Firm Dynamics and the Macroeconomy: Basics

Toshihiko MukoyamaGeorgetown University

University of Tokyo, July 2021 (Lecture 2)

From today, we focus on L

I If the capital is fixed,

Ai(Kαi L

1−αi )γ = BiL

βi

whereBi = AiK

αγi

andβ = γ(1− α) ∈ (0, 1).

From today, we focus on LI If the rental market for capital is perfectly competitive,

maxKi

Ai(Kαi L

1−αi )γ − rKi

imply

Ki =(αγr

) 11−αγ

A1

1−αγi L

γ−αγ1−αγi

Plugging this solution into the production function

Ai(Kαi L

1−αi )γ =

(αγr

) αγ1−αγ

A1

1−αγi L

γ−αγ1−αγi

Thus we can write a new production function

BiLβi

where

Bi =(αγr

) αγ1−αγ

A1

1−αγi

and

β =γ − αγ1− αγ

∈ (0, 1)

MisallocationFirst, continuing with the discussion in the last class, let us talkabout the misallocation. An example:

I There are two firms, firm 1: Y1 = A1Lα1 and firm 2:

Y2 = A2Lα2 where α ∈ (0, 1).

I Let A1 = 1 and A2 = 2. α = 1/2.I Assume that there are firm-specific distortions τi (i = 1, 2).

We can think of τi as a tax.I Firm i maximizes the profit

(1− τi)AiLαi − wLi.

Because the actual output is AiLαi , the firm’s decision

problem is distorted.I Assume that the total labor is fixed at 1. Thus w is

deterimined by L1 + L2 = 1.I The total output is computed as

Y = Y1 + Y2 = A1Lα1 +A2L

α2

Misallocation

-0.5 0 0.5

1.9

1.95

2

2.05

2.1

2.15

2.2

2.25

I The total output is reduced the most with positive correlationbetween the distortion (discouragement) and productivity(Restuccia and Rogerson, 2008)

Misallocation

-0.5 0 0.5

0.5

1

1.5

2

2.5

3

3.5

I MPL (αAiLα−1i ) dispersion is the source of the productivity

loss.

Situations where misallocations can occur: ExamplesI Tax rates are different depending on firm

identity/characteristicsI Regulations that depend on firm sizeI Firing/hiring taxes/subsidiesI Entry/exit taxes/subsidies, some other frictionsI Financial frictionsI Contract enforcement

Notes:I The cost of reduced entry depends on the post-entry

importance of entrants.I The importance of financial frictions depends on the

persistence of shocks (whether the firm can overcome thefriction by self-financing); see Moll (2014). The shock doesseem to be persistent; see Lee and Mukoyama (2015). Therestill can be effects for young firms and potential entrants.

I The importance of contract enforcement at the industry levelpositively correlates with industry productivity (Mukoyamaand Popov, 2020)

Entry barriersFrom Moscoso Boedo and Mukoyama (2012)

0.01 0.1 1

25

50

100

200

400

800

GNI per capita relative to US

Tim

e: D

ays

Start a Business

0.01 0.1 1

25

50

100

200

400

800

GNI per capita relative to US

Tim

e: D

ays

Dealing with Licenses

0.01 0.1 110

0

101

102

103

104

105

GNI per capita relative to US

Cost: %

GN

I per

capita

0.01 0.1 010

0

101

102

103

104

105

GNI per capita relative to US

Cost: %

GN

I per

capita

I Why? Political economy considerations (Mukoyama andPopov, 2014)

Misallocation as a theory of TFP

I Misallocation can change the measured TFP (measured byY/Lα, for example) without changing A1 and A2.

I The effect can be sizable but not as much as 10-foldsdifferences between rich and poor countries.

I For the development questions, the determination of Ai(growth of productivity at the firm level) is still important.

Firm growth

On firm growthTwo small points first:I First, note that individual firm growth is not necessary or

sufficient for aggregate growth.I Second, the loss from missing entry can be large if we take

firm growth into account.

An example:I Labor supply is elastic (employment is demand-determined).

One firm hires one worker.I The production of a firm who enters at time τ and age a

(today is t = a+ τ) is Aτeγa. γ > 0 is the firm growth rate.

Assume that Aτ = A0egτ .

I The surviving firms at age a is e−δa. Assume δ > γ.I The mass of entrants is 1.I Outcome: The total employment is

∫∞0 e−δada = 1/δ. The

aggregate production is A0egt/(δ + g − γ).

I If ∆ units of entrants are lost, the immediate loss is ∆Aτdtbut the present value of loss is ∆Aτ/(ρ+ δ − γ), where ρ isthe discount rate.

Some facts on firm growthFigures from Haltiwanger et al. (2013)

I The growth rate of a firm is independent of size: “Gibrat’sLaw” (mixed supports in the data)

Some facts on firm growth

Figures from Haltiwanger et al. (2013)

Some facts on firm growth

Figures from Haltiwanger et al. (2013)

Some facts on firm growth

Figures from Haltiwanger et al. (2013)

References

Cao, D., H. R. Hyatt, T. Mukoyama, and E. Sager (2020). Firm Growth through New Establishments.https://toshimukoyama.github.io/MyWebsite/CHMS.pdf.

Haltiwanger, J., R. S. Jarmin, and J. Miranda (2013). Who Creates Jobs? Small versus Large versus Young.Review of Economics and Statistics.

Hopenhayn, H. and R. Rogerson (1993). Job Turnover and Policy Evaluation: A General Equilibrium Analysis.Journal of Political Economy 101, 915–938.

Lee, Y. and T. Mukoyama (2015). Productivity and Employment Dynamics of US Manufacturing Plants.Economics Letters 136, 190–193.

Moll, B. (2014). Productivity Losses from Financial Frictions: Can Self-Financing Undo Capital Misallocation?American Economic Review 104, 3186–3221.

Moscoso Boedo, H. and T. Mukoyama (2012). Evaluating the Effects of Entry Regulations and Firing CostsonInternational Income Differences. Journal of Economic Growth 17, 143–170.

Mukoyama, T. (2009). On the Establishment Dynamics in the United States and Japan. Monetary and EconomicStudies 27, 53–74.

Mukoyama, T. and S. Osotimehin (2019). Barriers to Reallocation and Economic Growth: The Effects of FiringCosts. American Economic Journal: Macroeconomics 11, 235–270.

Mukoyama, T. and L. Popov (2014). The Political Economy of Entry Barriers. Review of Economic Dynamics 17,383–416.

Mukoyama, T. and L. Popov (2020). Industrialization and the Evolution of Enforcement Institutions. EconomicTheory 69, 745–788.

Restuccia, D. and R. Rogerson (2008). Policy Distortions and Aggregate Productivity with HeterogeneousEstablishments. Review of Economic Dynamics 11, 707–720.