Model setup

In the model, there are two types of firms and two types of labor input: Final-good firms use production line workers P and administrative services S to produce the fi-nal good according to a Cobb-Douglas production function with decreasing returns to scale. Production line workers are employed directly at the firm. The administrative service input can either be produced inhouse by employing administrative workers A^I, or obtained as an intermediate input S^O from external service providers. The total output of final-good firmi is therefore given by:

F_i,t = x_i,tP_i,t^α(A^I_i,t+S_i,t^O

| {z }

Si,t

)^β (1.10)

The production function implies that administrative inhouse staff and external services are perfect substitutes. The latter are produced by a continuum of external service

16For the special case ofρ= ¹₂ this can be shown analytically by taking the derivative of equation 1.9 with respect toχ: This derivative is negative as long asχ <q_κw

L¯ .

providers j which produce intermediate services with the same linear production func-tion as inhouse employees:

S_j,t = A_j,t (1.11)

External service providers have to pay a variable cost χ for each unit of services they produce. This exogenous cost component stands in as a proxy for the transaction costs of procuring external services and integrating them with services provided by in-house employees. The market for external services is perfectly competitive such that the price of one unit of externally procured service is the equilibrium wage and the variable cost accruing to the external provider: w_A +χ. An alternative approach to introduce a higher marginal cost of services produced by external providers would be to treatχas an endogenous price markup charged by service firms. That would require heterogeneous service input goods that are combined through a CES-aggregator to an intermediate input good. As the marginal cost wedge is treated as exogenous here, thus ruling out any feedback effects, I use the computationally more feasible reduced form of linear transaction costs rather than introducing explicit price markups through a CES-aggregator.

Final-good firms face idiosyncratic productivity shocks, x_i,t, that are independent across firms and follow an AR-1 process:

x_i,t = ρ_xxi,t−1+^x_i,t (1.12)

The innovations ^x_i,t are log-normally distributed with mean zero and variance σx re-spectively. There is no aggregate risk.

At the beginning of the period each firm learns its idiosyncratic shock. Taking wages {w_P, w_A} as given, final-good firm i chooses optimally the amount of production line workers P_i,t, administrative workers employed inhouse A^I_i,t and administrative services bought from external providers S_i,t^O. Final-good firms face worker type-specific fixed costs κ_P and κ_A when adjusting the amount of production line workers or administra-tive employees. Production line workers can only be employed inhouse and therefore the firm can only adjust their amount through hiring and firing at fixed cost κ_P. In contrast to that, the final-good firm can avoid the fixed cost of adjusting its stock of administrative employees by decreasing or increasing the amount of services it procures from external providers at the marginal costw_A+χ. The marginal cost component χ therefore introduces a trade-off between hiring and firing administrative staff employed inhouse or adjusting the amount of services procured from external providers.

Final-good firms cannot sell intermediate services themselves (S_i,t^O ≥ 0).¹⁷ The

17Without that constraint, final-good firms which receive negative productivity shocks would utilize their redundant administrative workers to sell intermediate service inputs to other final-good firms.

final-good firm’s value function is therefore given by:

V_F(Pi,t−1, A^I_i,t−1, x_i,t) = F_i,t −w_PP_i,t−w_AA^I_i,t−(w_A+χ)S_i,t^O (1.13)

−κ_P1_Pi,t6=P_i,t−1 (1.14)

−κ_A1_AI

i,t6=A^I_i,t−1 (1.15)

+E

1

1 +rV_F(P_i,t, A^I_i,t, x_i,t+1)

(1.16) External services can be adjusted flexibly in every period and are therefore not a state variable. Hence, they are chosen optimally as a function of the amount of inhouse employees:

S_i,t^∗ = βx_i,t(P_i,t^∗)^α w_A+χ

!_1−β¹

−A^I∗_i,t (1.17)

The first term on the right-hand side is the total amount of service inputs demanded by firmi at time t.

The model is closed by assuming a fixed supply of labor for both types, where the share of production line workers in the economy isλ∈(0,1) and the total amount of workers is normalized to 1. The two wages are then pinned down by

λ = ^X

i

ω_i^IP_i^∗ (1.18)

1−λ = ^X

i

ω_i^I(A^I∗_i +S_i^O∗) (1.19) Here,ω^I_i denotes the weights of final good firms in the joint distribution of the stochastic states x_i.

Worker reallocation

Due to the decreasing returns to scale technology in the final-goods sector (α+β <1), there will be a stationary distribution of final-good firms with heterogeneous labor choices. Reallocation arises endogenously as final-good firms move between different idiosyncratic states and adjust the amount of employees. There is no matching friction and therefore workers move directly from firms with negative productivity shocks to firms with positive productivity shocks without going through unemployment. The aggregate reallocation rates in the final goods sector are therefore given by:

π_P = 1

2^P_iω_i^IP_i^I∗

X

i

X

i⁰

ω_i^Iπ_i,i⁰|P_i^∗0 −P_i^∗| (1.20)

π_A = 1

2^P_iω_i^IA^I∗_i

X

i

X

i⁰

ω^I_iπ_i,i⁰|A^∗_i0 −A^∗_i| (1.21) The reallocation is given by the sum of individual adjustment decisions of all firms weighted with their respective weights ω_i^I and the stochastic transition probabilities π_i,i⁰. Note that in the stationary distribution each transiting worker is counted twice

-as a separation in the previous firm and -as a hire in the new firm. The absolute sum of labor adjustments therefore has to be divided by two.

In contrast to the final-goods sector, service firms do not face idiosyncratic shocks and have a linear production function. Hence, there is no scope for endogenous real-location. In order to match aggregate reallocation rates in the data, there will be a fractionπS of service firms exiting the market in every period which are replaced by new service firms. That triggers exogenous reallocation of workers in the service sector. As there are no aggregate shocks, wages and the aggregate amount of worker reallocation are constant over time. The following analysis therefore concerns the steady state in the economy rather than the reaction to aggregate shocks.

Mechanics of the model

Before laying out the calibration, it is helpful to reconsider how the final-good firm reacts to idiosyncratic TFP-shocks compared with the simple example in the stylized model. The case for production line workers is equivalent to thebenchmark scenario in the stylized model: Adjusting the amount of these employees comes at a fixed adjust-ment cost κP. That creates an inaction region in which shocks are sufficiently small that the firm refrains from paying κ_P and holds on to its production staff from last period.

For administrative workers the basic mechanism from the stylized model holds as well.

However, the dynamic setting implies that the firm might choose to always procure a positive amount of administrative services externally. The rationale for procuring a

“buffer stock” of external services stems from the asymmetry outlined in the stylized model: The firm can always buy more external services but cannot utilize its overca-pacity of service workers in case of negative shocks to sell their services as intermediate input to other firms. Through buying a certain amount of services externally the firm hedges against negative shocks as it provides the opportunity to cut down on adminis-trative labor input in case of negative shocks without having to pay the fixed adjustment costs.

Nonetheless, this additional feature does not alter the main intuition from the stylized static model: The possibility to outsource administrative service inputs to external providers makes it more profitable for firms to respond to positive (negative) TFP-shocks by procuring more (less) external services rather than hiring (firing) inhouse staff. Administrative workers employed with final-good firms are therefore shielded from idiosyncratic shocks and reallocate less often between firms.