The Log-Linearized System of the CEE Model

All equations that are relevant for simulating the model developed in this thesis are log-linearized with one of the two methods outlined in section A.2 of the appendix. Christiano et al. (2001) use thirteen Euler and other equations to solve the dynamics of the system. Moreover, the Dynare code uses seven auxiliary equations that facilitate finding the solution. The following subsections describe all these equations. For the sake of brevity, only the more complex derivations are shown in detail for the CEE model. The numeric listing is identical to the Dynare code provided in section A.4.

A.3.1. The Inflation Phillips Curve

To facilitate a coherent characterization of the intermediate goods sector problem, the inflation Phillips curve was already discussed in section 3.5. Therefore, no further details will be given here. For ease of reference, equation 3.5-10 is quoted again:

𝜋̃_𝑡= 1

1 + 𝛽𝜋̃_𝑡−1+ 𝛽

1 + 𝛽𝐸_𝑡𝜋̃_𝑡+1+(1 − 𝛽𝜉_𝑝)(1 − 𝜉_𝑝)

𝜉_𝑝 𝐸_𝑡𝑠̃_𝑡 A.3.1

A.3.2. The Money Demand Function

Equation 3.6-9 describes the household’s FOC for nominal cash balances. Log-linearization yields, after dropping constant terms and solving for cash balances:

𝑞̃_𝑡 = − 1 𝜎_𝑞[ 𝑅

𝑅 − 1𝑅̂_𝑡+ 𝜓̃𝑐,_𝑡] A.3.2 A.3.3. The “Wage Phillips Curve”

The equation involving the aggregate wage rate is derived as the log-linearized equation 3.7-5, together with 3.7-2. For ease of reference, these equations are quoted again:

Ε_𝑡∑(𝛽𝜉_𝑤)^𝑙𝜓_{𝑐,𝑡+𝑙}(𝑊̂_𝑡𝑋_𝑡𝑙

𝑃_𝑡+𝑙 − 𝜆_𝑤2𝜓₀(𝑛_{𝑖,𝑡+𝑙})

𝜓_{𝑐,𝑡+𝑙} ) 𝑛_{𝑖,𝑡+𝑙} = 0

∞

𝑙=0

,

𝑛_𝑖𝑡 = (𝑊_𝑡 𝑊_𝑖𝑡)

𝜆_𝑤 𝜆𝑤−1

𝑙_𝑡. CEE use a few additional definitions in their working paper:

𝑞̅_𝑡 = 𝑄_𝑡

𝑃_𝑡−1, 𝑤̅_𝑡= 𝑊_𝑡

𝑃_𝑡−1 , 𝑤̂_𝑡= 𝑊̂_𝑡

𝑊_𝑡. A.3.3.i

In percentage terms, the deviation of the household’s real wage rate from its non-stochastic steady-state value is given by 𝑤̂̃_𝑡+ 𝑤̃_𝑡. Moreover, 𝑤̅̃_𝑡= 𝑤̃_𝑡− 𝜋̃_𝑡, where 𝜋_𝑡 corresponds to the time 𝑡 inflation rate, denoted by 𝜋_𝑡 = 𝑃_𝑡/𝑃_𝑡−1. This result follows from taking logs of the term in the middle of the equation A.3.3.i, which yields

55 ln(𝑤̅_𝑡) = ln(𝑊_𝑡) − ln(𝑃_𝑡−1).

The term ln(𝑊_𝑡) can be rewritten using the definition of real variables, 𝑤_𝑡 = 𝑊_𝑡/𝑃_𝑡. Therefore, ln(𝑤̅_𝑡) = ln(𝑤_𝑡) − ln (𝑃_𝑡) − ln(𝑃_𝑡−1).

The time 𝑡 inflation rate in log-linearized terms has the form 𝜋̃_𝑡 = 𝑃̃_𝑡− 𝑃̃_𝑡−1. Consequently,

𝑤̅̃_𝑡 = 𝑤̃_𝑡− 𝜋̃_𝑡. A.3.3.ii

Log-linearization of equation 3.7-5, using equation 3.7-2 results in 𝑤̂̃_𝑡+ 𝑤̃_𝑡 = ∑(𝛽𝜉_𝑤)^𝑙

∞

𝑙=1

(𝜋̃_𝑡+𝑙− 𝜋̃_{𝑡+𝑙−1}) +(1 − 𝛽𝜉_𝑤)(𝜆_𝑤 − 1)

2𝜆_𝑤− 1 ∑(𝛽𝜉_𝑤)^𝑗

∞

𝑗=𝑙

[𝐿̃ _𝑡+𝑗+ 𝜆_𝑤

𝜆_𝑤 − 1𝑤̃_𝑡+𝑗− 𝜓̃_{𝑐,𝑡+𝑗}].

A.3.3.iii

Comparable to equation 3.5-7, equation 3.7-3 can be rewritten as

𝑊_𝑡 = [(1 − 𝜉_𝑤)𝑊̂

𝑡 1

1−𝜆𝑤+ 𝜉_𝑤(𝜋_𝑡−1𝑤_𝑡−1)

1 1−𝜆_𝑤]

1−𝜆_𝑤

.

Dividing this equation by the price level of the current period, and log-linearizing, yields (1 − 𝜉_𝑤)𝑤̂̃_𝑡 = 𝜉_𝑤𝑤̃_𝑡− 𝜉_𝑤(𝑤̃_𝑡−1(𝜋̃_𝑡− 𝜋̃_𝑡−1)). A.3.3.iv Merging equation A.3.3.iii with A.3.3.iv results, after some rearrangement, in the final equation

𝑤̃_𝑡−1 =𝑏_𝑤(1 + 𝛽𝜉_𝑤²) − 𝜆_𝑤

𝑏_𝑤𝜉_𝑤 𝐸_𝑡𝑤̃_𝑡− 𝛽𝐸_𝑡𝑤̃_𝑡+1

− 𝐸_𝑡[𝛽(𝜋̃_𝑡+1− 𝜋̃_𝑡) − (𝜋̃_𝑡− 𝜋̃_𝑡−1)]

−1 − 𝜆_𝑤

𝑏_𝑤𝜉_𝑤 𝐸_𝑡(𝜓̃_𝑐,𝑡− 𝑙̃_𝑡),

A.3.3

where 𝑏_𝑤 = ^{2𝜆𝑤−1}

(1−𝜉𝑤)(1−𝛽𝜉𝑤).

A.3.4. The Household‘s Intertemporal Euler Equation

Equation 3.6-10 corresponds to the FOC with respect to the household’s beginning of period 𝑡 + 1 money stock, which has the log-linearized form

𝐸_𝑡(𝜓̃_𝑐,𝑡+1+ 𝑅̃_𝑡+1− 𝜋̃_𝑡+1− 𝜓̃_𝑐,𝑡) = 0 A.3.4

56 A.3.5. The Capital Euler Equation

The fifth equation is the log-linearized Capital Euler equation. To derive this equation, it helps to collect several definitions and functional form assumptions here.

The following two steady-state constraints are imposed on the capital utilization function 𝑎(𝑢_𝑡)

𝑈 = 1; 𝑎(𝑈) = 0. A.3.5.i

Moreover, at the steady-state, the real rental rate on capital equals 𝑟^𝑘 = 1

𝛽− (1 − 𝛿) = 𝑎^′(𝑢), A.3.5.ii where the last equality follows directly from equation 3.6-11. Function 3.6-15 describes the evolvement of the capital stock, which is restricted to

𝑆(1) = 𝑆^′(1) = 0; 𝜂_𝑘 ≡ 𝑆^′′(1) > 0; 𝐹₁ = 1; 𝐹₂ = 0 A.3.5.iii Given these functional form assumptions, equation 3.6-14 implies that at the steady-state

𝑃_𝑘^′_,𝑡 = 𝑃_𝑘^′_,𝑡+1 = 1. A.3.5.iv Equation 3.6-17 is also repeated here for convenience:

Ε_𝑡𝜓_𝑐,𝑡+1[𝑢_𝑡+1𝑟_𝑡+1^𝑘 + 𝑃_{𝑘′,𝑡+1}(1 − 𝛿) − 𝑃_𝑘^′_,𝑡𝑎(𝑢_𝑡+1)] − Ε_𝑡𝜓_𝑐,𝑡𝑃_𝑘^′_,𝑡 = 0.

Taking logs and ignoring the expectations operator to keep notation concise results in ln(𝜓_𝑐,𝑡) + 𝑙𝑛(𝑃_𝑘^′_,𝑡) = ln(𝛽) + 𝑙𝑛(𝜓_𝑐,𝑡+1) + ln [𝑢_𝑡+1𝑟_𝑡+1^𝑘 + 𝑃_{𝑘′,𝑡+1}(1 − 𝛿) − 𝑃_𝑘^′_,𝑡𝑎(𝑢_𝑡+1)].

Linearization with a first-order Taylor expansion yields 1

𝜓(𝜓_𝑐,𝑡 − 𝜓) + 1

𝑃_𝑘^′(𝑃_𝑘^′_,𝑡 − 𝑃_𝑘^′)

= 1

𝜓(𝜓_𝑐,𝑡+1− 𝜓)

+ 1

𝑈𝑟^𝑘− 𝑎(𝑈) + 𝑃_𝑘^′(1 − 𝛿)[𝑟^𝑘(𝑢_𝑡+1− 𝑈)𝑈

𝑈+ 𝑈(𝑟_𝑡+1^𝑘 − 𝑟^𝑘)𝑟^𝑘 𝑟^𝑘 + (1 − 𝛿)(𝑃_𝑘^′_,𝑡+1− 𝑃_𝑘^′)𝑃_𝑘^′

𝑃_𝑘^′− 𝑃_𝑘^′𝑎^′(𝑈)(𝑢_𝑡+1− 𝑈)𝑈 𝑈 + 𝑎(𝑈)(𝑃_𝑘^′_,𝑡− 𝑃_𝑘^′)𝑃_𝑘^′

𝑃_𝑘^′ ], where the steady-state relations were already omitted.

57 This can be greatly simplified using equations A.3.5.i to A.3.5.iv. The term preceding the square brackets can be reformulated with the following steps:

1

𝑈𝑟^𝑘− 𝑎(𝑈) + 𝑃_𝑘^′(1 − 𝛿)= 1

𝑟^𝑘+ (1 − 𝛿)= 1 1

𝛽− (1 − 𝛿) + (1 − 𝛿)

= 𝛽. A.3.5.v

Consequently, the total equation can be written as 𝜓̃_𝑐,𝑡+ 𝑃̃_𝑘^′_,𝑡 = 𝜓̃_𝑐,𝑡+1

+ 𝛽[𝑟^𝑘𝑈𝑢̃_𝑡+1+ 𝑈𝑟^𝑘𝑟̃_𝑡+1^𝑘 + (1 − 𝛿)𝑃_𝑘^′𝑃̃_𝑘^′_,𝑡+1− 𝑃_𝑘^′𝑎^′(𝑈)𝑈𝑢̃_𝑡+1 + 𝑎(𝑈)𝑃_𝑘^′𝑃̃_𝑘^′_,𝑡].

Moreover, the term in square brackets can be simplified further. Since 𝑃_𝑘^′ = 1, the first and fourth term cancel each other due to A.3.5.ii. The fifth drops out because of A.3.5.i. Thus,

𝜓̃_𝑐,𝑡 + 𝑃̃_𝑘^′_,𝑡 = 𝜓̃_𝑐,𝑡+1+ 𝛽[𝑟^𝑘𝑟̃_𝑡+1^𝑘 + (1 − 𝛿)𝑃̃_𝑘^′_,𝑡+1].

Applying similar steps as the ones carried out in equation A.3.5.v, this expression can be written as

𝜓̃_𝑐,𝑡+ 𝑃̃_𝑘^′_,𝑡 = 𝜓̃_𝑐,𝑡+1+ [1 − 𝛽(1 − 𝛿)]𝑟̃_𝑡+1^𝑘 + (1 − 𝛿)𝛽𝑃̃_𝑘^′_,𝑡+1.

Substituting for the return on capital, given by 𝑟̃_𝑡+1^𝑘 = 𝑤̃_𝑡+1+ 𝑅̃_𝑡+1+ 𝑙̃_𝑡+1− 𝑘̃_𝑡+1, results in the final solution, which after reinserting the expectations operator is identical to the equation provided by CEE

0 = 𝐸_𝑡{−𝑃̃_𝑘^′_,𝑡 − 𝜓̃_𝑐,𝑡+ 𝜓̃_𝑐,𝑡+1+ (1 − 𝛽(1 − 𝛿))[𝑤̃_𝑡+1+ 𝑅̃_𝑡+1+ 𝑙̃_𝑡+1− 𝑘̃_𝑡+1]

+ 𝛽(1 − 𝛿)𝑃̃_𝑘^′_,𝑡+1}. A.3.5

A.3.6. The Aggregate Resource Constraint

For convenience, the derivation is postponed to subsection A.4.1. This is useful because the steps are identical except for an additional term that accounts for vacancy posting costs in the augmented model. The log-linearized form of the resource constraint provided in equation 3.9-2 is given by

[1

𝛽− (1 − 𝛿)]𝑠_𝐾

𝑠_𝑐 𝑢̃_𝑡+ 𝑐̃_𝑡+𝛿𝑠_𝑘

𝑠_𝑐 𝑖̃_𝑡 = 𝛼

𝑠_𝑐𝑘̃_𝑡+1 − 𝛼

𝑠_𝑐 𝑛̃_𝑡. A.3.6

A.3.7. The Loan Market Clearing Condition

Equation 3.9-1 in real terms has the form 𝜇_𝑡𝑚_𝑡− 𝑞_𝑡 = 𝑤_𝑡𝑙_𝑡. Log-linearization results in 𝑤̃_𝑡+ 𝑙̃_𝑡 = 1

𝜇𝑀 − 𝑄(𝑀(𝜇_𝑡− 𝜇)𝜇

𝜇+ 𝜇(𝑚_𝑡− 𝑀)𝑀

𝑀− (𝑞_𝑡− 𝑄)𝑄 𝑄).

At the steady-state, 𝑤𝐿 = 𝜇𝑀 − 𝑄.

58 Therefore, combining terms and rearranging yields

𝜇𝑀(𝜇̃_𝑡+ 𝑚̃_𝑡) − 𝑄𝑞̃_𝑡− 𝑤𝐿(𝑤̃_𝑡+ 𝑙̃_𝑡) = 0. A.3.7 A.3.8. The Money Growth Rate

The money growth rate is defined as 𝜇_𝑡−1= ^𝑀𝑡

𝑀_𝑡−1. Log-linearization gives

𝜇̃_𝑡−1+ 𝑚̃_𝑡−1− 𝜋̃_𝑡− 𝑚̃_𝑡= 0, A.3.8 where again, prior to log-linearization, the definition of real variables, 𝑤_𝑡 = 𝑊_𝑡/𝑃_𝑡, was used to rewrite the equation as

𝜇_𝑡−1= 𝑚_𝑡𝑃_𝑡 𝑚_𝑡−1𝑃_𝑡−1. A.3.9. The Generalized Habit Formation

CEE define the utility of consumption as 𝑐_𝑡− 𝐻_𝑡, with 𝐻_𝑡= 𝜒𝐻_𝑡−1+ 𝑏𝑐_𝑡−1.⁵² Log- linearization yields

𝐻̃_𝑡 = ¹

𝜒𝐻+𝑏𝐶(𝜒(𝐻_𝑡−1− 𝐻) + 𝑏(𝑐_𝑡−1− 𝑐)^𝐶

𝐶), where the steady-state relations ¹

𝜒𝐻+𝑏𝐶 = 𝐻 and 𝐻(1 − 𝜒) = 𝑏𝑐 can be used to rewrite it as 𝐻̃_𝑡− 𝜒𝐻̃_𝑡−1− (1 − 𝜒)𝑐̃_𝑡−1= 0. A.3.9 However, CEE estimate a value of 𝜒 = 0, which essentially renders this equation futile.

Equation A.3.9 is only kept in the system of equations to keep notation in line with CEE.

A.3.10. The Generalized Consumption Euler Equation

Equation 3.6-7, together with equations 3.6-8 and A.3.9 can be log-linearized to the form

𝐸_𝑡 {

−𝛽𝜒𝜓̃_𝑐,𝑡+1 + 𝜎̃_𝑐^𝑖[𝑐̃_𝑡− 𝑏 1 − 𝜒𝐻̃_𝑡]

−(𝑏 + 𝜒)𝛽𝜎̃_𝑐^𝑖[𝑐̃_𝑡+1− 𝑏

1 − 𝜒𝐻̃_𝑡+1] + 𝜓̃_𝑐,𝑡 }

= 0, A.3.10 with 𝜎̃_𝑐^𝑖 = ^1−𝜒

1−𝜒−𝑏 1−𝛽𝜒 1−𝛽𝜒−𝛽𝑏 .⁵³

52 Note that the working paper of CEE erroneously writes 𝐻𝑡= 𝜓𝐻𝑡−1+ 𝑏𝑐𝑡−1.

53 Note that given 𝜒 = 0, equation A.3.A.3.10 collapses to the standard version because 𝐻̃_𝑡= (1 − 𝜒)𝑐̃_𝑡−1. Therefore, 𝑐̃𝑡− 𝑏/(1 − 𝜒)𝐻̃_𝑡 becomes 𝑐̃𝑡− 𝑏𝑐̃_𝑡−1.

59 A.3.11. The Investment Euler Equation

The log-linearized form of 3.6-14 is

𝐸_𝑡𝑃̃_𝑘^′_,𝑡 = 𝜂_𝑘𝐸_𝑡{𝑖̃_𝑡− 𝑖̃_𝑡−1− 𝛽[𝑖̃_𝑡+1− 𝑖̃_𝑡]}, A.3.11 where equations A.3.5.iii and A.3.5.iv were used for the derivation.

A.3.12. The Capital Accumulation Equation

Capital evolves according to equation 3.6-3, which using the equation 3.6-15, together with the corresponding functional form assumptions provided in A.3.5.iii yields the log-linearized form

𝑘̅̃_𝑡+1 = (1 − 𝛿)𝑘̅̃_𝑡+ 𝛿𝑖̃_𝑡, A.3.12 where the steady-state relation 𝑖 = 𝛿𝑘 was used.

A.3.13. The Household’s Capital Utilization Decision

The Euler equation linked to the household’s capital utilization decision is given by equation 3.6-11. Using the functional form assumptions of equation A.3.5.i and the inverse of the elasticity of the utilization rate to the rental rate of capital as defined in equation 6.3-1, the log-linearized form becomes

𝐸_𝑡[𝑟̃_𝑡^𝑘− 𝜎_𝑎𝑢̃_𝑡] = 0 A.3.13 This completes the derivation of the thirteen equations used by CEE. The following seven auxiliary equations augment the system in a way that facilitates the solution process in Dynare.

A.3.14. The Definition of Capacity Utilization

Recall from section 3.6 that the physical capital stock, 𝑘̅_𝑡, is associated with capital services via the relationship 𝑘_𝑡= 𝑢_𝑡𝑘̅_𝑡, with 𝑢_𝑡 symbolizing the utilization rate of capital. Log- linearization results in

𝑢̃_𝑡 = 𝑘̃̅_𝑡− 𝑘̃_𝑡, A.3.14 where the Dynare code takes into account that the physical capital stock is assumed to be predetermined in the system.

A.3.15. The Return on Capital

As already mentioned in the discussion of equation A.3.5, the return on capital has the following log-linearized form

𝑟̃_𝑡+1^𝑘 = 𝑤̃_𝑡+1+ 𝑅̃_𝑡+1+ 𝑙̃_𝑡+1− 𝑘̃_𝑡+1. A.3.15

60 A.3.16. The Taylor Rule

Using the properties of the natural logarithm, the Taylor rule described in equation 6.2-1 can be expressed as

ln(𝑅_𝑡) − ln(𝑅)

= 𝜌_𝑟[ln(𝑅_𝑡−1) − ln(𝑅)]

+ (1 − 𝜌_𝑟)[𝜌_𝜋(ln(𝜋_𝑡) − ln(𝜋)) + 𝜌_𝑌(ln(𝑌_𝑡) − ln(𝑌))] + 𝜖_𝑡.

A first order Taylor expansion around steady-state yields, after dropping constant terms, the form

𝑅̃_𝑡 = 𝜌_𝑟𝑅̃_𝑡−1+ (1 − 𝜌_𝑟)[𝜌_𝜋𝜋̃_𝑡+ 𝜌_𝑌Ỹ_t] + 𝜖_𝑡. 0.a Consequently, the CEE variant described in section 3.8 has the log-linearized form

𝑅̃_𝑡= 𝜌_𝑟𝑅̃_𝑡−1+ (1 − 𝜌_𝑟)[𝜌_𝜋Ε_𝑡𝜋̃_𝑡+1+ 𝜌_𝑌Ỹ_t] + 𝜖_𝑡. 0.b A.3.17. Real Marginal Cost

As derived in appendix A.1, cost minimization in the intermediate goods sector implies that real marginal costs have the form of equation 3.5-2. Log-linearization yields

𝑠̃_𝑡= 𝛼𝑟̃_𝑡^𝑘+ (1 − 𝛼)𝑤̃_𝑡𝑅̃_𝑡. A.3.17 A.3.18. The Cobb-Douglas Production Function

Equation 5.4.2-1 describes the production function in the intermediate goods sector, which has the following log-linearized form:

𝑦̃_𝑡= 𝛼𝑘̃_𝑡+ (1 − 𝛼)𝑙̃_𝑡. A.3.18 A.3.19. The Real Wage as Defined in CEE

CEE define the wage in their thirteen variable system as shown in equation A.3.3.i. For ease of reference, the corresponding log-linearized form, equation A.3.3.ii, is quoted again:

𝑤̅̃_𝑡 = 𝑤̃_𝑡− 𝜋̃_𝑡. A.3.19

A.3.20. Real Cash Balances

Consistent with equation A.3.19, real cash balances in log-linearized form give

𝑞̅̃_𝑡 = 𝑞̃_𝑡− 𝜋̃_𝑡. A.3.20

Im Dokument Unemployment, the business cycle and monetary policy: augmenting a medium sized new keynesian DSGE model with labor market dynamics / Alexander Koll (Seite 55-61)