The Log-Linearized System of the DMP Model

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

61 A.4.1. The Aggregate Resource Constraint

In addition to the standard resource constraint used in CEE, vacancy posting costs need to be taken into account. Moreover, the Search and Matching framework now determines the aggregate amount of the homogeneous labor input used in the intermediate production process.

A few considerations are needed to derive the final equation. As explained in section 3.5, capital and labor is rented in perfectly competitive factor markets. Consequently, each producer faces identical factor prices and constant returns to scale in the production function imply that there is no need to distinguish between firms. Thus, the unweighted average of output can be expressed as

𝑌^∗ = ∫ 𝑌_𝑗𝑑𝑗 = ∫ 𝑘_𝑗^𝛼𝑙_𝑗^1−𝛼𝑑𝑗 = 𝑘^𝛼𝑙^1−𝛼,

1

0 1

0

A.4.1.i

where 𝑘 and 𝑙 represent the aggregate amount of capital and of the homogenous labor input:⁵⁴ 𝑘 = ∫ 𝑘_𝑗𝑑𝑗

1

0

; 𝑙 = ∫ 𝑙_𝑗𝑑𝑗.

1

0

A.4.1.ii

Time subscripts are omitted in order to be consistent with the derivation provided in the working paper of Christiano et al. (2001). Eventually, time subscripts will be reinserted to provide a correct representation of the log-linearized resource constraint. The authors list two reasons why this is not the best way to express the aggregate resource constraint. Firstly, total output should be related to total labor, calculated as in equation 5.5-1. This is preferable because 𝑛_𝑡 characterizes labor as measured in the data, whereas 𝑙_𝑡 is the amount of the homogenous labor input used in the intermediate goods production. Secondly, 𝑌^∗ does not have any meaningful economic interpretation, because it is simply a sum of differentiated intermediate goods. The latter problem can be solved by substituting optimal demand for intermediate good 𝑗, characterized by equation 3.4-2, into A.4.1.i:

𝑌^∗ = ∫ 𝑌_𝑗𝑑𝑗 = ∫ (𝑃 𝑃_𝑗)

𝜆_𝑓 𝜆𝑓−1

𝑌𝑑𝑗 = 𝑌𝑃

𝜆_𝑓 𝜆_𝑓−1(𝑃^∗)

𝜆_𝑓 1−𝜆_𝑓,

1

0 1

0

where 𝑃^∗ denotes the weighted average of all individual prices.⁵⁵ Solving for 𝑌 and substituting out 𝑌^∗ with the last term in equation A.4.1.i. gives

𝑌 = (𝑃^∗ 𝑃)

𝜆_𝑓 𝜆𝑓−1

𝑘^𝛼𝑙^1−𝛼.

Instead of the unweighted average output, this equation now contains the aggregate output, which is allocated between consumption, investment, the resources used up in capital utilization

54 CEE use capital letters, whereas here lower case letters are used to avoid confusion with steady-state values.

55 Note that the weights are not identical to the ones used in equation 3.4-3 because 𝑃^∗= [∫ 𝑃_𝑗

𝜆𝑓 1−𝜆𝑓 1

0 𝑑𝑗]

1−𝜆𝑓 𝜆𝑓

.

62 as well as the total vacancy posting costs. Consequently, the model economy is facing the following modified resource constraint:

𝑐 + 𝑖 + 𝑎(𝑢)𝑘̅ + 𝜅𝑣 ≤ (𝑃^∗ 𝑃)

𝜆𝑓 𝜆𝑓−1

𝑘^𝛼𝑙^1−𝛼.

The last modification needed regards the total labor force. Substituting equation 5.1-2, using definitions 5.5-1 and A.4.1.ii, gives

𝑐 + 𝑖 + 𝑎(𝑢)𝑘̅ + 𝜅𝑣 ≤ (𝑃^∗ 𝑃)

𝜆𝑓 𝜆𝑓−1

𝑘^𝛼(𝐴𝑛)^1−𝛼, A.4.1.iii

which corresponds to the sought-after resource constraint of equation 5.5-2. CEE state that the term in front of the production function is similar to the so called Solow residual, which is essentially the TFP part, denoted as 𝑧_𝑡, of the production function described in equation 6.5-1.

Yun (1996) showed that in a first order approximation, as used here, this “Solow residual” is a constant. As CEE highlight, this result can be found by obtaining expressions for 𝑊^∗/𝑊 and 𝑃^∗/𝑃 respectively.

Equivalent to the way equation 3.4-3 was rewritten in the form of equation 3.5-7, 𝑃_𝑡^∗ and , 𝑊_𝑡^∗can be expressed as

𝑃_𝑡^∗ = [(1 − 𝜉_𝑝)𝑃̃_𝑡

𝜆𝑓 1−𝜆_𝑓

+ 𝜉_𝑝(𝜋_𝑡−1𝑃_𝑡−1^∗ )

𝜆_𝑓 1−𝜆_𝑓]

1−𝜆𝑓 𝜆𝑓

.

Dividing this term by 𝑃_𝑡 yields

𝑝_𝑡^∗= [(1 − 𝜉_𝑝)𝑝̂_𝑡

𝜆𝑓 1−𝜆_𝑓

+ 𝜉_𝑝(𝜋_𝑡−1𝑝_𝑡−1^∗ )

𝜆_𝑓 1−𝜆𝑓]

1−𝜆𝑓 𝜆𝑓

, A.4.1.iv

where 𝑝^∗= 𝑃^∗/𝑃_𝑡. Log-linearizing this expression around the steady-state results in 𝑝̃_𝑡^∗ = (1 − 𝜉_𝑝)𝑝̂̃_𝑡+ 𝜉_𝑝(𝜋̃_𝑡−1− 𝜋̃_𝑡+ 𝑝̃_𝑡−1^∗ ).

After substituting equation 3.5-8 into the log-linearized expression one obtains 𝑝̃_𝑡^∗ = 𝜉_𝑝𝑝̃_𝑡−1^∗ .

Let’s supddpose 𝑝̃₀^∗ = 0. Therefore, 𝑝̃_𝑡^∗ = 0 for all future time periods and consequently, 𝑃_𝑡^∗ = 𝑃_𝑡 for all possible realizations of t. Since the almost identical steps apply for 𝑊_𝑡^∗ and this part is not required in the augmented model, this derivation is left out here.⁵⁶ During log-linearization, the transformations above can be used to rationalize considering 𝑃^∗/𝑃 as constant

56 CEE provide this part in their working paper appendix (2001).

63 equal to unity. For this reason, the following steps obtain the log-linearized resource constraint of equation A.4.1.iii. Using the properties of the natural logarithm, this constraint can be written as

ln(𝑎(𝑢)𝑘̅ + 𝑐 + 𝑖 + 𝜅𝑣) = 𝜆_𝑓

𝜆_𝑓− 1ln (𝑃^∗

𝑃) + (1 − 𝛼) ln(𝐴) + (1 − 𝛼) ln(𝑛) + 𝛼 ln(𝑘), where the inequality sign was switched to an equality sign because at the optimum all output is used up. After reinserting the omitted time subscripts, a linear first order Taylor expansion yields,

ln(𝑎(𝑈)𝐾̅ + 𝐶 + 𝐼 + 𝜅𝑉) +1

𝑌(𝑎^′(𝑢)𝐾̅𝑈𝑢̃_𝑡+ 𝑎(𝑈)𝐾̅𝑘̅̃_𝑡+ 𝑐𝑐̃_𝑡+ 𝑖𝑖̃_𝑡+ 𝜅𝑉𝑣̃_𝑡)

= (1 − 𝛼) ln(𝐴) + (1 − 𝛼)𝐴̃_𝑡+ (1 − 𝛼) ln(𝑁) + (1 − 𝛼)𝑛̃_𝑡+ 𝛼 ln(𝐾) + 𝛼𝑘̃_𝑡 Using the relationship 𝑘_𝑡 = 𝑢_𝑡𝑘̅_𝑡 and equation A.3.A.3.5.i, this can be rewritten as

𝑎^′(𝑢)𝐾

𝑌𝑢̃_𝑡+𝑐 𝑌𝑐̃_𝑡+ 𝑖

𝑌𝑖̃_𝑡+𝜅𝑉

𝑌 = (1 − 𝛼)(𝐴̃_𝑡+ 𝑛̃_𝑡) + 𝛼𝑘̃_𝑡,

where the steady-state terms of the log-linearized resource constraint are already cancelled out. It helps to define

𝑠_𝑐 = 𝑐/𝑌, 𝑠_𝑖 = 𝑖/𝑌 = 𝛿 𝐾/𝑌, 𝑠_𝑘 = 𝐾/𝑌, 𝑠_𝑘𝑣 = 𝑘𝑉/𝑌.

Thus, in combination with equation A.3.A.3.5.ii, the log-linearized resource constraint can be written in the final version, which has the following form:

[1

𝛽− (1 − 𝛿)]𝑠_𝑘

𝑠_𝑐𝑢̃_𝑡+ 𝑐̃_𝑡+𝛿𝑠_𝑘

𝑠_𝑐 𝑖̃_𝑡+𝑠_𝜅𝑣

𝑠_𝑐 𝑣̃_𝑡 = 𝛼

𝑠_𝑐𝑘̃_𝑡+1 − 𝛼

𝑠_𝑐 (𝐴̃_𝑡+ 𝑛̃_𝑡). A.4.1 A.4.2. The Labor Augmenting Technology Shock

Log-linearizing the labor augmenting technology shock of equation 5.1-3 yields

𝐴̃_𝑡 = 𝛿𝐴̃_𝑡−1 – 𝜖_𝑡. A.4.2 A.4.3. The Price Charged for the Aggregate Labor Input

Equation 5.2-9 in log-linearized form yields

𝑝̃_𝑡^𝑤 = 𝑚𝑐̃_𝑡^𝑤 A.4.3

A.4.4. The DMP Production Function

The homogenous aggregate labor input is produced according to equation 5.1-2. Using equation A.4.1.ii one obtains the following log-linearized form:

𝑙̃_𝑡 = 𝐴̃_𝑡+𝑛̃_𝑡. A.4.4

64 A.4.5. The Law of Motion for Employment

The employment accumulation equation is described by equation 4-9, which has the log-linearized form

𝑁𝑛̃_𝑡= (1 − 𝜌)𝑁𝑛̃_𝑡−1+ 𝑀𝑚̃_𝑡−1. A.4.5 A.4.6. The Unemployment Equation

Log-linearizing equation 4-1 results in

(𝑁/𝑈)𝑛̃_𝑡= −𝑢̃_𝑡. A.4.5

A.4.7. The Matching Function

Since the matching function defined in equation 4-2 has the form of a standard Cobb-Douglas function, the log-linearization was already described in section A.2 of the appendix. Applied to the matching function, log-linearization yields

𝑚̃_𝑡= 𝛾̃_𝑡+ 𝜉𝑢̃_𝑡+ (1 − 𝜉)𝑣̃_𝑡 A.4.7 A.4.8. Labor Market Tightness

Equation 4-3 has the following log-linearized form:

𝜃̃_𝑡= 𝑣̃_𝑡− 𝑢̃_𝑡 A.4.8

A.4.9. The Wage Equation

As already noted in section 4.3, the equilibrium real wage 𝑤_𝑡 is obtained by maximizing the generalized Nash product denoted by

max𝑤𝑡

(𝐽_𝑡− 𝑃_𝑡)^1−𝜂(𝐻_𝑡− 𝐼_𝑡)^𝜂. For convenience, all relevant equations are stated again:

 Value of employment 4.1-1: 𝐻_𝑡= 𝑤_𝑡+ Ε_𝑡𝛽[(1 − 𝜌)𝐻_𝑡+1+ 𝜌𝐼_𝑡+1]

 Value of unemployment 4.1-2: 𝐼_𝑡 = 𝑜 + Ε_𝑡𝛽[𝑓_𝑡𝐻_𝑡+1+ (1 − 𝑓_𝑡)𝐼_𝑡+1]

 Value of a vacancy 4.2-1: 𝑃_𝑡 = −𝜅 + Ε_𝑡𝛽[𝑞(𝜃_𝑡)𝐽_𝑡+1+ (1 − 𝑞(𝜃_𝑡))𝑃_𝑡+1]

 Value of a job 4.2-2: 𝐽_𝑡 = 𝐴_𝑡− 𝑤_𝑡+ Ε_𝑡𝛽[(1 − 𝜌)𝐽_𝑡+1+ 𝜌𝑃_𝑡+1]

 Free entry of firms: 𝑃_𝑡 = 0

Since the free entry condition is imposed, the value of a job equation reduces to the relationship 𝐽_𝑡 = 𝐴_𝑡− 𝑤_𝑡+ Ε_𝑡𝛽(1 − 𝜌)𝐽_𝑡+1. Therefore, it becomes evident that equation 5.2-8 is almost identical to equation 4.2-2. The only difference is the additional marginal cost term as well as the stochastic discount factor, which links the NK model to the DMP model, as described in section 5.2.

65 For ease of reference, equation 5.2-8 is quoted again:

𝐽_𝑡= 𝑚𝑐_𝑡^𝑤𝐴_𝑡− 𝑤_𝑡+ (1 − 𝜌)𝐸_𝑡𝛽^{𝜓𝑐,𝑡+1}

𝜓_𝑐,𝑡 𝐽_𝑡+1.

In the Nash product above, taking the first order condition with respect to the real wage 𝑤_𝑡 yields

(1 − 𝜂)(−1)(𝐽_𝑡− 𝑃_𝑡)^−𝜂(𝐻_𝑡− 𝐼_𝑡)^𝜂+ 𝜂(𝐽_𝑡− 𝑃_𝑡)^1−𝜂(𝐻_𝑡− 𝐼_𝑡)^𝜂−1= 0.

This expression can be simplified to

𝜂(𝐽_𝑡− 𝑃_𝑡) = (1 − 𝜂)(𝐻_𝑡− 𝐼_𝑡). A.4.9.i Imposing the free entry condition and rearranging yields the form

𝐽_𝑡 = 1 − 𝜂

𝜂 (𝐻_𝑡− 𝐼_𝑡), A.4.9.ii

which is identical to the Nash Sharing Search specification in Christiano et al. (2013). Taking the stochastic discount factor into account, inserting the value functions into A.4.9.i results in

𝜂 (𝑚𝑐_𝑡^𝑤𝐴_𝑡− 𝑤_𝑡+ Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [(1 − 𝜌)𝐽_𝑡+1])

= (1 − 𝜂) (𝑤_𝑡+ Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [(1 − 𝜌)𝐻_𝑡+1+ 𝜌𝐼_𝑡+1] − 𝑜

− Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [𝑓_𝑡𝐻_𝑡+1+ (1 − 𝑓_𝑡)𝐼_𝑡+1]).

Multiplying out the terms that involve 𝑤_𝑡 gives 𝜂𝑚𝑐_𝑡^𝑤𝐴_𝑡− 𝜂𝑤_𝑡+ ηΕ_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [(1 − 𝜌)𝐽_𝑡+1]

= (1 − 𝜂)𝑤_𝑡

+ (1 − 𝜂) {Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [(1 − 𝜌)𝐻_𝑡+1+ 𝜌𝐼_𝑡+1] − 𝑜

− Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [𝑓_𝑡𝐻_𝑡+1+ (1 − 𝑓_𝑡)𝐼_𝑡+1]}.

Solving for 𝑤_𝑡 yields

𝑤_𝑡= 𝜂𝑚𝑐_𝑡^𝑤𝐴_𝑡+ ηΕ_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [(1 − 𝜌)𝐽_𝑡+1] + (1 − 𝜂)𝑜

− (1 − 𝜂)Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [(1 − 𝜌)𝐻_𝑡+1+ 𝜌𝐼_𝑡+1] + (1 − η)Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [𝑓_𝑡𝐿_𝑡+1+ (1 − 𝑓_𝑡)𝐼_𝑡+1].

66 Note that after imposing the free entry condition and forwarding by one period, equation A.4.9.i has the form

𝜂Ε_𝑡𝐽_𝑡+1 = (1 − 𝜂)Ε_𝑡(𝐻_𝑡+1− 𝐼_𝑡+1). A.4.9.iii The wage equation can be simplified using A.4.9.iii, together with equation 5.2-7, reformulated

as ^𝜅

Ε_𝑡𝛽^{𝜓𝑐,𝑡+1}

𝜓𝑐,𝑡 𝐽_𝑡+1= 𝑞_𝑡. Using the fact that ^𝑓𝑡

𝑞𝑡= 𝜃_𝑡, the equations can be combined to the form 𝜅𝜃_𝑡 = 𝑓_𝑡Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 𝐽_𝑡+1. A.4.9.iv

Using equation A.4.9.iii, one can obtain 𝜂Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 (1 − 𝜌)𝐽_𝑡+1 = (1 − 𝜂)Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 (1 − 𝜌)(𝐻_𝑡+1− 𝐼_𝑡+1), which can be rewritten as

ηΕ_𝑡𝛽^{𝜓𝑐,𝑡+1}

𝜓𝑐,𝑡 (1 − 𝜌)𝐽_𝑡+1= (1 − 𝜂)Ε_𝑡𝛽^{𝜓𝑐,𝑡+1}

𝜓𝑐,𝑡 ((1 − 𝜌)𝐻_𝑡+1− (1 − 𝜌)𝐼_𝑡+1), which in turn is identical to

ηΕ_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 (1 − 𝜌)𝐽_𝑡+1

= (1 − 𝜂)Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 ((1 − 𝜌)𝐻_𝑡+1+ 𝜌𝐼_𝑡+1) − (1 − 𝜂)Ε_𝑡𝛽𝜓_𝑐,𝑡+1 𝜓_𝑐,𝑡 𝐼_𝑡+1. Likewise, (1 − 𝜂)Ε_𝑡𝛽^{𝜓𝑐,𝑡+1}

𝜓𝑐,𝑡 [𝑓_𝑡𝐻_𝑡+1+ (1 − 𝑓_𝑡)𝐼_𝑡+1] can be written as (1 − 𝜂)Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [𝑓_𝑡𝐻_𝑡+1− 𝑓_𝑡𝐼_𝑡+1] + (1 − 𝜂)Ε_𝑡𝛽𝜓_𝑐,𝑡+1 𝜓_𝑐,𝑡 𝐼_𝑡+1 . Thus, the wage equation has the form

𝑤_𝑡= 𝜂𝑚𝑐_𝑡^𝑤𝐴_𝑡+ (1 − 𝜂)Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [(1 − 𝜌)𝐻_𝑡+1+ 𝜌𝐼_𝑡+1] − (1 − 𝜂)Ε_𝑡𝛽𝜓_𝑐,𝑡+1 𝜓_𝑐,𝑡 𝐼_𝑡+1

−(1 − 𝜂)Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [(1 − 𝜌)𝐻_𝑡+1+ 𝜌𝐼_𝑡+1]+(1 − 𝜂)𝑜 + (1 − η)Ε_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 [𝑓_𝑡𝐻_𝑡+1− 𝑓_𝑡𝐼_𝑡+1] + (1 − 𝜂)Ε_𝑡𝛽𝜓_𝑐,𝑡+1 𝜓_𝑐,𝑡 𝐼_𝑡+1 The terms (1 − 𝜂)Ε_𝑡𝛽^{𝜓𝑐,𝑡+1}

𝜓_𝑐,𝑡 𝐼_𝑡+1 as well as (1 − 𝜂)Ε_𝑡𝛽^{𝜓𝑐,𝑡+1}

𝜓_𝑐,𝑡 [(1 − 𝜌)𝐻_𝑡+1+ 𝜌𝐼_𝑡+1] cancel.

Moreover, using A.4.9.iii and A.4.9.iv results in (1 − η)Ε_𝑡𝛽^{𝜓𝑐,𝑡+1}

𝜓𝑐,𝑡 [𝑓_𝑡𝐻_𝑡+1− 𝑓_𝑡𝐼_𝑡+1] = 𝜂𝜅𝜃_𝑡.

67 Combining these considerations, the final solution to the Nash bargaining process is

𝑤_𝑡= 𝜂(𝑚𝑐_𝑡^𝑤𝐴_𝑡+ 𝜅𝜃_𝑡) + (1 − 𝜂)𝑜, A.4.9.v which is identical to equation 5.3-1.

Taking logs results in

ln(𝑤_𝑡) = ln( 𝜂(𝑚𝑐_𝑡^𝑤𝐴_𝑡+ 𝜅𝜃_𝑡) + (1 − 𝜂)𝑜), which after a linear first order Taylor expansion yields

ln(𝑊) + 1

𝑊(𝑤_𝑡− 𝑤)

= ln(𝜂(𝑀𝐶^𝑤 ∗ 𝐴 + 𝜅𝜃) + (1 − 𝜂)𝑜) + 1

𝑊(𝜂𝑀𝐶^𝑤 ∗ 𝐴𝐴̃_𝑡+ 𝐴 ∗ 𝑀𝐶^𝑤𝑚𝑐̃_𝑡^𝑤+ 𝜅𝜃𝜃̃_𝑡).

Canceling the steady-state terms and rearranging results in the final solution

𝑊𝑤̃_𝑡= 𝜂𝑀𝐶^𝑤 ∗ 𝐴(𝐴̃_𝑡+ 𝑚𝑐̃_𝑡^𝑤) + 𝜂𝑘𝜃𝜃̃_𝑡. A.4.9 A.4.10. The Value of a Job Equation

For ease of reference, equation 5.2-8 is quoted again:

𝐽_𝑡 = 𝑚𝑐_𝑡^𝑤𝐴_𝑡− 𝑤_𝑡+ (1 − 𝜌)𝐸_𝑡𝛽𝜓_𝑐,𝑡+1 𝜓_𝑐,𝑡 𝐽_𝑡+1. Taking the natural logarithm results in

ln (𝐽_𝑡) = ln (𝑚𝑐_𝑡^𝑤𝐴_𝑡− 𝑤_𝑡+ (1 − 𝜌)𝐸_𝑡𝛽𝜓_𝑐,𝑡+1

𝜓_𝑐,𝑡 𝐽_𝑡+1 ), which after a linear first order Taylor expansion yields

ln(𝐽) + 𝐽̃_𝑡 = ln (𝑀𝐶_𝑡^𝑤𝐴 − 𝑊 + (1 − 𝜌)𝐸_𝑡𝛽𝜓_𝑐 𝜓_𝑐𝐽) +1

𝐽(𝑀𝐶 ∗ 𝐴(𝐴̃_𝑡+ 𝑚𝑐̃_𝑡^𝑤) + 𝑊𝑤̃_𝑡+ (1 − 𝜌)𝐸_𝑡𝛽1

𝜓𝐽𝜓_𝑐(𝜓_𝑐,𝑡+1− 𝜓_𝑐 𝜓_𝑐 ) + (1 − 𝜌)𝐸_𝑡𝛽𝜓_𝑐

𝜓_𝑐²𝐽(𝜓_𝑐,𝑡− 𝜓_𝑐) + (1 − 𝜌)𝐸_𝑡𝛽𝜓_𝑐

𝜓_𝑐𝐽 (𝐽_𝑡+1− 𝐽 𝐽 )).

Cancelling steady-state terms and combining terms results in the final equation

𝐽𝐽̃_𝑡 = 𝑀𝐶 ∗ 𝐴(𝐴̃_𝑡+ 𝑚𝑐̃_𝑡^𝑤) + 𝑊𝑤̃_𝑡+ (1 − 𝜌)𝐸_𝑡𝛽𝐽(𝜓̃_𝑐,𝑡+1− 𝜓̃_𝑐 + 𝐽̃_𝑡+1) A.4.10

68 A.4.11. The Job Creation Condition

Equation 5.2-7 states

𝜅

𝑞(𝜃_𝑡)= 𝐸_𝑡𝛽𝜓_𝑐,𝑡+1 𝜓_𝑐,𝑡 𝐽_𝑡+1. Taking logs results in

ln(𝜅) − ln(𝑞(𝜃_𝑡)) = ln(𝛽) + ln(𝜓_𝑐,𝑡+1) − ln(𝜓_𝑐,𝑡) + ln (𝐽_𝑡+1) Linearization yields

ln(𝜅) − [ln(𝑞(𝜃)) + 1

𝑞(𝜃)𝑞^′(𝜃)𝜃 (𝜃_𝑡− 𝜃 𝜃 )]

= ln(𝛽) + ln(𝜓_𝑐) + 1

𝜓_𝑐(𝜓_𝑐,𝑡+1− 𝜓_𝑐) − [ln(𝜓_𝑐) + 1

𝜓_𝑐(𝜓_𝑐,𝑡− 𝜓_𝑐)] + ln(𝐽) +1

𝐽(𝐽_𝑡+1− 𝐽).

Dropping steady-state terms and using the elasticity of the job filling rate with respect to the labor market tightness, defined as – 𝜉 =^{𝜕𝑞(𝜃)}

𝜕𝜃 𝜃

𝑞(𝜃), the function can be simplified to

𝜉𝜃̃_𝑡 = 𝜓̃_𝑐,𝑡+1− 𝜓̃_𝑐,𝑡+ 𝐽̃_𝑡+1. A.4.11

Im Dokument Unemployment, the business cycle and monetary policy: augmenting a medium sized new keynesian DSGE model with labor market dynamics / Alexander Koll (Seite 61-69)