , the project should be rejected. Note that an

expected 0≥NPV does not necessarily imply that the proposed project has to be

taken. Obstacles in computing the expected NPV stem from the uncertainty about T, I,

R, C and TS .

The Internal Rate of Return (IRR) Criterion

Investment projects vary in size and the development budget (say, of a ministry) is

limited. It is useful to calculate the IRR of each the proposed competing investment

projects and to use the IRRs for ranking in descending order. The internal rate of

return on an investment project is defined as the discount rate ρ for which

( ) 0NPV ρ = . It is found by solving

0

1 1( ) 01 1

t TT

t t t Tt

NPV R C I Sρ ρ=

= − − + = + + ∑

for ρ . If rρ < , the project should be rejected.

Tutorial questions

1. Develop the formula of IRR ( ρ ) for the case where T=1.

2. Develop the formula of IRR ( ρ ) for the case where T=2.

The Dynamics of Capital Stock

Assuming linear depreciation, the capital stock evolution (or net investment) from an

initial value oK can be displayed as:

1 1t t t t tNI K K I Kδ− −≡ − = − , (1)

where 0 1δ< < is, for simplicity, a time-invariant (fixed) depreciation rate.

The evolution of the capital stock can be portrayed as a first order linear difference

equation:

1(1 )t t tK K Iδ −− − = (1’)

Let us solve the initial-value problem and demonstrate the evolution of capital stock

for the special case of a time-invariant gross investment, 1, 2,3,...tI I t= ∀ =

Step 1: the stationary capital stock ( ssK )

In steady state,

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1t t ssK K K−= = .

Hence,

ssK Iδ =

and

/ssK I δ= . (2)

Step 2: the solution to the homogeneous part

The homogeneous part of (1’) is:

1(1 ) 0h ht tK Kδ −− − = (3)

Assuming that h ttK γ= , then

1 [ (1 )] 0 1 tγ γ δ γ δ− − − = ⇒ = − (4)

and hence

(1 )h ttK δ= − . (5)

Step 3: the general solution

/ (1 )h tt ss tK K cK I cδ δ= + = + − (6)

Step 4: Finding c

At 0t = , 0/ (1 )oK I cδ δ= + − and hence

0 /c K I δ= − . (7)

Step 5: the solution to the initial value problem

/ ( / )(1 )tt o ssK I K Iδ δ δ= + − − . (8)

Property: Recalling that 0 1δ< < , lim(1 ) 0t

tδ

→∞− = and hence lim /t sst

K I Kδ→∞

= = .

That is, in this particular case, the stationary capital stock is asymptotically stable—it

is attainable from any initial level, as depicted in the figure below.

ssK

K

t 0

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Lecture 2 (Week 2)

The Dale Jorgenson’s User-Cost Approach to Computing the

Desired Capital Stock

The user (or rental) cost of capital ( ci )

The cost of a dollar directed to hiring production capital for a period of time (e.g., a

year) is the foregone periodical (e.g., annual) real interest rate [i.e.,

( ) / (1 )r i π π= − + , where i is the periodical nominal interest rate and π the expected

periodical inflation rate1] plus the periodical (e.g., annual) depreciation rate (δ ) of

that capital; namely,

δ+= ric .

The firm

The firm employs only two inputs: capital (K) and labor (L). Its production process is

represented by a concave function ),( LKF displaying positive, but diminishing,

marginal products of capital and labor (i.e., positive first partial derivatives,

0, >LK FF , and negative second non-cross derivatives, 0, <LLKK FF ). The firm is a

small player (i.e., has no monopsonistic power) in the markets of capital and labor.

Being a price-taker in these markets, its user cost ( ci ) and worker’s wage (w) are

positive scalars.

The Firm’s objective

Finding the combination of inputs that minimizes the cost of producing a

predetermined output level.

Isocost

The locus of all the combinations of capital and labor that generate the same level cost

(C ) for the firm. That is, the set of all (K,L) for which

CwLKic =+ .

1 1 (1 )(1 ) (1 ) ( ) / (1 )i r i r r i r r i+ = + + ⇒ = + + ⇒ − = + ⇒ = − +π π π π π π π .

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A total differentiation of the above expression while holding the level of cost

unchanged can help finding the slope of the isocost curve in the plane spanned by L

and K. Namely,

0==+ dCwdLdKic

and by rearranging terms,

ciw

dLdK

−= .

The slope of any isocost is the negative of the labor-capital price ratio and as the firm

is a price-taker in these inputs’ markets, this ratio does not vary with the combination

of capital and labor employed by the firm but is fixed. Thus, an isocost is negatively

sloped and linear in the plane spanned by L and K and the greater is distance from the

origin the higher the associated input-hiring cost.

L

K

0 w/ic

For countries where labor is relatively scarce (e.g., North-Western Europe, Australia,

Canada and USA) the isocosts are steeper than for countries where labor is relatively

abundant (e.g., Bangladesh, India and China).

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Isoquant

The locus of all the combinations of capital and labor producing the same level of

output. That is, the set of all (K,L) for which

yLKF =),( .

By total differentiation, while holding the output level constant,

0),(),( ==+ dydLLKFdKLKF Lk .

Consequently,

),(),(

LKFLKF

dLdK

K

L−= .

As 0, >LK FF and 0, <LLKK FF the slope of an isoquant in the L-K plane is negative

and diminishing as displayed in the following figure (excluding the special cases

where labor and capital are prefect substitutes or perfect complements).

L

K

0

The Production-Input Choice

The Dale Jorgenson’s user-cost approach for determining a firm’s desired capital

stock suggests that the firm chooses its mix of inputs, capital and labor for simplicity,

efficiently so as to minimize the costs of producing a predetermined quantity of

output during a given period. The decision problem of the firm, in the Jorgenson’s

context, can be formally displayed as:

)(min,

wLKicLK+

subject to yLKF =),( .

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Graphical solution

The graphical presentation of the problem and its solution is based on isocosts and

isoquants. The solution ( ** , LK ) of the firm’s problem is found at the tangency point

between an isocost and the isoquant associated with the production level y.

L

K

0 w/ic

K*

L*

y

In the cost-minimizing combination of capital and labor ( *, *)K L the following

optimality conditions prevail:

1. * *

* *

( , )( , )

L

K c

F K L wF K L i

=

2. yLKF =),( **

These optimality conditions constitute a system of two equations with two unknowns

whose solution is the firm’s demand for production inputs:

the desired capital stock

),,(* yiwfK c=

and the desired labor level

),,(* yiwgL c= .

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Lagrange-method solution

The optimality conditions can also be obtained by applying the Lagrange method to

)(min,

wLKicLK+ subject to yLKF =),( . The Lagrange function ( L ) associated with

this problem is:

[ ( , )]= + + −ci K wL y F K LλL

where λ is the Lagrange multiplier and is interpreted as the marginal cost of

production.

Since ( , )F K L is jointly concave and ( )ci K wL+ is linear in K and L, then L is

jointly convex in K and L, the second-order conditions for minimum are satisfied and

an interior solution exists. The first order conditions (FOCs) for minimum are:

( *, *) 0 ( *, *)∂ = − = ⇒ = ∂ L Lw F K L F K L w

Lλ λL

( *, *) 0 ( *, *)∂ = − = ⇒ = ∂ c K K ci F K L F K L i

Kλ λL

( *, *) 0 ( *, *)∂ = − = ⇒ = ∂ y F K L F K L y

λL

The division of the first condition by the second implies the first optimality condition:

( *, *)( *, *)

L

K c

F K L wF K L i

λλ

= .

Illustration: The desired capital stocks for two countries endowed with the same

technology but with different input-price ratio, BcAc i

wiw

>

.

L

K

0

K*A

L*A

y

A

B K*B

L*B

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Example

Consider the case where the firm’s production function is Cobb-Douglas and

displaying constant return to scale, i.e., 10,0,1 <<>= − ααα ALAKF . The positive

parameter A represents the firm’s technological level and the parameters α and α−1

are the elasticities of the firm’s production with respect to capital and labor,

respectively (i.e., αη =∂∂

=KF

KF

k / and αη −=∂∂

= 1/LF

LF

L ).

In this case, the optimality condition of equality between the ratio of the marginal

products of labor and capital ( KL FF / ) and the input price ratio ( ciw / ) is

ciw

LAKLAK

=−

−−

−

αα

αα

αα

11 ****)1( .

By rearranging terms,

* *c

c c

i(1 )K * w K * w 1( )( ) L ( )( )KL* i L* 1 i w

−α α −α = ⇒ = ⇒ = α −α α .

By substituting this expression into the second optimality condition for *L ,

yKwi

AK c =

−

−αα

αα 1

** ))(1(

By rearranging terms, the desired capital stock is

α

αα −

−

= 1*

))(1(wi

A

yKc

.

This expression proposes that the desired capital stock:

1. proportionally increases with the output level to be produced,

2. decreases with the firm’s technological level,

3. increases with the capital production elasticity,

4. decreases with the labor production elasticity,

5. increases with the wage rate, and

6. decreases with the user cost.

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Net Investment and the Baumol’s Partial Adjustment Model Due to various possible constraints (e.g., funds, floor space, training complementary

labor, and/or objection of trade union to labor-replacing capital) there might not be an

immediate adjustment of the capital stock to the desired level. William Baumol’s

partial adjustment model reflects that at every period the net investment in capital is

proportional to the difference between the desired stock and the actual stock of capital

at the end of the previous period:

10),( 1*

1 <<−=−= −− φφ tttt KKKKNI .

By collecting term, Baumol’s model can be rendered as a first-order, linear difference

equation *

1)1( KKK tt φφ =−− −

whose solution is obtained by using the following five-step method.

Step 1: The stationary capital stock, ssK

In steady state, 1t tK K −= and hence

*(1 )ss ssK K Kφ φ− − =

which in turn implies: *

ssK K= .

Step 2: The solution to the homogeneous part 1(1 ) 0h ht tK Kφ −− − =

Suppose h ttK γ= .

Then 1

1h ttK γ −− = .

By substitution into the homogeneous part: 1(1 ) 0t tγ φ γ −− − = .

In turn, 1 [ (1 )] 0 (1 )tγ γ φ γ φ− − − = ⇒ = − .

Hence,

(1 )h ttK φ= − .

Step 3: the general solution

* (1 )h tt ss tK K cK K c= + = + −φ , where c is a scalar.

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Step 4: the value of c

Recalling the 0( 0)K t K= = , and substituting 0t = into the general solution:

00 * (1 ) *K K c K cφ= + − = +

and hence

0 *c K K= − .

Step 5: the solution to the initial value problem

By substituting 0 *c K K= − into the general solution,

tt KKKK )1)(( *

0* φ−−+= .

Since tt KKKK )1)(( *

0* φ−−+= and t)1( φ− diminishes over time (i.e., as ∞→t ),

the actual capital stock asymptotically converges to the desired capital stock from any

possible initial level that is different from the desired one, as displayed by the

trajectories in the following figure.

*K

K

t 0

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Exercise 1 (To be handed to the lecturer at the beginning of the lecture in week 3)

Firm A invest every year 1,000,000 dollars in on premises’ production capital. Its

production capital depreciates at a rate of 0.1 per annum. Its present production

capital stock is 5,000,000 dollars. Use 1 1t t t t tNI K K I Kδ− −≡ − = − to compute

1. firm A’s steady-state capital stock; and

2. the first three years levels of this stock.

Consider a firm whose production process can be depicted by the following function 0.6 0.44000Y K L= . The annual salary of any worker employed by the factory is $

30,000. The annual nominal interest rate is 0.07. There is no inflation expected by the

firm’s manager. The annual capital-depreciation rate is 0.03. The value of the initial

capital stock in the factory is $ 10,000,000. The manager adjusts the capital stock in

the factory to the desired level by 50 per cent of the discrepancy in any given year.

The objective of the firm’s manager is to minimize the costs of producing $

120,000,000 value of goods in any given year. Use the Jorgenson and Baumol models

to answer the following questions.

3. What is the firm’s user cost?

4. What is the firm’s desired capital stock (measured in dollars)?

5. What is the firm’s net investment in the first year?

6. What is the firm’s net investment in the second year?

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Lecture 3 (Week 3)

The Role of Interest Rate in the Optimal Choice of Saving and Consumption:

A Two-Period Analysis Consider a person who:

1. lives two periods: present (0) and future (1);

2. has certain incomes: 0y at present and 1y in the future;

3. can transfer financial resources from one period to another in a fixed interest

rate )(r : saving in the case of a transfer from period 0 to period 1, or

borrowing in the case of a transfer from period 1 to period 0;

4. leaves neither bequest nor debt at death;

5. who derives utility from consumption at present and in the future in

accordance to a concave lifetime utility function ),( 10 CCu ; and

6. who simultaneously chooses consumption in period 0 ( 0C in nominal values)

and consumption in period 1 ( 1C in nominal values) at the beginning of period

0 so as to maximize his/her utility subject to his/her inter-temporal budget

constraint.

Formally, the decision problem of this person is:

0 1max ( , )u C C

subject to the intertemporal budget constraint

0

0011 )()1(S

CyryC −++= .

As displayed by the intertemporal budget constraint, the price of present consumption

comprises the purchasing price ($1) and the forgone interest on that dollar (i.e., 1 r+ ).

Due to the concavity of the utility function and linearity of the budget constraint, the

corresponding Lagrange function

0 1 1 0 0 1( , ) [ (1 )( ) ]= + + + − −u C C y r y C CλL

is concave in 0C and 1C and hence there exists an interior solution ( **1

*0 ,, λCC ).

The Lagrange multiplier (λ ) can be interpreted in the present case as the shadow

value of the budget available for the second period consumption. It is expressed in

utility terms. It is the marginal lifetime utility of funds available in the second period.

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The interior solution satisfies the following necessary conditions:

* *

*0 1

0 0

( , ) (1 ) 0∂∂= − + =

∂ ∂u C C r

C CλL (1)

* *

*0 1

1 1

( , ) 0∂∂= − =

∂ ∂u C C

C CλL (2)

* *

1 0 0 1(1 )( ) 0∂= + + − − =

∂y r y C C

λL . (3)

From equations (1) and (2) [i.e., dividing both sides of equation (1) by *λ while noting that * * *

0 1 1( , ) /u C C Cλ = ∂ ∂ ], the marginal rate of substitution of future consumption for present consumption is equal to the price ratio of present and future consumption:

rCCCuCCCu

+=∂∂∂∂

1/),(/),(

1*1

*0

0*1

*0 (4)

From equation (3)

))(1( *001

*1 CyryC −++= . (5)

The system comprising equations (4) and (5) determines *

0C and *1C and subsequently

the individual’s level of saving (when *0 0 0y C− > ), or borrowing (when

0*00 <−Cy ).

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Illustration: Two-period consumption and saving for two different interest rates ( r r′> ) Extension: Two period consumption and saving with borrower lenderr r>

0C

1C

0y

1y

1C

0y

1y

1 r+ 1 r′+

0C

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Tutorial questions 1. Let,

5.01

5.00 9.0 CCu +=

0 1 $ 60,000y y= =

0.08r = .

1. Prove that the optimality condition leads to:

2* *

1 00.45 (1 )0.5

C r C = +

2. Substitute the above expression into the budget constraint and show that

* 1 00 2

(1 )0.45 (1 ) (1 )0.5

y r yCr r

+ += + + +

3. Compute 010 ,, SCC . 4. Suppose, instead, that 10.0=r and all other things remain the same. Re-compute

010 ,, SCC and draw a conclusion on the relationship between saving (or borrowing) and interest rate.

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Saving (m) with Fixed Income (y), Consumption (c) and Interest Rate (r): An Analysis in Continuous Time

The change in the individual’s saving account is given by:

( ) ( ) ( )dm m t rm t y cdt

≡ = + −

It can be expressed as a first order linear differential equation (FOLDE):

( ) ( ) ( )m t rm t c y− = − − whose solution, as explained by the five steps below, is

0( ) rtc y c ym t m er r− − = + −

where 0( 0)m t m= = . Step 1: The stationary level of the saving account

In steady state, 0m = . The substitution of this property in the FOLDE implies:

( )ssrm c y− = − −

Hence,

ssc ym

r−

= .

Step 2: The solution to the homogeneous part ( ) ( ) 0h hm t rm t− =

Assume that

( ) thm t eγ= .

Then,

( ) thm t eγγ= .

By substitution into the homogeneous part:

0t te reγ γγ − =

which implies:

rγ = .

Hence,

( ) rthm t e= .

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Step 3: the general solution

( ) ( ) rtss h

c ym t m m t er

β β−= + = +

Step 4: The value of β

By substituting 0t = into the general solution:

0

0

( 0)ss h

r

m m m tc y c ye

r r

β

β β

= + =− −

= + = +

Hence,

0 0ssc ym m m

rβ −= − = − .

Step 5: the solution to the initial value problem

By substituting the value of β into the general solution:

0( ) rtc y c ym t m er r− − = + −

.

The continuous-time discount factor ( rte− )

Assuming no withdrawals and deposits, the level of the saving account can be

obtained by substituting 0t tc y= = for every instance t into the above solution to the

initial value problem:

0( ) rtm t m e= .

Dividing both sides by rte ,

0 ( )rtm e m t−= .

The term rte− can be interpreted as the continuous-time discount factor. Tutorial questions On her retirement at 65 years of age Rosemary received from her superannuation (pension) fund a sum of 400,000 dollars, which she deposited in a saving account with a fixed 6 percent annual interest rate. Rosemary expects her annual spending to be 30,000 dollars.

1. Denote by t the date (instance) in which Rosemary’s saving account is exhausted and substitute this date into the solution to the saving account FOLDE and prove that

0

1 /ˆ ln/

= −

c rtr c r m .

2. To which age ( ˆ65 t+ ) Rosemary’s saving will be sufficient?

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Lecture 4 (Week 4)

Intertemporal Analysis of Efficient Investment in Production Capital with Costs of Adjustment

1. The model’s building blocks Consider a firm with the following characteristics. 1.1 Production: The firm produces a single product with capital and, for simplicity,

with a fixed quantity of labour in a time-invariant technology that can be depicted by

the following concave production function:

( ( ), )ty F K t L=

where 0KF > and 0KKF < .

1.2 Markets: The firm operates in perfectly competitive markets of its product and

inputs (i.e., it is a price taker). The price of its product (P), wage rate (w) and

purchasing price of capital (q) may change over time due to changes in demand and

supply in the respective markets. 1.3 Adjustment Costs: The instantaneous costs of adjustment (e.g., instalment costs)

associated with new capital are depicted by a convex function ( )tC I with ( ) 0tC I′ >

and ( ) 0tC I′′ > .

1.4 Capital motion equation: The firm’s initial capital stock is 0K and, despite

having a stock with a variety of vintages, the rate of depreciation of the firm’s capital

is taken, for simplicity, to be a scalar 0 1δ< < . Consequently,

( ) ( ) ( )K t I t K tδ= − . 1.5 Planning Horizon and Time Preferences: The firm is infinitely lived and has a

time-invariant (fixed) rate of time preference (discount), 0ρ > .

1.6 Objective: The firm chooses its gross investment path I so as to maximize the

sum of its discounted instantaneous net cash-flows subject to the motion equation of

its capital stock. This postulate is consistent with maximising the value of the firm in

an efficient stock market (i.e., in a stock market where all agents—buyers and sellers

of stocks—are rational and informed).

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2. The optimal control problem and the sufficient and necessary conditions The firm’s objective can be formally expressed as:

0

max [ ( ) ( ( ), ) ( ) ( ) ( ( )) ( ) ]t

Ie P t F K t L q t I t C I t w t L dtρ

∞− − − −∫

subject to the firm’s capital’s motion equation:

( ) ( ) ( )K t I t K tδ= − and the initial condition:

0(0)K K= . The (present value) Hamiltonian associated with this optimal control problem is:

( ) [ ( ) ( ( ), ) ( ) ( ) ( ( )) ( ) ] ( )[ ( ) ( )]−= − − − + −tt e P t F K t L q t I t C I t w t L t I t K tρ λ δH where ( )tλ is the co-state variable. It converts physical units of net capital investment into nominal units. Hence, it can be interpreted as the shadow value (expressed in present value) of capital. The second-order (curvature) conditions require that the present value Hamiltonian is jointly concave in the state variable and the control variable. That is, the 2 × 2 Hessian matrix of H with respect to (K,I) must be negative semidefinite. Noting that the Hessian matrix of H is

( , ) 00 ( )

tKK

t

e PF K LH

e C I

−

−

=

′′−

ρ

ρ

and recalling assumptions 1 and 3, the principal minors of H are negative (i.e.,

( , ) 0tKKe PF K L− <ρ and ( ) 0te C I− ′′− <ρ ) and det ( , )[ ( )] 0t t

KKH e PF K L e C I− − ′′= − >ρ ρ , as required. Hence, the following necessary conditions for maximum are also sufficient:2

( )( ) ( ) ( ( ), ) ( )−∂= − = − +

∂

tK

tt e P t F K t L tK

ρλ λ δH (the adjoint equation)

( ) [ ( ) ( ( ))] ( ) 0−∂ ′= − + + =

∂tt e q t C I t t

Iρ λH (the optimality condition)

2The maximum principle was developed by Pontriagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V. and Mishchenko, E.F. (The Mathematical Theory of Optimal Process, Wiley, New York, 1962). It shows that if the control function [I*(t)] and the corresponding path of the state variable [K*(t)] maximize the objective function subject to the constraints, there exists a continuously differentiable function ( )tλ such that I*, K* and λ simultaneously satisfy the necessary conditions. Mangasarian (1966) provided the basic sufficiency theorem for the maximum principle (Mangasarian, O.L, “Sufficient Conditions for the Optimal Control of Nonlinear Systems,” SIAM Journal Control 4, 1966: 139-152).

25

( ) ( ) ( )K t I t K tδ= − (the state equation)

lim ( ) ( ) 0t

t K t→∞

=λ (the tranversality condition).

In an infinite horizon dynamic optimization, the transversality condition requires the present value of the state variables, capital stock in the present case, to converge to zero as time goes to infinity. It is the boundary condition in an infinite horizon dynamic optimization. Combined with the initial condition, 0( 0)K t K= = , it determines a solution to the problem's first-order conditions. “The first-order and transversality conditions are sufficient to identify an optimum in a concave optimization problem. Given an optimal path, the necessity of the transversality condition reflects the impossibility of finding an alternative feasible path for which each state variable deviates from the optimum at each time and increases discounted utility (i.e., value of the firm in the case under consideration, A. Levy).” (Robert A. Becker, “Transversality Condition”, The New Palgrave Dictionary of Economics, Second Edition, 2008, Edited by Steven N. Durlauf and Lawrence E. Blume) The optimality condition implies that along the profit maximizing path of investment the shadow value of capital is equal to the cost of acquiring an additional unit of capital—the purchasing price and the marginal adjustment cost:

( ) [ ( ) ( ( ))]tt e q t C I tρλ − ′= + . Dividing both sides of the adjoint equation by ( )tλ , while noting that

( ) [ ( ) ( ( ))]tt e q t C I tρλ − ′= + , the rate of change of the shadow value of capital is given by:

( ) ( ( ), )( )( ) [ ( ) ( ( ))]

KP t F K t Ltt q t C I t

λ δλ

= −′+

.

Compatible with the association of value with scarcity, the first term on the right-hand side (RHS) of the above equality implies that higher the capital’s depreciation rate, the greater the firm’s appreciation of its remaining capital. The second term on the RHS indicates the potential accumulation of capital generated by the last unit of capital added to the firm’s stock—the market value of the firm’s capital’s marginal product divided by the cost of acquiring additional capital. The larger this potential accumulation of capital, the lower the scarcity of capital in the firm and, in turn, the firm’s rate of appreciation of its capital. 3. The singular control and the no-arbitrage rule While the firm’s decision problem is intertemporal, the Hamiltonian and, in turn, the first order condition for maximum are constructed for a point in time t. The optimality condition can be perpetuated by differentiation with respect to time and setting the resultant derivative to be equal to zero:

26

( ) 0 0∂ = = ∂ d tdt I

H (singular control).

More specifically,

[ ( ) ( ( ))] ( ) 0 0td e q t C I t tdt

ρ λ− ′− + + = =

which implies that the singular control equation is:

[ ( ) ( ( ))] [ ( ) ( ( )) ( )] ( ) 0t te q t C I t e q t C I t I t tρ ρρ λ− −′ ′′+ − + + =

. The substitution of the adjoint equation, ( ) ( ) ( ( ), ) ( )t

Kt e P t F K t L tρλ λ δ−= − + , into the singular control equation implies:

[ ( ) ( ( ))] [ ( ) ( ( )) ( )] ( ) ( ( ), ) ( ) 0t t tKe q t C I t e q t C I t I t e P t F K t L tρ ρ ρρ λ δ− − −′ ′′+ − + − + =

The substitution of ( ) [ ( ) ( ( ))]tt e q t C I tρλ − ′= + implies further:

[ ( ) ( ( ))] [ ( ) ( ( )) ( )] ( ) ( ( ), )[ ( ) ( ( ))] 0

t t tK

t

e q t C I t e q t C I t I t e P t F K t Le q t C I t

ρ ρ ρ

ρ

ρ

δ

− − −

−

′ ′′+ − + −

′+ + =

which can be rendered as: ( )[ ( ) ( ( ))] [ ( ) ( ( )) ( )] ( ) ( ( ), ) 0Kq t C I t q t C I t I t P t F K t Lρ δ ′ ′′+ + − + − =

. Consequently, the firm’s gross investment evolution is:

( )[ ( ) ( ( ))] [ ( ) ( ( ), ) ( )]( )( ( ))

Kq t C I t P t F K t L q tI tC I t

ρ δ ′+ + − +=

′′

. (No arbitrage rule)

The change in the level of gross investment from one instance to the following should be equal to the difference between the opportunity cost of (i.e., the foregone return on distributed revenues due to) investment in an additional unit of capital (namely, the product of the user cost of capital and the full price of the additional unit of capital) and the marginal return on the additional unit of capital (namely, the market value of the firm’s capital’s marginal product plus capital gain), divided by the associated rise in the marginal adjustment cost. If ( )[ ( ) ( ( ))] ( ) ( ( ), ) ( )Kq t C I t P t F K t L q tρ δ ′+ + > + gross investment is accelerated over time. If ( )[ ( ) ( ( ))] ( ) ( ( ), ) ( )Kq t C I t P t F K t L q tρ δ ′+ + < + gross investment is decelerated over time.

27

Lecture 5 (Week 5)

Intertemporal Analysis of Efficient Investment in Production Capital

with Costs of Adjustment (cont.) 4. The phase-plane diagram of capital stock and gross investment The solution to the firm’s optimal control problem is given by the capital motion equation:

( ) ( ) ( )K t I t K tδ= − and the no-arbitrage rule:

( )[ ( ) ( ( ))] [ ( ) ( ( ), ) ( )]( )( ( ))

Kq t C I t P t F K t L q tI tC I t

ρ δ ′+ + − +=

′′

.

This system of two differential equations articulates the possible joint trajectories of capital stock and investment. The system, the possible trajectories and the stationary levels of investment and capital stock can be portrayed in a phase-plane spanned by the state variable K and the control variable I. The phase-plane is divided into several phases (or regions) by two imaginary curves called isoclines. Along one of this isoclines 0K = , and along the other 0I = . Each phase has its own dynamic characteristics. The intersection point(s) of the isoclines is a steady state(s). To facilitate the construction of the phase-plane diagram we assume that the firm takes (expects) the purchasing price of capital (q) and the price of its product (P) to be time-invariant (fixed): 0q P= = . 4.1 The isoclines The formula of the isocline 0K = is obtained by setting the left hand side of the state equation ( ) ( ) ( )K t I t K tδ= − to be equal to zero: I Kδ= . Recalling that 0 1δ< < , the isocline 0K = is depicted by a positively sloped line in the plane spanned by K and I as displayed by the green line in Figure 1 below. The isocline 0I = is the locus of all the combinations of K and I for which the right hand side of the no-arbitrage rule is equal to zero. In turn, each of these combinations of K and I satisfy the following equality: ( )[ ( )] ( , ) 0Kq C I PF K Lρ δ ′+ + − = . By total differentiation:

28

( ) ( ) ( , ) 0KKC I dI PF K L dKρ δ ′′+ − = . Hence, the slope of the isocline 0I = is given by:

( , )( ) ( )

KKPF K LdIdK C Iρ δ

=′′+

.

Recalling the assumptions 0KKF < and 0C′′ > :

( , ) 0( ) ( )

KKPF K LdIdK C Iρ δ

= <′′+

That is, the isocline 0I = is negatively slopped in the plane (K,I) as depicted by the brown line in Figure 1 below for the case where 0KKKF C′′′= = . The steady state (SS) is in the intersection of the two isoclines. It is unique and interior.

Figure 1: The isoclines and steady state

0K =

K

I

0I =

SS

0

29

4.2 The motions of K and I in the phases and the nature of the steady state The isoclines 0K = and 0I = divide the plane spanned by K and I into four phases. Each phase has its own dynamic characteristics. The following procedure is made to identify the dynamic characteristics of the various phases. By differentiating K I Kδ= − with respect to I:

1 0dKdI

= >

Consequently, 0K > in the region above the isocline 0K = (since moving to that region from any point is made by increasing I) and 0K < in the region below the isocline 0K = (since moving to that region from any point is made by decreasing I) as depicted by the horizontal arrows.

Figure 2: The horizontal forces above and below the isocline 0K =

By differentiating the no-arbitrage rule ( )[ ( )] [ ( , ) ]( )

Kq C I PF K L qIC I

ρ δ ′+ + − +=

′′

with

respect to K and recalling that 0KKF < and 0C′′ > :

( , ) 0( )

KKPF K LdIdK C I

−= >

′′

.

Consequently, 0I > in the region to the right of the isocline 0I = (since moving rightward from any point on the isocline 0I = is facilitated by a larger K) and 0I < in the region to the left of the isocline 0I = (since moving leftward from any point on the isocline 0I = is facilitated by a smaller K) as displayed by the vertical arrows.

0K =

K

I

0K <

0

0K >

30

Figure 3: The vertical forces on the right and left hand sides of the isocline 0I =

Figure 4 is constructed by combining Figure 2 and Figure 3. The two isoclines divide the plane into four phases. The motion in each phase is indicated by the combination of the horizontal and vertical arrows. As depicted in this phase-plane diagram, there are only two trajectories leading to steady state. They are called the convergent arms (or branches) of the stable manifold. All other possible trajectories do not converge to the steady state. In such a case, the steady state is not asymptotically stable and can be classified as a saddle point. (See Appendix for mathematical proof.) It can be reached from any initial level of capital by adjusting the gross investment level so as to embark on the relevant arm of the stable manifold.

Figure 4: The phase-plane diagram of capital stock and gross investment

0I >

K

I

0I =0

0I <

0K =

K

I

0I =

SS

0K

0I

0

31

Tutorial questions Suppose that:

2 , 0C cI c= >

, 0F aLK a= >

0( )P t P=

0( )q t q= .

1. Ignoring the curvature condition on the Hamiltonian, prove that the no-arbitrage rule for this case is

0 0( )[ 2 ( )]( )2

q cI t P aLI tc

ρ δ+ + −= .

2. Show that the steady state levels of K is

0 0( )2 ( )ss

P aL qIc

ρ δρ δ

− +=

+

1. Show that the solution to the no-arbitrage rule is

( )

0( ) ( ) tss ssI t I I I e ρ δ+= + − .

32

Lecture 6 (Week 6)

Intertemporal Analysis of Efficient Investment in Production Capital

with Costs of Adjustment (cont.) 5. The steady state capital stock and investment and comparative statics 5.1 The steady state values of K and I In steady state, 0I = . By substituting these steady state properties into the no-arbitrage rule: ( )[ ( )] ( , ) 0ss K ssq C I PF K Lρ δ ′+ + − = That is, the foregone distributed revenue due to investment in an additional unit of capital (i.e., the marginal opportunity cost of capital investment) is equal to the market value of the marginal product of capital. Furthermore, in steady state 0K = and by substituting this condition into the state equation: /ss ssK I δ= . Thus, the stationary investment level should satisfy the following equality: ( )[ ( )] ( / , ) 0ss K ssq C I PF I Lρ δ δ′+ + − = . This equality might not yield a close form solution for ssI . Numerical simulations of this equation with assumptions about the explicit functional forms of C and F and values of the model parameters can generate the value of ssI . 5.2 Comparative statics: the effects of the model parameters on steady state The total differentiation of ( )[ ( )] ( / , ) 0ss K ssq C I PF I Lρ δ δ′+ + − = and the assumptions about the firm’s production and costs of adjustment functions gives the effects of changes in the firm’s product price, time-preference rate, depreciation rate and capital purchasing price on the steady-state levels of gross investment and capital stock. The effect of the planner’s rate of time preference on the steady-state level of gross investment is developed in detail. By total differentiation, [( ) ( ) ( / ) ( / , )] [ ( )] 0ss KK SS ss ssC I P F I L dI q C I dρ δ δ δ ρ′′ ′+ − + + = implying, [( ) ( ) ( / ) ( / , )] [ ( )]ss KK SS ss ssC I P F I L dI q C I dρ δ δ δ ρ′′ ′+ − = − + and, consequently,

33

[ ( )] 0( ) ( ) ( / ) ( / , )

ss ss

ss KK SS

dI q C Id C I P F I Lρ ρ δ δ δ

′− += <

′′+ −.

The following effects of the rest of the model’s parameters on the steady-state level of gross investment are obtained in a similar manner:

2[ ( )] ( / ) ( / , ) 0( ) ( ) ( / ) ( / , )

ss ss ss KK ss

ss KK SS

dI q C I P I F I Ld C I P F I L

δ δδ ρ δ δ δ

′− + += <

′′+ −

( ) 0

( ) ( ) ( / ) ( / , )ss

ss KK SS

dIdq C I P F I L

ρ δρ δ δ δ

− += <

′′+ −

( / , ) 0

( ) ( ) ( / ) ( / , )ss K ss

ss KK SS

dI F I LdP C I P F I L

δρ δ δ δ

= >′′+ −

As /ss ssK I δ= , then 0ssdKdρ

< , 0ssdKdδ

< , 0ssdKdq

< , 0ssdKdP

> .

The effects of these parameters on the steady state can be illustrated by shifts in the isoclines. Note that while changes in , ,q Pρ shift only the isocline 0I = , a change in δ shifts both isoclines. Specific Appendix: Identification of the local asymptotic stability nature of steady state of the aforesaid firm’s optimal control problem Let us linearise the no-arbitrage rule

( )[ ( ) ( ( ))] [ ( ) ( ( ), ) ( )]( )( ( ))

Kq t C I t P t F K t L q tI tC I t

ρ δ ′+ + − +=

′′

in the vicinity of steady state under the assumption that 0C′′′ = (i.e., C′′ is a scalar):

1

[( ) / ][ ( )] ( / ) ( , )[( ) / ] ( ) ( / ) ( )( ) ( / )

ss K ss

ss KK ss

KK

I C q C I P C F K LC C I I P C F K K

I P C F K b

ρ δρ δρ δ

′′ ′ ′′= + + −′′ ′′ ′′+ + − − −

′′= + − +

where

1 ( )[ / ] ( / )SS KK ssb q C I P C F Kρ δ ′′ ′′≡ + − + . Consequently, the linearised no-arbitrage rule

1( ) ( / ) KKI I P C F K bρ δ ′′= + − + and the state-equation:

34

( )K t I Kδ= −

constitute a system of first order linear differential equations:

1( ) ( / )1 0

KKP C F bI IKK

ρ δδ′′ + −

− = −

.

A solution to the homogeneous part of this system is the form:

2 1

tIA e

Kµ

×

=

In which case,

2 1

tIA e

Kµµ

×

=

.

By substituting into the homogeneous part:

2 0 ( ) 0t tAe aAe a I Aµ µµ µ− = ⇒ − = where

( ) ( / )1

KKP C Fa

ρ δδ′′+ −

= − (the state-transition matrix).

A non-trivial solution (i.e., 0A ≠ ) to this equality requires the matrix 2( )a Iµ− to be singular. Namely,

2det( ) 0a Iµ− = . Consequently, 1µ and 2µ are the characteristic roots (eigenvalues) of the state-transition matrix:

21,2

2

0.5 ( ) [ ( )] 4det( )

0.5 4[ ( ) ( / ) ]KK

tr a tr a a

P C F

µ

ρ ρ ρ ρ δ

= ± −

′′= ± + + −

and the general solution of the differential equation system is:

1 2

1 1 2 2

= + +

t ss t t

t ss

I Ic A e c A e

K Kµ µ

.

35

Since 0C′′ > and 0KKF < , then 2 4[ ( ) ( / ) ]KKP C Fρ ρ ρ δ ρ′′+ + − > and, in turn,

1 0µ > and 2 0µ < . In which case, the steady state is a saddle point. That is, it is not locally asymptotic stable, yet there are two converging arms (or branches) in the vicinity of the steady state. These two converging arms comprise the stable manifold. The corresponding characteristic vectors can be found, up to an arbitrary constant, by a consecutive substitution of the characteristic roots into the general solution at 0t = and while taking into account the initial condition. General Appendix: Identification of the local asymptotic stability nature of a steady state of an infinite horizon optimal control problem A solution to a general infinite planning horizon optimal control problem with one state variable and one control variable is a set of two differential equations: one for the state variable (the state equation, i.e., the motion equation of the stock variable) and the other for the control variable (the Euler equation). Following a linearization in the steady state combination, this system can be displayed as

( ) ( )X t aX t b− = − where a is a 2x2 Jacobian (or state-transition) matrix and b a 2x1 column vector. The solution to this system is found as follow. Step 1. In steady state 0X = and hence 1

ssX a b−= . Step 2. Let ( ) tX t Aeλ= be a solution to the homogeneous part is ( ) ( ) 0X t aX t− = . Then,

0t tAe aAeλ λλ − = Hence, ( ) 0ta I Aeλλ− = The non-trivial solution to this equation implies that ( )a Iλ− is a singular matrix and hence det( ) 0a Iλ− = . Namely,

11 12 211 22 11 22 21 12

21 22

( )0 ( ) ( ) 0

( )a a

a a a a a aa a

− = ⇒ − + + − = −

λλ λ

λ

Consequently, 2

1,2 0.5[ ( ) 4det ]tra tra aλ = ± − Step 3. The general solution is

1 21 2( ) t t

ssX t X A e A eλ λ= + + . Consequently,

1 21 2lim ( ) lim( )t t

sst tX t X A e A eλ λ

→∞ →∞− = + .

Conclusions:

36

If 1 2, 0λ λ < , ssX is locally asymptotically stable (node). If 1 2, 0λ λ > , ssX is not locally asymptotically stable. If 1 20 & 0λ λ> < , ssX is a saddle point – there exists a singular stable manifold comprising two convergent branches (arms). If ( 4det ) 0tra a∆ ≡ − < , then 1 2&λ λ are complex (having a real part and an imaginary part) and the trajectory of X oscillates around ssX (focus). The oscillations are: explosive if 0tra > (diverging spiral), damped if 0tra < (converging spiral), or orbiting if 0tra = (centre). Comment: For any exponentially discounted, infinite horizon, one control variable and one state variable optimal control problem the eigenvalues of the pertinent Jacobian matrix sum to the discount rate. This implies that, as long as the discount rate is positive, the steady state cannot be locally asymptotically stable. (See Michael R. Caputo, 2005, Foundations of Dynamic Economics Analysis, chapter 18 for proof and explanation.) Tutorial: Discussion of Possible Extensions 1. Credit-financed investment with:

a. a flat interest rate (compatible with Modigliani-Miller Proposition 1 on the independence of the firm’s value from its debt-equity structure)

b. an interest rate that rises with the firm’s leverage (incompatible with Modigliani-Miller Proposition 1)

2. Uncertainty about revenues and costs and irreversibility of investment: the new investment theory and the delay option 3. Uncertainty about the firm’s life-expectancy (duration) Extension 1.a: Efficient Investment and Borrowing with Adjustment and Transaction Cost and with Tax Considerations The firm’s objective is postulated to be maximizing the sum of the discounted instantaneous net cash-flows:

1 2 , 0

max (1 )[ ( ) ( ( ), ) ( ) ( ( )) ( ( )) ( ) ( )]

( ) ( ) ( ) ( )

t

I Be P t F K t L w t L C I t C B t r t D t

q t I t D t B t dt

ρ τ

µ

∞− − − − − −

− − +

∫

subject to:

( ) ( ) ( )K t I t K tδ= − ( ) ( ) ( )D t B t D tµ= −

where,

( )B t is borrowing at t, ( )D t is debt at t,

τ is a flat time-invariant tax rate,

37

1C is the adjustment cost function,

2C is the transaction cost function, ( )r t is the average contracted instantaneous interest rate, and µ is the instantaneous debt-repayment rate. The necessary conditions for maximum and the singular control equations imply that the no-arbitrage rules of investment and borrowing are:

1

1

( )[ ( ) (1 ) ( ( ))] [(1 ) ( ) ( ( ), ) ( )]( )(1 ) ( ( ))

Kq t C I t P t F K t L q tI tC I t

ρ δ τ τ

τ

′+ + − − − +=

′′−

2

2

[(1 ) ( ) ] ( )[1 (1 ) ( ( ))]( )(1 ) ( ( ))

r t C B tB tC B t

τ µ ρ µ ττ

′− + − + − −=

′′− .

Comment: In this setting where the firm’s contracted interest rate is flat, the efficient investment and borrowing trajectories are independent. In contrast, the firm’s efficient investment and borrowing trajectories are interdependent if the firm’s contracted interest rate rises with the firm’s current financial leverage (cf. Stiglitz, 1972, 1975; Levy et al., 1989)3:

( ) ( ( )), 0r t r l t r′= > where

( )( )( ) ( )

D tl tK t D t

≡−

.

Steady state By setting 0K = and 0D = the following steady state levels are obtained:

/ss ssK I δ= (stationary capital stock)

/ss ssD B µ= (stationary debt)

// /

ss ssss

ss ss ss ss

D BlK D I B

µδ µ

≡ =− −

(stationary leverage)

where ssI and ssB are found by setting ( ) 0I t = and ( ) 0B t = in the no-arbitrage rules and assuming that the prices and interest rate are time-invariant. For example, suppose that:

21 1 1, 0C c I c= >

3 Stiglitz, J.E. (1972), “Some aspects of pure theory and corporate finance: bankruptcy and turnovers” The Bell Journal of Economics 3, pp. 458-483. (See also Reply in Bell vol. 6, 1975, pp. 711-714.) Levy, A. Justman, M. and Hochman, E. (1989),”The implications of financial cooperation in Israel’s semi-ccoperative villages’, Journal of Development Economics 30, pp. 24-46.

38

22 2 2, 0C c B c= >

, 0F aLK a= >

0( )P t P=

0( )q t q=

0tr r= .

Then (as was shown earlier):

0 0

1

( )2 ( )ss

P aL qIc

ρ δρ δ

− +=

+.

By setting ( ) 0B t = in the borrowing’s no-arbitrage rule:

0 2[(1 ) ] ( )[1 2 (1 ) ] 0ssr c Bτ µ ρ µ τ− + − + − − = and hence:

0

2

1 [(1 ) ] /( )2 (1 )ss

rBc

τ µ ρ µτ

− − + +=

−.

39

Lecture 7 (Week 8)

Modigliani-Miller Approach to Valuation of Firms, Leverage and

Cost of Capital (American Economic Review Vol. XLVII, 1958, 261-297)

Suppose that firm j belongs to a homogeneous class k of firms in which the capitalization (discount) rate is common and equal to a scalar kρ (which also may reflect the category of risk associated with the uncertain returns on the firm’s operation). Also suppose that firm j is infinitely lived and has a time-invariant expected return,

jx , on its activity every instance. If information is perfect and all agents are rational, the price (value) of firm j is given by:

[ ]k

jkk

k

jj

tk

kj

tkj

xee

xxedtxep

ρρρρρρρ =−−=−=∫= −∞−∞−

∞− )(1 0

00

.

In order to understand the following propositions it is helpful to consider the balance sheet of the firm. Firm’s j Balance Sheet

Assets Liabilities Assets-in-place

and growth opportunities

Stocks of Firm j ( )jS Bonds of Firm j ( )jD

Value of Firm j ( jV )

Modigliani-Miller First Proposition: If the market of capital is perfectly

competitive, then jk

jjjj p

xDSV ≡=+≡

ρ.

(This equality is an equilibrium, no-arbitrage, condition.) This proposition suggests that, as long as the capital market is perfectly competitive, the value of firm j is not affected by the firm’s capital-financing method (composition of equity and debt). Suppose that the rate of interest on loans is a scalar r. That is, the rate of return on all bonds is equal to r. In contrast, the (expected) return on stocks (shares) is uncertain. That is, while bonds are taken to be risk-free stocks are risky. Since stocks are risky

40

assets, their rate of return (i) should compensate rational, risk averse holders for the cost of risk bearing. Namely,

ri j > for every firm j. Modigliani-Miller Second Proposition: The (expected) rate of return on firm’s j stocks is

j

jkkj S

Dri )( −+= ρρ . ( rk >ρ )

That is, ji exceeds the capitalization rate kρ by the difference between the capitalization rate and the interest rate, compounded by the firm’s leverage.

Proof: By definition, j

jjj S

rDxi

−= . From M-M first proposition ( jj

k

j DSx

+=ρ

),

)( jjkj DSx += ρ . Hence, ( )

( )k j j j jj k k

j j

S D rD Di r

S Sρ

ρ ρ+ −

= = + − .

Comment: M-M second proposition implies that the firm cannot gain from substituting cheap securities (bonds bearing a certain rate of return r for holders) for expensive securities (stocks whose expected rate of return is larger than r). As can be seen from M-M second proposition ji rises with the firm’s leverage ( jj SD / ) proportionally to )( rk −ρ . By rearranging terms in M-M second proposition and recalling that jjj VDS =+ :

( ) ( )j j jj k k j j j j k j k j

j j jjV

D S Di r i S S D rD i r

S V Vρ ρ ρ ρ

= + − ⇒ = + − ⇒ = +

.

Hence, kρ can also be interpreted as the Weighted Average Cost of Capital (WACC).

Tutorial questions Suppose that the current interest rate is 0.06 per annum, the current market values of the bonds and stocks of firm A are $ 40,000,000 and $ 60,000,000, respectively, and the expected rate of return on firm A’s stocks is 0.10 per annum. 1. Compute firm A’s leverage. 2. Compute firm A’s WACC. 3. Are the M-M propositions 1 and 2 valid when the market of credit is not perfectly competitive? Can the firm’s leverage affect the firm’s value?

41

Houses’ Values and Prices and the Cobeweb Phenomenon Value of a House Houses constitute a major part of most individuals’ portfolios. The value of a house is the sum of its discounted net returns over its remaining life expectancy (0,T):

0

[Re ]T

rtt t tV e nt Capital Gains Depreciaion dt−= + −∫ .

As in Modigliani-Miller’s (1958) formulation of the firm price, let us assume that the house is infinitely lived and that the discounting (interest) rate and the annual rent, capital gains, and depreciation of the house are time-invariant (fixed), the value of the house is given by:

0

(Re )

Re

rtV e nt Capital Gains Depreciaion dt

nt Capital Gains Depreciationr

∞−= + −

+ −=

∫

Example: Consider a house whose full-tenancy rent is estimated to be $ 600 per week (i.e., $ 31,200 per annum), capital gain is estimated to be $ 35,000 per annum and depreciation (or maintenance cost) is estimated to be $ 20,000 per annum. The annual interest rate is 0.06. The present value of this house is:

000,77006.0

000,20000,35200,31=

−+=V dollars.

If the market of houses is efficient, the price of a house should be equal to V. Market Price of a House under Perfect Foresight Let the market of houses be perfectly competitive and agents be endowed with perfect foresight. Demand: 0 1 2 3 4

dt t t t t t tQ a a LOC a POP a Y a i Pα ε= + + + − − +

where, t is a time index, LOC is an index of how thought after (reputable) is the location (due to physical, environmental, social and commercial characteristics of the neighbourhood), POP denotes population, Y aggregate income, i interest rate, P price and ε random disturbance, and where all the parameters are positive scalars. Supply: 0 1

st t t tQ b b w Pβ ν= − + +

where, w is an aggregate index of construction-input price, ν is a random disturbance and the parameters are all positive.

42

The equilibrium condition is:

0 1 2 3 4 0 1t t t t t t t t ta a LOC a POP a Y a i P b b w Pα ε β ν+ + + − − + = − + + implying that the market price of a house is given by:

* 0 0 1 2 3 1 4( ) [ ]t t t t t t tt

a b a LOC a POP a Y b w a iP ε να β

− + + + + − + −=

+.

Market Price of a House under Naïve Expectation: A Cobweb Phenomenon There is usually a lag between listing a house for sale and actual sale. Vendors listing their houses for sale have expectations about the price of houses. Suppose that the vendors’ price expectations are naïve:

1exp

−= tt PP . That is, vendors expect the sale price to be equal to the price observed at the time of listing their properties for sale. In this case, the supply schedule is given by:

tttts PwbbQ νβ ++−= −110 .

The equilibrium condition is:

ttttttttt PwbbPiaYaPOPaLOCaa νβεα ++−=+−−+++ −11043210 or, equivalently,

[ ]][)(141321001 ttttttttt iawbYaPOPaLOCabaPP νε

ααβ

−+−++++−

=

+ − .

This first-order linear difference equation can be solved for the case where all variables, except price, are time invariant:

[ ]][)(141321001 νε

ααβ

−+−++++−

=

+ − iawbYaPOPaLOCabaPP tt .

In this case, the stationary (long-run) price of a house is given by:

βανε

+−+−++++−

=)()( 4132100 iawbYaPOPaLOCaba

Pss .

The deviation of the market price from the stationary level is given by

43

t

sssst PPPP

−−=−

αβ)( 0 .

which displays damped oscillations when αβ < (i.e., when the supply curve is steeper than the demand curve), explosive oscillations when αβ > (i.e., when the demand curve is steeper than the supply curve), and constant oscillations when αβ = (i.e., when the demand curve is as steep as the supply curve). The following figures display the possibilities of damped and explosive price fluctuations for a market that used to be in equilibrium, but following a hike in the interest rate (represented by a downward shift of the demand curve) is not in equilibrium due to the vendors’ naïve (sticky rather than perfect, or rational) price expectations.

Explosive oscillations ( αβ > ) following a hike in the interest rate ( 1 0i i> )

Damped oscillations ( αβ < ) following a hike in the interest rate ( 1 0i i> )

flateS

1( )steepD iQ

P

0( )steepD i

0Q

0P

steepS

1( )flateD i

Q

P

0( )flateD i

44

Lecture 8 (Week 9)

Risk, Expected Utility and Optimal Insurance Von Neuman-Morgenstern utility function: When facing uncertainty (e.g., about income) a rational consumer maximizes expected utility (e.g., from income). Let u be the consumer’s utility, then E(u) is the consumer’s expected utility (or the Von Neuman-Morgenstern utility. Suppose that: 1. the consumer derives utility from x [i.e., )(xu ], 2. x is random variable (e.g., income and hence consumption level), 3. there can be S mutually exclusive realizations of x (depending on the state of his world—combination of circumstances such as weather, political crisis, employment,

personal health …), each with a probability 10 << sp , 11

=∑=

S

ssp .

In this case, the expected utility (Von Neuman-Morgenstern utility) of the consumer is given by:

1( )

S

s ss

Eu p u x=

= ∑ .

(The Savage theorem provides an extension to this expected utility theorem.) As long as )(xu concave, the consumer expected utility is lower than her/his utility from the expected value of x (i.e., a risk averse prefers having the expected value of a gamble to taking the gamble). For illustration, consider the case where there are only two mutually exclusive states of the world (S=2):

−=

pxpx

x12

1

In this case,

)()1()())(( 21 xupxpuxuE −+= and

))1(())(( 21 xppxuxEu −+=

45

x1 x2 E(x)

u(E(x))

E(u(x))

u

The more concave the utility curve the larger the difference between ))(( xEu and

))(( xuE . A measure of the concavity of the utility curve is the Arrow-Pratt degree of absolute risk aversion: )(/)( xuxu ′′′− . Note that, by itself, u ′′ is not a good measure of the concavity of the utility curve. The reason is that any affine transformation of u is also represents the utility of the consumer. Thus, if, for instance, u is multiplied by 2, also u ′′ is multiplied by 2. But deflated by u′ , the concavity of the curve is unchanged. Thus, the measure proposed by Arrow and Pratt is a standardized one. It is positive for a risk-averse and negative for a risk-lover. Optimal Insurance (Coverage) Level Let the value of a property, W, be a random variable due to uncertainty of the state of the world and assume for simplicity that there are only two mutually exclusive states of world – heavy storm with a probability p, or calm weather with a probability 1-p. In the event of a heavy storm, the property damage/loss is L dollars. Let π be the insurance premium rate and q the level of insurance. The distribution of the property owner wealth is given by:

−−−+−

=pqW

pqLWW

1)1(

0

0

ππ

The property owners’ optimal level of insurance is found by:

)()1())1((max))((max 00 qWupqLWpuWuEqq

ππ −−+−+−⇔

The necessary condition for maximum expected utility is:

0)()1())1(()1(0))(( 00 =−′−−−+−′−⇒= qWupqLWupdq

WudE ππππ

46

)1()1(

*)(*))1((

0

0

ππ

ππ

−−

=−′

−+−′p

pqWu

qLWu.

To facilitate a close-form solution, let βWu = with 10 << β , then:

*])1([)1(

)1(* 01

1

0 qLWp

pqW ππππ

β−+−

−

−=−

−.

Subsequently, the property owner’s demand for insurance is given by:

ππππ

ππ

β

β

+−

−

−

−

−

−−

=−

−

)1()1(

)1(

)()1(

)1(

*1

1

01

1

0

pp

LWp

pWq .

Supply Side and the Actuarially Fair Premium Rate Considering the supply side, note that the profit of an insurance company is random:

−−−

=Πpq

pq1

)1(π

π

If the competition in the insurance market is strong, the expected profit can be driven to zero:

)1()1(0)1()1( ππππ −=−⇒=−−− ppqpqp which implies:

p=π . That is, the actuarially fair premium rate is equal to the probability of the property being damaged. Substituting this rate into the insurance demand function:

L

pppppp

LWppppW

q =

+−

−

−

−

−

−−

=−

−

)1()1(

)1(

)()1(

)1(

*1

1

01

1

0

β

β

.

When the actuarially premium rate is fair, a policy with full coverage is sought and contracted. Appendix Let us now consider instead the polar case where the insurance industry is a monopoly or stable cartel. For simplicity, let us assume that this monopoly is profit oriented and risk neutral. In which case, it sets the insurance premium rate so as to maximize its expected profit, while taking into account the adverse effect of the insurance premium rate on the aggregate demand for insurance:

47

*

1( ) ( )

ND

ii

Q qπ π=

=∑

where N is the number of clients. The monopoly expected profit is:

* * *

1 1 1( ) (1 ) ( ) (1 ) ( ) ( ) ( )

N N Nm

i i ii i i

E p q p q p qπ π π π π π= = =

Π = − − − = −∑ ∑ ∑

The premium rate that maximizes the monopoly expected profit should satisfy the following necessary condition:

**

1 1

( )( ) ( ) ( ) 0mm N N

m m ii

i i

dqdE q pd d

ππ π

π π= =

Π= + − =∑ ∑ .

To simplify matters even further, let us consider the case where the clients are identical. In this case, * * 1,2,3,...,iq q i N= ∀ = and the necessary condition is:

** ( )( ) ( ) 0

mm m dqNq p N

dππ ππ

+ − =

or, alternatively: *

* ( )( ) ( ) 0m

m m dqq pdππ ππ

+ − = .

Recalling that 1

10 0

11

(1 ) ( )(1 )

*

(1 ) (1 )(1 )

m

m

mm m

m

pW W Lp

q

pp

β

β

ππ

π π ππ

−

−

−− −

− =

−− +

−

mπ and its comparative static properties can be found by numerical simulations.

Moral Hazard: This term reserved to the case where the probability of the bad event (e.g., fire, accident, break in, hospitalisation) is affected by one’s negligence (or “free-riding”), which eventually raises the premium rate to all. This can be eliminated by monitoring the individuals’ behaviour by the insurer. Tutorial questions John derives utility from the market value (W) of his prime asset - a bungalow in Paradise Beach - separately from his other utility-generating assets. His utility from the market value of the bungalow is given by:

u W= . John assesses the value of the market to be 1,000,000 dollars. Alas, the bungalow is threatened by the Haloa Tsunami. It is common knowledge that the Haloa Tsunami hits Paradise Beach at random once in fifty years and inflicts total property loss. John is aware of that risk. 1. Calculate John’s expected-utility maximising annual level of insurance when:

1a. The per annum premium rate is 0.05. 1b. The per annum premium rate is 0.02.

2. Can the premium rate be 0.015?

48

Lecture 9 (Week 10)

Uncertainty, Risky Assets (Activities) and Portfolio Choice

1. Basic Concepts 1.1 Portfolio A portfolio is the composition of one’s wealth. If there are N assets in one’s world, one’s portfolio is:

1 2 3( , , ,..., )NP A A A A= where

1 2 31

... 1N

N ii

A A A A A=

+ + + + = =∑

and 0 1iA≤ ≤ is the fraction of one’s wealth held in asset i. 1.2 Rate of return on asset i The rate of return on asset i in period t is given by:

1

1

it it itit

it

P P drP

−

−

− +=

where itP is the price of asset i at the end of period t , and

itd is dividend on asset i received by the owner (holder) during period t. At the beginning of period t the price of asset i at the end of the period and the dividend on asset i during the period are unknown. Hence, at the beginning of period t the rate of return on asset i in period t ( itr ) is random and asset i is risky. The expected rate of return on asset i is:

( )i iE rµ ≡ . The variance of the rate of return on asset i is:

2 2var( ) ( )i i i ir E rσ µ≡ = − . The covariances of the rate of return on asset i and the rate of return on any other asset j is:

cov( , ) ( )( )ij i j i i j jr r E r rσ µ µ≡ = − − .

49

1.3 Rate of return on a portfolio ( pr ) In view of the definition of portfolio, the rate of return on a portfolio is a weighted average of the rates of return on the various assets with the weights being equal to the fractions of one’s wealth held in these assets:

1 1 2 2 3 31

...N

p N N i ii

r A r A r A r A r A r=

= + + + + =∑

Since the rates of return on the various assets are random, pr is random. The expected rate of return on the portfolio is:

1 1 2 2 3 3

1 1 2 2 3 3

1

( ) ( ... )

( ) ( ) ( ) ... ( )

.

p p N N

N N

N

i ii

E r E A r A r A r A rA E r A E r A E r A E r

A

µ

µ=

≡ = + + + +

= + + + +

=∑

The variance of pr is:

( )21 1 2 2 3 3

2 2

1 1

var( ) var ...

.

p p N N

N N N

i i i j iji i j i

r A r A r A r A r

A A A

σ

σ σ= = ≠

≡ = + + + +

= +∑ ∑∑

2. Efficient Portfolios: Mean-Variance Approaches Rational investors acquire information about assets (e.g., past performance and market development and trend) for assessing the distribution of their rates of return. If the distributions of the rates of return on all the assets are believed to be normal, it is sufficient to consider the perceived means, variances and covariances of the rates of return on the various assets for constructing efficient portfolios. This is known as the mean-variance approach. Otherwise, efficiency can be gained by considering higher moments of the joint distribution of the rates of return on the assets. Yitzhaki and Shalit (Journal of Finance, 1984) proposed a Mean-Gini approach, as the Gini is a distribution-free measure of disperssion. 2.1 The Markowitz Efficient Portfolio Set Harry Markowitz (1952) has viewed the expected rate of return on a portfolio as the benefits from holding the portfolio and the variance (hence the standard deviation) of the rate of return on the portfolio as a measure of the level of the risk associated with holding the portfolio. Therefore, he has argued:

a. that the combination ( 1 2 3, , ,..., NA A A A ) that minimizes the variance of the rate of return on the portfolio for a predetermined expected rate of return on the

50

portfolio is an efficient portfolio (i.e., 2

min

ipA

σ subject to 1

N

i i pi

Aµ µ=

=∑ and

1

1N

ii

A=

=∑ ); and

b. that the set of all possible efficient portfolios can be found by repeating the

above procedure for every predetermined expected rate of return on a portfolio and this set of efficient portfolios displays a trade-off between expected rate of return and risk.

Fig. 1: The Markowitz curve

Investors with high degree of aversion toward risk would prefer portfolios on the lower section of the Markowitz curve. The Markowitz’s approach can be displayed as a quadratic programming problem:

1

2 2

( ,..., )1 1

min ( )N

N N N

i i i j ijA A i i j i

A A Aσ σ= = ≠

+∑ ∑∑

subject to: 1

N

i i pi

Aµ µ=

=∑

1

1N

ii

A=

=∑ .

The Lagrange function associated with this problem is:

pµ

pσ0

The Markowitz set of efficient portfolios

Unattainable portfolio

Inefficient portfolio

51

2 21 2

1 1 1 1

( ) (1 )= = ≠ = =

= + + − + −∑ ∑∑ ∑ ∑N N N N N

i i i j ij p i i ii i j i i i

A A A A Aσ σ λ µ µ λL . Since the Lagrange function is convex in 1( ,..., )NA A ,4 the second order conditions for minimum are satisfied and the efficient portfolio * *

1( ,..., )NA A is obtained by solving the following set of 2N + first order conditions:

2 * *1 22 2 0 1,...,

≠

∂= + − − = ∀ =

∂ ∑N

i i ij j ij ii

A A i NA

σ σ λ µ λL

*

11

0=

∂= − =

∂ ∑N

p i ii

Aµ µλL

*

12

1 0=

∂= − =

∂ ∑N

ii

AλL .

From the solution of this set of first order conditions, the fraction of wealth that is efficiently held in the form of asset i is some function of the model parameters:

*1( ,..., , , )i i N ij pA f µ µ σ µ= .

The first-order conditions imply that for every pair of assets k and l:

2 * * 2 * *1 12 2 2 2

N N

k k kj j k l l lj j lj k j l

A A A Aσ σ λ µ σ σ λ µ≠ ≠

+ − = + −∑ ∑

or, equivalently:

2 * * 2 * *1 1(2 / )[ ] (2 / )[ ]

N N

k k k kj j l l l lj jj k j l

A A A Aµ λ σ σ µ λ σ σ≠ ≠

− + = − +∑ ∑ .

That is, for any two assets included in a Markowitz efficient portfolio there should be equality between the expected rates of return net of the marginal contributions to the portfolio’s risk level.

4 Note that 2 2

1 1

'N N N

i i i j iji i j i

A A A A Aσ σ= = ≠

+ = Ω∑ ∑∑ , where A is an 1N × column vector of the wealth

fractions held in the N assets and Ω is an N N× variance-covariance matrix of the rates of return on

the N assets and hence positive semi-definite. Therefore, 2 2

1 1

N N N

i i i j iji i j i

A A Aσ σ= = ≠

+∑ ∑∑ is convex in

1,..., NA A . Being a linear combination of a convex and a linear functions of 1,..., NA A , L is convex

in 1,..., NA A .

52

2.2 Tobin’s Capital Market Line (CML) James Tobin (1958) suggested the existence of a risk-free asset. Denoting by fr the rate of return on this risk-free asset, he has argued that the set of efficient portfolio is given by a line in the p pσ µ− plane with an intercept fr and that is tangent to the Markowitz curve as displayed in the Figure 2. He called this line the capital market line (CML). Investors with high degree of aversion toward risk would prefer portfolios on the lower section of the CML.

53

Fig. 2: The capital market line

The tangency point between the CML and the Markowitz curve can be interpreted as the market portfolio of risky assets. Its associated combination of risk and expected rate of return is ( , )m mσ µ . The CML formula is:

m fp f p

m

rr

µµ σ

σ−

= +

.

The term m f

m

rµσ−

can be interpreted as the market price of risk.

Tutorial: Construction of the CML The CML can be analytically constructed by noting that any efficient portfolio (represented by a point on the CML) is a linear combination of the risk-free asset and the market portfolio. Let A denote the fraction invested in the risk-free asset and 1-A the fraction invested in the market portfolio, then the expected rate of return on an efficient portfolio is given by:

(1 )p f mAr Aµ µ= + − and the standard deviation of the rate of return on an efficient portfolio is given by:

(1 )p mAσ σ= − . Since,

pµ

pσ0

Markowitz curve

CML

fr

mσ

mµ

The market portfolio

54

(1 ) p mp f m

f m

Ar A Arµ µ

µ µµ

−= + − ⇒ =

− and (1 ) m p

p mm

A Aσ σ

σ σσ−

= − ⇒ =

then

p m m p

f m mrµ µ σ σ

µ σ− −

=−

.

By rearranging terms:

( ) m p m fp m f m f m p

m m

rr r

σ σ µµ µ µ µ σ

σ σ− −

− = − = − +

.

Consequently, the CML formula is:

m fp f p

m

rr

µµ σ

σ−

= +

.

3. William Sharpe’s Capital Asset Pricing Model (CAPM) Let iβ denote the risk associated with a unit of risky asset i relative to the risk associated with a unit of investment in the market portfolio of risky asset. That is,

i i m/β = σ σ . The market of assets is assumed to be in equilibrium. In which case, the risk adjusted rate of return should be the same for all assets. The expected rate of return on any risky asset i should be equal to the rate of return on the risk-free asset plus an adequate compensation (premium) for bearing the risk associated with holding this risky asset ( mii σβσ = ). Recalling that the market price for risk is mfm r σµ /)( − , the compensation for bearing the risk of holding a unit of asset i should be:

)( fmim

fmmi r

r−=

−µβ

σµ

σβ

Thus, if the market of assets is in equilibrium, the expected rate of return on every asset i is:

)( fmifi rr −+= µβµ . This equality is the capital asset pricing model (CAPM). The CAPM suggests that the expected rate of return on the entire market of risky assets positively affects the expected rate of return on asset i and it does so proportionally to the relative risk embedded in holding asset i. Having time-series observations on the rates of return on risky asset i and on the market portfolio, the relative risk associated with holding asset i can be estimated by regressing the rate of return on asset i onto the rate of return on the market portfolio. Using, for example, ordinary least squares (OLS) estimation method:

55

)var(),cov(ˆ

fm

fmii rr

rrr−

−=β .

56

Lecture 10 (Week 11)

Uncertainty, Risky Assets (Activities) and Portfolio Choice (cont.)

4. Expected Utility Maximizing Approach to Portfolio Choice Economic theory suggests that a rational individual facing uncertainty makes a choice that maximizes her/his expected utility. The following analysis applies this approach to the choice of portfolio. Postulate 1: The portfolio holder derives utility (u) from the rate of return on her/his portfolio ( pr ) and in view of the uncertainty about pr makes her/his portfolio choice so as to maximize her/his expected utility subject to the wealth constraint. That is,

1 ,...,max [ ( )]

NpA A

E u r

subject to:

1

1N

ii

A=

=∑ .

Postulate 2: The marginal utility from the rate of return on the portfolio is positive but diminishing. That is,

0>∂∂

pru and 02

2<

∂

∂

pru .

Postulate 3: The individual utility from her/his rate of return is given by the following negative exponential function

pRreu −−= 1 where uuR ′′′−= / indicates her/his degree of absolute risk aversion. (The more concave the utility function, the greater the premium that one is willing to pay for avoiding risk.)

57

Figure: The negative exponential utility function In view of Postulate 3:

( ) (1 ) 1 1 ( )p pRr Rrp pEu r E e Ee mgf r R− −= − = − = − −

where ( )pmgf r R− is the moment generating function of the random variable pr assessed at –R. Assumption: The rate of return on every asset is normally distributed [i.e.,

2( , ) 1,...,∀ =i i ir i Nµ σN ].

Recalling that 1

N

p i ii

r A r=

=∑ , this assumption implies:

2( , )p p pr µ σN

The moment generating function of the normally distributed pr is:

2 2 2( ) exp 0.5 (0.5 ) exp 0.5 ( 0.5 )p p p p pmgf r R R R R Rµ σ µ σ− = − + = − − . Consequently,

1 1

2

,..., ,...,max [ ( )] max ( 0.5 )

N Np p pA A A A

E u r Rµ σ⇔ − .

pr

u

1

0

58

Recalling that 1

N

p i ii

Aµ µ=

=∑ and 2 2 2

1 1

N N N

p i i i j iji i j i

A A Aσ σ σ= = ≠

= +∑ ∑∑ , then the investor

decision problem can be displayed as:

1

2 2

,..., 1 1 1

max 0.5 [ ]N

N N N N

i i i i i j ijA A i i i j i

A R A A Aµ σ σ= = = ≠

− +∑ ∑ ∑∑

subject to:

1

1N

ii

A=

=∑ .

The Lagrange function associated with this maximization problem is:

2 2

1 1 1 1

0.5 [ ] (1 )= = = ≠ =

= − + + −∑ ∑ ∑∑ ∑N N N N N

i i i i i j ij ii i i j i i

A R A A A Aµ σ σ λL

where the Lagrange multiplier λ can be interpreted as the shadow value of wealth. Recalling that the variance of pr is a quadratic form in 1,..., NA A that is based on a

positive semi definite variance-covariance matrix ijσ , 2 2

1 1

N N N

i i i j iji i j i

A A Aσ σ= = ≠

+∑ ∑∑ is

convex in 1,..., NA A . Consequently, L is concave in 1,..., NA A and the N+1 necessary conditions:

* 2 * *( ) 0 1,...,≠

∂= − + − = ∀ =

∂ ∑N

i i i j ijj ii

R A A i NA

µ σ σ λL

*

1

1 0=

∂= − =

∂ ∑N

ii

AλL

lead to an interior solution:

*1( ,..., , , ) 1,...,i i N ijA f R i Nµ µ σ= ∀ =

*

1( ,..., , , )N ijg Rλ µ µ σ= . The necessary conditions imply equality between the risk-adjusted expected rates of return on assets included in the portfolio. That is,

* 2 * * 2 *( ) ( )N N

k k k j kj l l l j ljj k j l

R A A R A Aµ σ σ µ σ σ≠ ≠

− + = − +∑ ∑

59

for every pair of assets k and l. The second term on each side of this equality represents the extra cost of risk bearing for the investor resulting from a slight rise in the share of the wealth held by assets k and l, respectively. These extra costs of risk bearing are equal to the asset marginal contribution to the variance of pr compounded by the portfolio holder’s degree of absolute risk aversion. Claim (N=2): If all assets are risky and the number of assets is qual to two, and if

2 21 2 122 0σ σ σ+ − > , there exists an interior solution:

2

* 1 2 2 121 2 2

1 2 12

[( ) / ]2

RA µ µ σ σσ σ σ− + −

=+ −

.

Proof: In such a case the investor’s decision problem is:

1

2 2 2 21 1 1 2 1 1 1 2 1 1 12max (1 ) 0.5 [ (1 ) 2 (1 ) ]

AA A R A A A Aµ µ σ σ σ+ − − + − + −

The first order condition for maximum is:

2 2 * 21 2 1 2 12 1 2 12[( 2 ) ] 0R Aµ µ σ σ σ σ σ− − + − − + =

The second order condition for maximum is:

2 21 2 12( 2 ) 0R σ σ σ− + − <

Consequently, if 2 2

1 2 122 0σ σ σ+ − > , there exists an interior solution:

2* 1 2 2 12

1 2 21 2 12

[( ) / ]2

RA µ µ σ σσ σ σ− + −

=+ −

.

Tutorial Use the expected utility approach to construct the optimal portfolio under the assumption that 2 2

1 2 1 2 120.05, 0.1, 0.1, 0.15, 0.25, 1Rµ µ σ σ σ= = = = = = .

60

Exercise 2 To be handed at the beginning of the lecture at the beginning of week 12 lecture

The expected rate of return is 0.05 on asset 1 and 0.10 on asset 2 during the coming year. The variance of the rate of return is 0.20 on asset 1 and 0.30 on asset 2 and the covariance of their rates of return is -0.05 for the same year. 1. What will be the portfolio of an expected utility maximizing person who has such expectations, preferences on rate of return that can be displayed by a negative exponential utility function, and degree of absolute risk aversion equal to 1?

2. How will her/his portfolio change if her/his degree of absolute risk aversion has doubled? 3. Suppose now that asset 1 is risk free with rate of return 0.05. All other things are as in 1. What will be the portfolio?

61

Lecture 11 (Week 12)

FINANCIAL CRISES

Evolution of a Firm’s Debt Burden The evolution of the firm’s debt (D) can be compactly displayed by the following motion equation:

t t t tD r D π= −

where tπ is the firm’s profit at t and tr is the interest rate on the firm’s liabilities at t . A possible measure of the firm’s debt burden (d) is its debt-profit ratio:

tt

t

Ddπ

= .

By differentiation with respect to time, the evolution of the firm’s debt burden is expressed as:

2t t t t t t t t t

t tt t t t tt

D D D D Dd dπ π π ππ π π π ππ

−= = − = −

Consequently, the rate of change of the firm’s debt burden is given by:

( / )t t t t t

t t t t t t t

d D Dd D D

π ππ π π π

= − = −

In recalling the first equation describing the evolution of the firm’s debt,

t t t t t

t t t

d r Dd D

π ππ

−= −

and by rearranging terms:

t t tt

t t t

d rd D

π ππ

= − +

Additional insight can be gained by considering that:

( ; )t t t t tp y C y wπ = −

where ty is the firm’s output, tp its product-price and tw its input-price vector at t.

62

Developing Countries’ External Sovereign Debts: Structure Causes and Evolution Implication for Exchange Rate Policy External Debt and Economic Growth Debt Laffer Curve and Discount Repudiation and Debt’s Secondary Market Price Structure

Public and Publicly Guaranteed vs. Private

Commercial Creditors vs. Non-Profit International Crediting Institutions

Short Term vs. Long Term

Causes and Evolution Collapse of market price system and inefficient allocation: During the 1950s and 1960s there were expectations that the prices of exported primary goods would decline. Hence, tariff protected industrialization was recommended and pursued. It led to an erosion of the role of the market prices in conveying proper signals for producers and consumers and in turn to inefficient allocation of inputs. Bad and populist economic management: Subsidies on goods consumed by pressure groups (e.g., voters in the major cities) contributed further to the distortion of the internal price system and hence to inefficiency in the allocation of inputs. Initially cheap credit: The large increase in the oil revenues (“petrodollars”) led to a rise in some major commercial banks reserves during the early and mid 1970s. Developing countries were encouraged to take large loans in initially low, but variable, interest rate. Rising interest rate: The two oil crisis (1973/4 and 1978/9) and the grain crisis led to a hike in the inflation (two digit inflation by the end of the decade) and, due to contracted variable interest rate, to a rise in the cost of servicing the external debt. The 1985/6 oil glut: Some heavily indebted oil-exporting developing countries such as Mexico and Nigeria saw a decline in their oil revenues due to the 1985/6 glut that led to a fall in the price of crude oil from about USD 32 to USD 10 per barrel and hence experienced difficulties in servicing their external liabilities.

63

The David Howard’s Model of the Evolution a Country’s External Debt Burden The absolute level of a country’s external liabilities (D) increases with the accumulated periodical interest (rD) and decreases with the country’s trade balance (TB):

t t t tD r D TB= − (1) A measure of the external debt burden (d) is the debt-GNP ratio:

tt

t

DdY

= . (2)

The change in the external debt burden is given by differentiating (2) with respect to time:

2/t t t t t t tt t

t t t t

D D Y Y D D Yd t dY Y Y Y

−= ∂ ∂ = = −

(3)

The rate of change of the county’s external debt burden is obtained by dividing both sides of (3) by d:

( / )t t t t t

t t t t t t t

d D Y D Yd Y D Y Y D Y

= − = −

. (4)

Substituting (5.1) into (5.4):

t t t t t

t t t

d r D TB Yd D Y

−= −

(5)

Consequently,

t t tt

t t t

d Y TBrd Y D

= − +

. (6)

Tutorial question Consider a country with a stagnant economy, 40 billion dollars external debt, and a 2 billion dollar current account deficit per annum. The current interest rate on the country’s external liability is 10 percent per annum. What is the current rate of change in the country’s external debt burden?

64

Implication for Exchange Rate Policy

( / ) ( / , ) ( / , )for dom for dom for domt t t t t t t t t t t tTB EX e P P IM e P P Y TB e P P Y= − = (7)

where, e is the domestic currency nominal exchange rate (the domestic price of the foreign exchange),

domP is the domestic price level, forP is the foreign price level, and

0( / )for dom

t t

TBeP P∂

>∂

and 0TBY

∂<

∂.

Hence,

( / , )for domt t t t t

tt t t

d Y TB eP P Yrd Y D

= − +

. (8)

The sterility of devaluation (Rudiger Dornbusch): If a significant portion of the country’s external debt is public and publicly guaranteed and the tax system is mull-functioning, a devaluation of the local currency cannot improve the country’s balance of payment and reduce the external debt burden. A devaluation of the domestic currency ( e ↑ ) increases the government costs of servicing its external liabilities in domestic currency. Since the tax system is mull-functioning, the government resolves to printing domestic currency in a higher rate than before the devaluation for purchasing the external currency required for servicing its external liabilities. A higher rate of money supply increases the domestic price level ( domP ↑ ), which diminishes the improving effect of the devaluation on the real exchange rate and, in turn, the country’s trade balance and external debt burden:

fort

domt

e PP↑

↑

65

External Debt Burden and Economic Growth Determinants/Sources of Economic Growth Suppose that the aggregate output at period t is given by an aggregate production

function:

( , )t t t tY A F K L=

where,

0tA > is a technology shift parameter,

tK is the aggregate capital stock,

tL is the aggregate labour force,

0, 0K LF FF FK L∂ ∂

≡ > ≡ >∂ ∂

and 2 2

2 20, 0KK LLF FF F

K L∂ ∂

≡ < ≡ <∂ ∂

.

By differentiating with respect to time:

( , ) ( , ) ( , )K LY AF K L AF K L K AF K L L= + + The growth rate of this economy is obtained by dividing both sides by ( )Y t :

( , ) ( , ) ( , )K LY AF K L AF K L K AF K L L= + +

( , ) / ( , ) ( , ) / ( , ) ( , ) / ( , )K LY AF K L AF K L AF K L K AF K L AF K L L AF K LY= + +

Consequently,

[ / ] [ ( , ) / ( , )] [ ( , ) / ( , )]K LY A A F K L F K L K F K L F K L LY= + +

Recalling that production elasticities with respect to capital and labour are defined as:

( , )[ / ( , )]K KF K L K F K Lξ ≡

( , )[ / ( , )]L LF K L L F K Lξ ≡ then the growth rate of this economy can be expressed as:

( ) ( )t t t tK L

t t t t

Y A K Lt tY A K L

ξ ξ= + +

.

66

The substitution of this growth formula into equation (8) implies:

( / , )( ) ( )for dom

t t t t t t tt K L

t t t t t

d A K L TB eP P Yr t td A K L D

ξ ξ

= − + + +

Paul Krugman (1988) stressed that accumulation of public and publicly guaranteed external debt have led to expectations for increasing tax, which intensified capital flight and discouraged repatriation of capital flight and investment in new technologies. In some countries, sub Sahara African ones in particular, the HIV-AIDS, malaria and other epidemics, as well as natural catastrophes and civil wars, have shrunk the labour force and its technology absorptive capacity.

67

Lecture 12 (Week 13)

The Debt Laffer Curve (DLC) and Discount Being inspired by Laffer’s hypothesis of an inverted U-shaped relationship between tax payment and the tax rate, Paul Krugman (1988) suggested an inverted U-shaped relationship between a country’s expected external debt repayment (EDR) and the book-value of its external debt (D). The underlying rationale is that in the case of many developing countries a large portion of the external debt is public and publicly guaranteed. A sovereign state is unlikely to be declared bankrupt and its domestic assets are unlikely to be liquidized when it fails to service its external liabilities. If such an inverted U-shaped relationship between EDR and D exists and the indebted country is on the negatively sloped section of the debt Laffer curve, it is also in the interest of the creditor to write off part of the debt and give a debt relief of, say,

*0D D− .

While Krugman’s argument lends support for a debt relief Bullow and Rogoff (JEP 4.1, 1990) argue against giving a prise to governments that badly manage their countries and economies. Formal construction of the DLC: Consider the case where the entire external debt is either repudiated or repaid and where the probability of (total) repudiation rises with the sovereign’s external debt level: 0 ( ) 1p D< < with 0p′ > and 0p′′ > . In this case, the expected debt repayment is given by a quadratic form concave in D:

[1 ( )] ( )EDR p D D D p D D= − = − depicted by an inverted parabola—the debt Laffer curve. EDR peaks at *D . That is,

*D is found by solving the equation obtained from setting the first derivative of EDR to be equal to zero:

* * *1 ( ) ( ) 0p D D p D′− − = .

D

EDR

0D*D

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Repudiation and Debt’s Secondary Market Price Sovereign debt might be partially, or fully repudiated (e.g., Mexico 1982, Brazil 1985). A simple illustration is given by the following binomial distribution of a country’s debt repayment.

1(1 )D p

DRD pδ

−= −

where, 0 1p< < is the probability of repudiation, and 0 1δ< ≤ is the rate of repudiation. Consequently, the expected debt repayment is

( ) (1 ) (1 ) (1 )E DR p D p D p Dδ δ= − + − = − . In which case, we may expect that price of recycling the country’s external debt in the secondary market will be:

( ) (1 ) 1SME DR p DP p

D Dδ δ−

= = = − .

Tutorial question The external debt of country A is 200 billion dollars. Agents in the secondary market believe that there is a probability of 0.5 that country A will repudiate half of its external liabilities. What is the highest price that they would be will to pay for a one dollar external bond issued by country A? Appendix: Gross National Product, Debt Burden, Probability of Default and the Debt Laffer Curve It has been argued by Krugman (1988) that a country’s financial liabilities act like a high marginal tax rate which deters government from taking painful measures to improve the country’s economic performance and discourage capital formation. Thus when an indebted country is on the downwardly sloping side of the debt Laffer curve, both the debtor and creditor can benefit from a debt-reduction. Following Levy (Economic Analyses of Financial Crises, 1995), let us construct a debt Laffer curve for the case where: 1. the debt burden affects the probability of repudiation, and 2. the annual debt repayment affects the country’s gross national product. Assumption 1: The indebted country’s probability of default ( 0 1p< < ):

1. rises with the burden of servicing its (public and publicly guaranteed) external debt, which is measured by the ratio of the annual debt repayment (M) to the country’s gross national product (Y), due to public aversion to tax increase and subsidy cuts; and

69

2. decreases with the creditor’s ability to retaliate by limiting the country’s access to the international credit market, which is proportional to the country’s creditor’s (or syndicate of creditors’) market share (s)

This assumption is reflected by the following specification of the default probability:

, , 0Mp sY

α λ α λ = − >

. (1)

Assumption 2: The indebted country’s gross national product is adversely affected by the annual debt repayment.

This assumption is consistent with the debt-overhang hypothesis. Namely, the indebted country’s capital stock is adversely affected by the decline in the government development budget and by capital flight engendered by high tax rate.

This assumption can be formally represented by:

, 1Y Y Mδ δ= − ≥ (2)

Where Y is the (highest) level of gross national product attainable had the annual debt repayment been nil.

The substitution of equation (2) into equation (1) implies:

Mp s

Y Mα λ

δ = − −

(3)

By differentiation,

2 0( )M

YpY M

αδ

= >−

and

3

2 0( )MM

YpY M

δαδ

= >−

as long as 0Y Mδ− > .

Recalling that 0 1p< < and p rises with M, the annual debt repayment must be within the interval min maxM M M≤ ≤ , where min( ) 0p M = and max( ) 1p M = . The substitution of these boundary conditions into equation (3) implies:

70

minmin

min

0 M ss M YY M s

λα λδ α δλ

= − ⇒ = − +

maxmax

max

11 (1 )

M ss M YY M s

λα λδ α δ λ

+= − ⇒ = − + +

The feasible interval of the annual debt repayment min max( , )M M is shifted rightward by an increase in either Y or sλ and by decrease in either α or δ . This interval is also enlarged by Y . In view of the above, the country’s annual debt repayment (ADR) is perceived by a rational creditor to be binomially distributed:

1 /( )0 /( )M M Y M s

ADSRM Y M sα δ λ

α δ λ − − +=

− +

Consequently, the expected annual debt repayment is:

( )

1 [ /( ) ]p M

EADR M Y M s Mα δ λ= − − +

.

By differentiation,

1 [ ( ) ( ) 0MdEADR p M p M M as M M

dM

> <

< >= − + = =

where

1(1 )

YMs

αα δ λ δ

= − + +

An increase in the annual debt repayment reduces the probability of such payment and hence the expected annual debt repayment can be depicted by an inverted U-shaped curve. Beyond M an increase in the payment causes the probability of default to rise so much that the expected annual debt repayment falls.

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Figure: Debt Laffer curve

The Global Financial Crisis, 2007-present A review of the origin of the present Global Financial Crisis can be found by using the following link: http://en.wikipedia.org/wiki/Financial_crisis_of_2007%E2%80%9308 Recommended articles Pol, Eduardo, “The Preponderant Causes of the USA Banking Crisis 2007-08”, The Journal of Socio-Economics, 2012, 41, 519-528. Tularam, Gurudeo Anand; Subramanian, Bhuvaneswari, “Modeling of Financial Crises: A Critical Analysis of Models Leading to the Global Financial Crisis”, Global Journal of Business Research, 2013, 7 (3), 101-124. Bengtsson, Elias, “Shadow Banking and Financial Stability: European Money Market Funds in the Global Financial Crisis”, Journal of International Money and Finance, February 2013, 32(1), 579-94. Arouri, Mohamed; Jawadi, Fredj; Nguyen, Duc Khuong, “What Can We Tell about Monetary Policy Synchronization and Interdependence over the 2007-2009 Global Financial Crisis?” Journal of Macroeconomics, June 2013, 36, 175-87.

minM M maxM

EADR

45o