Errata for Amacher, Ollikainen, and Koskela, Economics of Forest Resources, The MIT Press, 2009

Transcription

1 Errata for Amacher, Ollikainen, and Koskela, Economics of Forest Resources, he MI Press, 9 Chapters 6, 8, his will be a continuously evolving document Please send queries to either Greg Amacher (gamacher@vtedu or Markku Ollikainen (markkuollikainen@helsinkifi An update of this errata adding the remaining chapters 7 and 9 will be posted during Spring of A note about first order conditions in Chapters 3, and 5, : In many cases, the nonzero term ( e e that is left premultiplying the entire Faustmann first order condition after its derivation has been intentionally deleted in order to ease interpretation In these cases, we used the language: the first order condition can be expressed or reexpressed as Similar instances occur in other chapters such as Chapter when Faustmann like first order conditions are derived for interpretation Chapter Page 6, equation in text below equation 3: A p is missing in the second term Page 7, equation 5: the correct equation is: [ f ( rf ( rf ( e [ pf ( rpf ( ] d ( e ] dp r( e dc pf ( V dv r Page 7, equations 6a 6c: Denominators are incorrect and should read: pf ( rpf ( Page 7, equation 6b should equal d F dv pf ( V r pf ( rpf ( hese errors were pointed out by Richard (Bin Mei

2 Page 7, text discussing price comparative static result, the first order condition should read f ( rf ( rf ( e ( e p [ rc( e ] Page 7, Real Interest Rate Effect here is no error, but we present the derivation for clarity: V r r pf ( V pf ( e ( e e ( pf ( e c ( e Rearrange next to obtain: rpf ( e pf ( V ( e re ( e ( r r V re And to find that ( pf ( V ( pf ( V ( e re = ( pf ( V ( e Divide the last bracket term by r e to obtain r r e r pf ( V = ( pf ( V r ( e ( e ( r Page 3, equation (9b here is no error but this result could use some clarification he Faustmann formula in the presence of the yield tax is defined by equation (9a, and we derive the comparative static effect in terms of the price, p, that does not include taxes he term in the denominator of equation (9b reflects the fact that the result is stated in terms of price A similar type of result appears in chapter 3, equation (3 o see this more formally, differentiate 9a with respect to to obtain the conventional harvesting rule: p( f ( rp( f ( r p( f ( e c( e Now, take derivatives separately with respect to the tax τ and price p to obtain, for the tax: p f ( rf ( rf ( e ( e

3 and for the timber price f ( rf ( rf ( e ( e ( his should make tax result visible: by multiplying the latter term by p /( we obtain the former Page 3 ( θ has been unintentionally opped from the site productivity tax derivative he first order condition ( θ V = indeed simplifies to V = as we have in the text, because the term ( θ ops out of the first order condition for, affect rotation age V = hus, as we note the site productivity tax does not Page 39, reference to Faustmann formula in first paragraph he formula should read: V ( e [ p F( ; q e c] Also, there is a typo concerning the interest rate impact f Differentiating with respect to r and evoking the envelope theorem produces e ( e pf ( ( e ( pf ( e c Chapter 3 Page 56, text before equation (36 he term e r ( A W should equal e W Page 56, 8 th line after equation (36 Intersecting from above should read intersecting from below Page 57, equation (37b, equation should read: d H pf ( V r W dv E r de Page 57, line 4 of text concerning the price effect should read F ( re Page 57, line of text concerning real interest rate effect he Faustmann part should dv equal pf ( V r Page 56, line 5 of text concerning real interest rate effect An exponent is missing in the term ( e in front of the integral term hese errors were pointed out by Richard (Bin Mei

4 Page 57, text concerning real interest rate (line 6 A minus one is missing for the last term: it should be rs ( e sf( s e ds When collecting the terms, the entire interest rate effect for amenities de r should be E r r r re ( e E, or like we have r r r( e E ( r ( e re ( e (that is, (+ r should multiply only the first r r r We can re express it as: term Finally, as demonstrated, the term de E r r r re ( e should be positive page 6, equation (3 he derivation here follows along the same lines as for equation (9b on page 3 described for chapter in this errata Page 7, unnumbered equation above equation (33a he discount rate is missing in the exponent of the first RHS term Page 74, equation (337 his should read ˆ ( e W e { f ( ( p q rf ( [ p q( ] rwˆ } f ( ( p q rq( f ( rpf ( rwˆ he LHS represents the marginal benefit (timber harvest, sequestered carbon, and interest on delayed carbon release he RHS represents the opportunity cost (interest on standing timber and land rent Page 69, bottom of the page he text should note that the interest rate has a negative effect on the indirect net present revenue function per parcel Chapter 4 3 Page 8, bottom of box 4 ext should have as equations, Q( t / Q /( Q K bdt, and [/ Q /( Q K] dq bdt Q t Notice that an x has replaced the K in these formulas Page 8, line 4 in box 4 he correct equation is: Q( t K( ce bt 3 he third error was pointed out by Selmin Creamer, and all other errors were raised by Richard (Bin Mei

5 Page 83, equation (46b, first order condition is missing = on RHS Page 85, equation (4 On the RHS, the term S x p should equal S x p Page 86, text below section title 44 he text price should read: pˆ p ( u Further, in the i i i i text below this, c and c should have been written as functions of x and x, ie, c pˆ x a s, to be consistent Also, an I subscript is missing on the tax rate here since we define it as periodic Page 87, line 5 in Box 4 Should read k 7 m 3 S S Page 89, equation (43a Should read x x a Page 9, equation (47 his should read: Rp ( g p v( k, and under constant prices u( c equation (47 would be written as p( r g v( k u( c Page, line 3 below equation (47d he first term in the V b t t equation should be u * ( c t b pt Page 3, line 4 below equation (49b Should read because is less than one Page 6, equation (43 the choice variables for this optimization problem should be x and x S S S Page 7, equation (434 his equation should read: x Q( f x ( f k( Q x /

6 Page 9, text above equation (48a When solving for the comparative statics results for current and future harvesting, x and x, the first order conditions (46b and (46c should be referred to in the text Also, on the top of the next page, we should refer to the Jacobian and not the Hessian matrix Page 93, text just before equation (49 the reference Koskela and Ollikainen 989 should be 997b in the references section Page 5, line 3 below title for section 44 he reference should be Amacher and Brazee 997a Note: here is a quick example of the application of comparative statics to current and future harvesting in the basic model of this chapter: Given first order conditions: U x p v( k ( g v( k U z p R v ( k he second order derivatives are U xx v( k ( g v( k g v( k U zz v( k U xz ( g v( k hus, the determinant of the Jacobian matrix is: U U xx xz Uzx U zz U xx U zz ( U zx Differentiate the first order conditions with respect to p: U xp U zp Using the determinant we obtain the effect of first period price on current and future harvesting

7 x p U xz v( k U zz U zz and z p U xx ( g v( k U zx U zx Chapter 5 4 Page 8, equation at bottom of page he first term in the brackets should be pf ( pf ( instead of Page 8, equation (58a, he notation x and z should be replaced by x and x Page 9, text just below (5b he equation reference is wrong, it should read future harvesting condition in (5b P9, in the text below equation 5 he text should read: Moreover, p R p ( g is larger in absolute value Page 3, text below equation (57 Amacher et al ( in references should be referenced here P38, equation (536b he equation should have m as subscripts, ie, G m H ( U X /( e m Page 39, second equation in the text below equation (538 he subscripts should equal m, ie, G m H ( Y X /( e m 4 Errors here were pointed out by Richard (Bin Mei

8 Page 4, equation (A5b, the second term in brackets on the RHS of this equation should equal pb b b Page 43, equation A53b he last two terms on the RHS should be p Page 43, equation A54b he terms in the first bracket of the RHS of this equation should be [ v( k t v( kt ( v( k t v( kt ] P44, equation A57b he last term in brackets on the RHS should be p Chapter 6 5 Page 58, text below equation (67 Should read c ( Q Page 6, line 3 in proposition 6, should read differs from the non corruption case Page 69, text below equation (5b Should read d / dn n Page 66, equation (6b A note for clarification: As discussed in the text, this is Reed s (984 interpretation of the Faustmann model under risk of catastrophic loss Reed modeled fire loss, but we model timber trespass and loss of rents from it Reed defined Net Present Value slightly differently because he assumed that initial planting costs are zero, and then planting is paid for at the second rotation and beyond In chapter 6 there is no difference in marginal cost of planting after trees are stolen, but in chapter there is a difference in planting marginal cost if a fire arrives compared to planting if a fire does not arrive, and thus the formula in chapter 6 and are slightly different to reflect this 5 Errors here were pointed out by Richard (Bin Mei

9 Chapter 8 (no known errors Chapter Page 7 In the two equations at the bottom of the page, the last term in these equations has the wrong subscript, that is, J should be defined as: J rx rx rx n E( e e e E( e n rx n Yn Page 73, equation (5 he upper limit of first integral should be instead of X page 74, equations 8a 8b here is no typo here, but note that the nonzero term r ( r r[ e ] has been deleted to facilitate discussion A similar simplification was made to the first order conditions for (a (b Chapter Page 39 3 In the comparative statics derivatives, c should be replaced bt c and by Page 3 Equation, EX X ( e should read E X ( e in the numerator Page 34, Second paragraph he ift parameter should be replaced by the symbol Chapter Page 38 he text at the bottom of the page should state g [ V ( t] is growth in the forest stock Page 33 In line seven from the top of the page, the term defining land rent a on the RHS

10 Should have fh( in place of f V ( Mathematics Appendix Page 358, line 5 in first paragraph he first order condition should read: V x ( x; and x