INTERTEMPORAL PRICE CAP REGULATION UNDER UNCERTAINTY By Ian M. Dobbs The Business School University of Newcastle upon Tyne, NE1 7RU, UK.

Σχετικά έγγραφα
( ) ( t) ( 0) ( ) dw w. = = β. Then the solution of (1.1) is easily found to. wt = t+ t. We generalize this to the following nonlinear differential

Appendix. The solution begins with Eq. (2.15) from the text, which we repeat here for 1, (A.1)

The Euler Equations! λ 1. λ 2. λ 3. ρ ρu. E = e + u 2 /2. E + p ρ. = de /dt. = dh / dt; h = h( T ); c p. / c v. ; γ = c p. p = ( γ 1)ρe. c v.

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

( ) ( ) ( ) Fourier series. ; m is an integer. r(t) is periodic (T>0), r(t+t) = r(t), t Fundamental period T 0 = smallest T. Fundamental frequency ω

APPENDIX A DERIVATION OF JOINT FAILURE DENSITIES

4.6 Autoregressive Moving Average Model ARMA(1,1)

The choice of an optimal LCSCR contract involves the choice of an x L. such that the supplier chooses the LCS option when x xl

Example Sheet 3 Solutions

On the solution of the Heaviside - Klein - Gordon thermal equation for heat transport in graphene

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

2 Composition. Invertible Mappings

Inverse trigonometric functions & General Solution of Trigonometric Equations

Section 8.3 Trigonometric Equations

is the home less foreign interest rate differential (expressed as it

Notes on the Open Economy

Lecture 12 Modulation and Sampling

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

Math221: HW# 1 solutions

Managing Production-Inventory Systems with Scarce Resources

Uniform Convergence of Fourier Series Michael Taylor

Solutions to Exercise Sheet 5

Finite Field Problems: Solutions

ECON 381 SC ASSIGNMENT 2

Homework 3 Solutions

= e 6t. = t 1 = t. 5 t 8L 1[ 1 = 3L 1 [ 1. L 1 [ π. = 3 π. = L 1 3s = L. = 3L 1 s t. = 3 cos(5t) sin(5t).

2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013

Matrices and Determinants

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Necessary and sufficient conditions for oscillation of first order nonlinear neutral differential equations

C.S. 430 Assignment 6, Sample Solutions

EE512: Error Control Coding

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

F19MC2 Solutions 9 Complex Analysis

Partial Differential Equations in Biology The boundary element method. March 26, 2013

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

The challenges of non-stable predicates

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

ORDINAL ARITHMETIC JULIAN J. SCHLÖDER

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007

Problem Set 3: Solutions

Approximation of distance between locations on earth given by latitude and longitude

Fourier Transform. Fourier Transform

Every set of first-order formulas is equivalent to an independent set

Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee

derivation of the Laplacian from rectangular to spherical coordinates

A Note on Intuitionistic Fuzzy. Equivalence Relation

Bounding Nonsplitting Enumeration Degrees

ST5224: Advanced Statistical Theory II

Χρονοσειρές Μάθημα 3

Second Order Partial Differential Equations

Reservoir modeling. Reservoir modelling Linear reservoirs. The linear reservoir, no input. Starting up reservoir modeling

Fractional Colorings and Zykov Products of graphs

Almost all short intervals containing prime numbers

Oscillation Criteria for Nonlinear Damped Dynamic Equations on Time Scales

The Simply Typed Lambda Calculus

Homework 8 Model Solution Section

Positive solutions for a multi-point eigenvalue. problem involving the one dimensional

Lanczos and biorthogonalization methods for eigenvalues and eigenvectors of matrices

Areas and Lengths in Polar Coordinates

Other Test Constructions: Likelihood Ratio & Bayes Tests

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

Section 7.6 Double and Half Angle Formulas

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

The one-dimensional periodic Schrödinger equation

ECE Spring Prof. David R. Jackson ECE Dept. Notes 2

CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD

w o = R 1 p. (1) R = p =. = 1

Lecture 15 - Root System Axiomatics

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

r t te 2t i t Find the derivative of the vector function. 19. r t e t cos t i e t sin t j ln t k Evaluate the integral.

Strain gauge and rosettes

Statistical Inference I Locally most powerful tests

LESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV. 18 February 2014

Areas and Lengths in Polar Coordinates

Fractional Calculus. Student: Manal AL-Ali Dr. Abdalla Obeidat

Lecture 2. Soundness and completeness of propositional logic

University of Washington Department of Chemistry Chemistry 553 Spring Quarter 2010 Homework Assignment 3 Due 04/26/10

ω = radians per sec, t = 3 sec

Math 6 SL Probability Distributions Practice Test Mark Scheme

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Section 9.2 Polar Equations and Graphs

One and two particle density matrices for single determinant HF wavefunctions. (1) = φ 2. )β(1) ( ) ) + β(1)β * β. (1)ρ RHF

Second Order RLC Filters

Differentiation exercise show differential equation

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with

Partial Trace and Partial Transpose

Risk! " #$%&'() *!'+,'''## -. / # $

Differential equations

b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!

Congruence Classes of Invertible Matrices of Order 3 over F 2

SOLVING CUBICS AND QUARTICS BY RADICALS

2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)

Assalamu `alaikum wr. wb.

Transcript:

INTERTEPORAL PRICE CAP REGULATION UNDER UNCERTAINTY By Ian. Dobbs The Bsiness Shool Universiy of Newasle on Tyne, NE 7RU, U. Eqaion Seion ADDITIONAL NOTES AND DERIVATIONS The blished aer omis deailed derivaions. These addiional noes give fll derivaions for all he reored resls. The following are very mh se by se derivaions, so eah se shold be easy and qik o follow. A. The Prie Proess The rie roess, on ime inervals on whih here is no enry/invesmen is given by he demand fnion: Q η A η (A. Wrie f ( A, Q Q η A η so f ( A, Q (A. Alying Iô s lemma, d f. da + f. dq + f da + f dq + f da dq (A.3 where A Q AA QQ AQ f ( A, Q η / A (A.4 A A (, / A η η f A Q + A (A.5 f ( A, Q η / Q (A.6 Q fq (, / Q A Q η η Q f A (, / Q A Q η Q A (A.7 (A.8 whils (sing he Iô rles, we ge da α Ad + σ A dϖ (A.9 da α Ad + σ A dϖ σ A d (A. dq θq d (A. dq θ Q d (A. and da dq. (A.3 Hene d f. da + f. dq + f da + f dq + f da dq A Q AA QQ AQ ( Q dq ( η Q A da dq ( A σ A d ( η / A da + ( η / Q dq + η( η + / A da + η( η / + / ( η / A α A d + σ A dϖ + ( η / Q θ Q d + η( η + / (A.4 Original Draf: /8/ Las daed: 4/7/3

Noes d η αd + σ dϖ θη d + η η + σ d (A.5 ( d η α + θ η + σ d ησ dϖ (A.6 Wriing µ ( η α + θ η + σ (A.7 hen his beomes d µ d ησ dϖ (A.8 whih is eqaion (5 in he aer. A Derivaion of he fndamenal differenial eqaion for vale: The arbirage eqaion was (reroded here for onveniene: θ + r vd d + E dv (A.9 Iô s lemma gives dv v d + v d + v d + v d + v dd (A. Now, v ψ ( x where x / (A. so v ψ ( / ψ ( / (A. ψ ( / v (A.3 v ψ ( / ψ ( / ψ ( / ψ ( / (A.4 v ψ ( / + ψ ( / + ψ ( / ψ 3 ( / 3 (A.5 v / ψ / (A.6 From (A.8 d µ d ησ dϖ (A.7 d µ d ησ dϖ η σ d (A.8 Sine d δ d i follows ha d. Hene he erm dv is given as dv v d + v d + v d + v d + v d + v dd ( / ψ ( / ( / ψ [ ] d + ψ ( / d + d + ψ ( / ψ ( / + ψ ( / 3 d d + dd and sbsiing frher (and abbreviaing he noaion a lile (A.9

ψ dv ψ d + d + ψ ψ d + ψ d + ψ dd ( / ψ µ ησ ϖ + ( η σ + ψ ψ ( δ ψ + + + 3 ψ d d d d dv ψ µ d ησ dϖ + η σ d + ψ ψ δ d Ths aking exeaions, η σ E( dv ψ + ( µ + δ ψ δ ψ d ψ η σ µ ψ δ ψ δ ψ d [ ησ ψ ] d ϖ θ + ψ + µ + δ ψ δ ψ (A.3 (A.3 Hene he arbirage eqaion beomes η σ ( r vd d ( d (A.3 η σ θ + r ψ ψ µ + δ ψ + δ ψ (A.33 ( ( σ x ψ + x µ + δ ψ θ + r + δ ψ + x (A.34 Noes 3 whih is eqaion (A. in he aendix o he aer. The general solion o (A.34 an be wrien as he sm of he general solion o he homogeneos eqaion ( σ x ψ + x µ + δ ψ θ + r + δ ψ (A.35 and a arilar solion o (A.34. A arilar solion, easily verified, is where ψ Bx (A.36 B ( r ( θ + r µ θ + + η α + θ η + σ, (A.37 and he general solion o he homogenos eqaion (A.35 an be wrien as ( x B x ψ + B x. (A.38 where he roos are defined as and where ( R R η σ + (A.39 / ( R R η σ, (A.4 / ( R µ + δ η σ. (A.4

Noes 4 R ( µ δ η σ η σ θ r δ + + + +. (A.4 Hene he solion o (A.34 akes he form ( x B x B x ψ + + B x. (A.43 Noie ha η σ ( θ r δ + + > if σ >, so he roos are real and of oosie sign when nerainy is resen. The arbirary onsans B, B are deermined by bondary ondiions. As x, if vale is o be finie, i ms be ha B or he solion exlodes (see Dixi, 993. The oher onsan is deermined by an analysis of smooh asing ondiions a he bondary (a whih new invesmen is riggered. A3 Analysis of smooh asing ondiions Given B, from (A.43, he solion is ψ ( x Bx + B x (A.44 where B is given by (A.37. (a The Comeiive ase: In his ase i was esablished ha he smooh asing ondiions are ψ ( (A.45 and ψ (. (A.46 From hese eqaions we ge ψ ( B + B, (A.47 ( B B ψ + B B /, (A.48 and hene ( ψ ( B, (A.49 ( r. (A.5 θ + µ whih is Resl (i. (b The monooly ase In he monooly ase, invesmen ommenes a a ime a whih rie reahes he level, where is he relaive rie a whih new aaiy is added. Sine is a free hoie by he firm, smooh asing involves firs and seond derivaive ondiions (see Dmas, 99. The firs derivaive ondiion is ha, wih rese o he onrol variable, he rae of hange of vale shold js eqal he rae of hange of os; V (,, Q / ( Q / (A.5 where V (,, Q ψ ( x Q, Q A γ and / x. Sbsiing hese ino (A.5 gives

!, (A.5 γ { [ ψ ] } A ( / / γ γ [ ] + whih gives γ A ψ ( / A ψ ( / (/ γ A { γ [ ψ ( / ] + ψ ( / } γ ψ ( x + x ψ ( x (A.53 (A.54 [ ] γ [ ψ ] + ψ ( (A.55 The seond derivaive ondiion is V (,, Q / Q / Noes 5 ( (A.56 { } (A.57 { γ ψ ψ ψ } γ { γ ψ ( x + x ψ ( x } + x {( + γ ψ ( x + xψ ( x } (A.58 γ γ ψ ( x γ x ψ ( x (A.59 + ( + γ x ψ ( x + x ψ ( x whih gives γ γ A γ ψ ( / + ψ ( / [ ] γ [ ] ( [ ] + ( + A ( / (/ + ( / + ( / (/ { } [ ] { } γ γ ψ ( + γ ψ ( + ( + γ ψ ( + ψ ( (A.6 [ ] ψ + γ ψ + γ ( γ [ ψ ] γ γ ψ + γ ψ + + γ ψ + ψ ( ( ( ( (A.6 Now, from (A.44, ψ ( x Bx + B x (A.6 so ψ ( x B B x + (A.63 ( x ( B x (A.64 ψ so, sing (A.6-(A.64, eqaion (A.55 gives γ [ ψ ( ] ψ ( " # + $ γ B + B + ( B + B " ( + γ B + B ( + γ γ (A.65 % γ ( + γ B B (A.66 + γ &( ' &( ' and, from (A.6, again sing (A.6-(A.64, ( B + γ B + B + γ ( γ B + B * & ( ' & ( B γ B γ B γ ( γ ( ' (A.67 + + + B + B + B ( + γ + γ γ + γ + γ γ B γ γ (A.68 { } { }

@ ( + γ + Sbsiing for B sing (A.66, his gives γ ( + γ B + { γ + γ ( γ } B γ ( γ (A.69 ( γ + γ ( γ { } ( γ ( + γ B ( + γ + γ ( γ + { γ + γ ( γ }( + γ B γ ( γ ( + γ B ( ( ( + γ ( + γ + γ + γ γ γ + γ γ + γ + γ γ + γ + γ γ + γ { } γ ( + γ ( ( γ γ ( γ γ γ ( γ + γ + γ γ (A.7 (A.7 + γ B γ (A.7 γ + γ ( ( γ + γ B γ (!# "# %&$! " #' (# ' ( % (A.73 γ γ + γ + γ ( γ ( γ B γ ( γ + ( + + (A.74 ( + γ B γ (A.75,* -/.+* -. Sbsiing for B sing (A.37, his gives γ, + ( r (A.76 θ + µ + γ whih is Resl (ii. 43 Noes 6 A4 Relaionshi beween alernaive oion vale mliliers (omeiive ase Sar wih he original resl (A.5 ha 5 748 6 ( r + θ µ (A.77 9 : > <?;> <? < >? σ σ σ 9 : >? 9 :;9 : R + R R R R R 4 and noe ha, from (A.39-(A.4 ha and ( θ + r + δ < σ θ r δ σ (A.78 + + / (A.79

@ Noes 7 Wrie R + R R R R + + σ σ σ ( µ δ σ + σ (A.8 ( µ + δ σ ( ( +. (A.8 ( r + θ µ ( r + θ + δ ( δ + µ (A.8 so, from (A.79 and (A.8 r + θ + δ δ + µ ( ( σ σ ( ( + ( σ (A.83 (A.84 σ From (A.79 σ θ + + δ (A.85 so r ( r (A.86 θ + + δ Tha is,!! "# ( r " # ( r (A.87 θ + + δ + θ µ $&% '( $% ' ( whih is reored in he aer in eqaions (, (3 and resl (i. Hene also, from (A.76, γ &* * ( r. (A.88 θ + + δ + γ A5 The limiing ase: Under erainy, he erainy relaive rie a whih enry akes lae is (,. θ + r + ax η α + θ, δ or eqivalenly, ha (a if δ η( α θ < +, ( θ r η( α θ + + +. (b if δ η( α + θ, hen θ + r + δ. This seion shows ha as σ. Tha is, wriing Lim Lim σ, ha Lim Lim θ + r + η ( α + θ ( η + σ ( (A.89 [, ] θ + r + ax η α + θ δ

Noes 8 Define where σ σ φ σ ( σ r ( (A.9 φ σ θ + + η α + θ η + σ (A.9 Clearly, from (A.9, Lim φ σ θ + r + η α + θ (A.9 Case (a δ < η( α + θ Firs, examine Lim and Lim (. From (A.39-(A.4, sbsiing for µ sing (A.7: µ + δ µ + δ η σ ( θ r δ η σ η σ ( r + + + + η σ η ( θ + α + η ( η + σ η ( θ + α + η ( η + σ + + η σ θ + + δ + δ η σ + δ η σ η σ (A.93 ( ( δ η ( θ α ( ( r δ η θ + α + ησ + δ η θ + α + ησ + η σ θ + + δ η σ (( δ η ( θ α ησ η σ θ r δ + + + + + + + η σ η σ (A.94 Now, if δ < η( α + θ hen ( δ η ( θ α + > (A.95 and, aking he limi in (A.94, as σ so + (whaever haens o he oher erms, hey are non-negaive, and he firs erm +. Ths, if δ η ( θ + α > hen Lim Lim (A.96 ( ( and hene Lim Lim Limφ θ + r + η( α + θ (A.97 Case (b δ > η( α + θ Then δ η θ + α <, (A.98 (

Noes 9 and behaves raher differenly in he limi. To see his, define ( ( r f ( σ δ η θ + α + ησ + δ η θ + α + ησ + η σ θ + + δ and so ha (A.99 g( σ η σ (A. ( σ f ( σ / g( σ (A. Now, when (A.98 holds, hen f ( and g ( so he limi for ( σ is no immediae from (A.. However, i an be obained by alying l Hoial s rle; hs g ( σ η and (( ( r ( δ η ( θ + α + ησ η + η ( θ + r + δ f ( σ η + δ η θ + α + ησ + η σ θ + + δ so ha and Lim g σ (A. η (A.3 Lim f ( σ η + ( η ( δ η ( θ + α + η ( θ + r + δ δ η ( θ + α ( ( ( r ( δ η ( θ + α η δ η θ + α + η δ η θ + α + η θ + + δ ( r ( δ η ( θ + α ( + r + ηδ + η θ + α + ηδ η θ + α + η θ + + δ η θ δ ( δ η θ + α Hene, by l Hoial s rle f Lim f Lim ( σ Lim g Lim g ( r ( σ ( σ ( σ ( σ η θ + + δ θ + r + δ + ( δ η θ + α η δ η ( θ α Ths given Lim ( σ exiss, i follows ha Lim Lim ( ( Lim θ + r + δ δ η ( θ + α θ + r + δ θ + r + δ θ + r + η ( θ + α δ η θ + α (A.4 (A.5 (A.6

Noes Hene σ φ Lim Lim Lim ( θ + r + δ + + + θ + r + η θ + α ( θ r δ + + ( θ r η ( θ α This omlees he derivaion for (A.89 and hene resl in he aer. (A.7 A6. onooly firm sbje o ineremoral rie a. Here ha denoes he demand rie (he rie whih lears he marke; ha is, sh Q A (A.8 η η However, he firm is reqired o se a rie whih saisfies he rie a ; s s denoing he se rie as, hen his reqires ha. Ths s in[, ] in[, ]. (A.9 where is a onsan hosen by he reglaor. For low vales of he relaive rie x, he monoolis ilises exising aaiy, whils he rie is nonsrained. One x reahes a erain level, he rie a binds, b invesmen may sill be deferred (so here is qaniy raioning. Finally, if x inreases sffiienly, i will be oimal for he firm o add o aaiy. There is hs a ransiion bondary beween he noinvesmen regimes, and a frher bondary a whih invesmen ommenes; a eah of hese bondaries, smooh asing ondiions aly. Le ψ denoe he solion when here is no invesmen and no rie onsrain and le ψ denoe he solion when he rie onsrain alies, b here is no invesmen. The firs ask is o haraerise he roess in eah of he regimes. Following his, smooh asing ondiions are sdied. Regime : Unonsrained rie, no invesmen. The solion here is idenial o ha already esablished for he nonsrained monooly firm. Tha is, from (A.34, reeaed for onveniene, σ x ψ + ( µ + δ xψ ( δ + θ + r ψ + x (A. and he solion is ( x B x B x ψ + + B x (A. where B is given by (A.37. As before, noe ha, as x, if vale is o be finie, i ms be ha B. Dixi (993 dissses his sor of bondary ondiion in more deail. Hene he solion beomes ψ ( x B x + B x (A.

. Noes Regime : Unonsrained rie, no invesmen. The arbirage ondiion in his ase is, sine in his regime, ha θ + r vd d + E dv (A.3 where v ψ ( /. Alying Io s lemma, aking exeaions and simlifying, ( + + + + + (A.4 σ x ψ µ δ xψ ( δ θ r ψ A arilar solion is learly ψ ( δ + θ + r (A.5 and so defining C /( θ + δ + r, (A.6 he general solion in his regime o akes he form ψ ( x C + C x + C x (A.7 where, are as before. The arbirary onsans C, C are deermined by a onsideraion of bondary ondiions (see below. Analysis of bondary ondiions: Le denoe a ime a whih here is a ransiion from a regime of nonsrained ries and no invesmen o a regime where ries are onsrained b here is sill no invesmen. Le denoe a ime where here is a ransiion from rie onsrained non-invesmen o a regime in whih here is rie-onsrained osiive invesmen. Regime / bondary: A he ime, smooh asing ondiions ms aly. Sine he roblem is no one in whih here is a free hoie for he level of he bondary (i is se by he reglaor, he smooh asing ondiions involve eqaliy only of vale and he firs derivaive of he vale fnions (Dmas, 99. Also, by definiion, he rie a binds, so x Ths v( v( and v ( v ( ms hold, and so ψ ( ψ (, (A.8 ψ ( ψ ( (A.9 where ψ ( B + B (A. B + B (A. ψ ψ ( C C C + + (A. C C ψ + (A.3 The analysis of hese ondiions is deferred nil he oher ondiions have been idenified. The osiive Invesmen bondary: A he hiing ime, smooh asing ondiions again aly. These are idenial wih hose for he nonsrained monooly roblem exe ha hey are now evalaed a he relaive rie. The firs derivaive ondiion is ha, [ ] γ ψ ( + ψ ( (A.4 The vale fnion an sill be wrien in his form in he rie onsrained region, simly bease he rie onsrain is also homogenos of degree in,.

Noes - omare wih (A.55 and he seond derivaive ondiion is ha. ψ ( + γψ ( + γ γ ψ ( (A.5 [ ] as in (A.6. Here C C ψ + + C (A.6 C C ψ + (A.7 ( C + ( C (A.8 ψ Analysis of smooh asing ondiions: From (A.8, sing (A. and (A., B + B C + C + C (A.9 Unknowns are B, C, C. From (A.9, sing (A. and (A.3, and mlilying by, B B C C + + (A.3 Unknowns are B, C, C. (A.3 From (A.4, sing (A.6 and (A.7, γ C + C + C + C + C Unknowns are C, C,. (A.3 From (A.5, sing (A.6, (A.7 and (A.8, ( C + ( C + γ C + C + γ γ C + C + C Unknowns are C, C,. Toal nknowns are B, C, C,. However he variable of ineres is. To find his, firs eliminae B. Eqaion (A.9 imlies B C B + C + C (A.33 whils from (A.3 C + C B B (A.34 C + C B B ( C B + C + C C + C B ( C B + C + C C + C B ( C B + C + C ( C B B C + (A.35

' Noes 3 This gives C. I does no deermine B, C b links hem bease B C C B ( C B + B B C B ( B C ( B C ( + ( ( C B B B ( C B B + Tha is, one C is esablished, so oo is B. Now se he osiive invesmen bondary ondiions: from (A.4-(A.8, γ C + C + C + C + C and (A.36 (A.37 (A.38 ( C + ( C + γ C + C + γ γ C + C + C Firs find C from (A.37: γ ( C γ + C C γ + Then se his in (A.38: ( C + ( C + γ C + γ C C C C + γ γ + γ γ + γ γ γ γ ( + γ + γ γ C + ( + γ + γ γ C + γ γ C (A.39 (A.4 # $ # $ # $ ( + γ ( γ C * + Sbsiing for C sing (A.39, % & γ ( C γ + C ( + γ + γ γ + ( + + ( C γ γ γ * + (* + ( γ + { C } C ( + γ + γ γ γ ( γ + C + γ + + γ + γ γ C + γ γ γ +

' ' ' 6 @ & A ( C ( + + ( ( + C ( + + ( ( γ ( γ γ ( γ C γ ( γ ( γ C γ γ γ γ γ γ γ γ + + + + + + ( γ + ( + γ + γ ( γ ( γ + ( + γ + γ ( γ C γ γ γ + C γ ( C ( + γ + γ ( γ ( C {( ( + ( ( } γ γ γ γ γ γ C Now, from (A.35, ( C B + B C so ( γ + ( + γ + γ ( γ ( γ + ( + γ + γ ( γ ( C ( ( ( + ( ( " ( γ ( γ γ ( γ % # ( C B & # + + + + B # ( γ + ( + γ + γ ( γ * { } γ γ γ γ γ γ! $ % &$ $ $ Noes 4 (A.4 (A.4 (A.43 where from (A.6, C /( θ + δ + r (A.44 and, from (A.37, B r + θ µ (A.45 / Simlifying (A.43, 6 γ ( C ( ( + γ. ( γ + ( + γ + γ ( γ (. ( γ + ( + γ + γ ( γ - +,. / / C + B / 4 5/ : > ;? γ ( C ( ( + γ ( γ ( γ ( ( γ + γ 7 8 9 + ( + ( 9 9 9 C + B + γ γ + + γ γ

6 6 6 6 Noes 5 ( C γ ( + γ ( γ γ + ( γ γ + C + B ( + ( γ + γ + γ ( γ ( + ( γ γ γ ( C γ ( + γ γ γ + γ + γ + C + B ( + γ + γ + γ γ γ + γ ( C γ γ γ γ ( + γ γ γ + C + B ( + γ γ γ ( C ( ( + γ #! " $ ( C + B ( + + ( + {} γ γ ( C γ ( + γ #! " $ (A.46 C + B γ + + γ + { } ( The aim now is o simlify his o obain he formla in resl 3. Denoe δ + θ + r (A.47 as he erainy omeiive enry rigger relaive rie, ( ( r ( ( r ( δ + θ + θ + µ (A.48 as he nerainy omeiive enry rigger relaive rie and γ ( + γ (A.49 as he nerainy nonsrained monooly enry rigger relaive rie, hen he aim is o show ha he rie a monooly relaive enry marke learing rie an be wrien as in resl 3; ha is, as % % & & / '' ' * ( * ( (. (A.5 So, rerning o (A.46, and dividing by C gives γ ( + γ (( / C ( B / C ( + -,.. (A.5 + γ + + γ + {} Dividing by ( and sbsiing for B, C hen gives

! Noes 6 ( θ + δ + r ( ( ( r + θ µ ( γ ( + γ {( γ γ ( } + + + ( r + ( + { } ( r + θ µ θ + δ + r γ + + γ + γ θ µ γ Using (A.47 and (A.48, his gives ( γ + + γ + ( γ ( + γ so ( γ ( + γ + ( + γ ( ( ( ( + η γ ( + γ (where η / γ. Wriing ( + η, hen ( ( + η ( (( + η ( + η " $! %# + η + η Hene (A.5 (A.53 (A.54 (A.55 ( ( " η η!& #("!# $ % " '! " $'% # # " $ %# + η + η + η + η " η + " # + η + η + η # whih is Resl 3. ( " ( η η # $ % + + η + * + + + ( - + η + η ( ( /. ( / (A.56

Noes 7 A7 Proof for Resl 4. From he formla for ( in Resl 3 was {( ( } / (A.57 ( / Differeniaing wih rese o gives d (( (A.58 d ( d d / where, sing he definiion for, d d (( ( ( d d ( (A.59 so hene ( / $!!$%$!!$%"!#" (A.6 ( d d ( + ( + ( + d > d < as > <. (A.6 This omlees he roof for resl 4(i. As in eqaion (A.57, he erm in brakes { } ; sine / <, i follows ha +, whih is resl 4(ii. Leing in (A.57, learly (, whih is resl 4(iii. Seing, from (A.57, { } (A.6 Now, ( > < as ( < > (sine <. Using (A.6 his imlies ( > < as ( ( < > & & ( < ( > > < In fa > and hene ( >, whih is Resl 4 (iv. To esablish Resl 4 (v, firs sbsie in (A.6 sing resl, for ( / ( / ( η and he definiion for, noe ha Lim σ + o ge ( / η, and so ( Limσ +. From resl 3, Lim σ. Also Limσ ( η Lim ( η / Hene Lim ( Lim ( η Lim whih is Resl 4 (v. σ σ σ. Hene from / / + +. + (A.63