A Quantitative Theory of TimeConsistent Unemployment Insurance Yun Pei and Zoe Xie

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Emil Bjerre Jensen

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Introduction

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Introduction Generally, during recessions, the U.S. government increases the duration of Unemployment Insurance (UI) benefits, meanwhile, unemployment rates are rising. Aim of the paper: 1. Why does the government extend UI benefits during recessions? 2. Does these increases lead to higher unemployment? 3. Would fixing durations improve welfare?

Two effects from increasing unemployment benefit durations • Negative job search effect: Moral Hazard problem due to time-inconsistent preferences, less incentive to search • Positive insurance effect: Insurance helps jobless workers smooth consumption while unemployed The Model • Endogenizes time-consistent UI policy in a stochastic general equilibrium search model – Using the concept of the Markov-perfect equilibrium • The government chooses UI benefit level, Taxes and Expected Duration each period. Findings; longer duration of unemployment benefit • Increases overall welfare (0.16%) through consumption of the unemployed workers • Reduces job search, leading to higher future unemployment (up to 55% of this is due to benefit durations)

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The Model environment •

Measure of workers and firms is normalized to one and they are both infinitely lived.

•

Workers can be either employed or unemployed, they are risk averse and maximizing 𝑡 expected lifetime utility 𝐸0 σ∞ 0 𝛽 𝑈 𝑐𝑡 − 𝑣(𝑠𝑡 ) – – – – –

•

The Unemployed workers – – –

•

Discount factor, 𝛽𝑡 Time is discrete. Utility of consumption Disutility of search effort Assumption: Cannot save or borrow

No on-the-job search, hence only unemployed workers choose 𝑠𝑡 > 0 Receive unemployment benefits, b, from the government. Produce leisure, home production, and welfare, jointly combined in h.

Firms – – –

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risk neutral maximizing expected sum of profits. Can either be matched (and producing) with a worker or vacant (at cost 𝜅)

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The Model environment •

Number of new matches is given by the matching function M(I,V) – – – –

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Job-finding probability per efficient unit of search intensity, f, is given by 𝑓 𝜃 = 𝑀(1, 𝜃) – –

•

search by unemployed workers, I Vacancy posting by firms, V M(I,V) exhibits constant returns to scale in both arguments Workers and firms are randomly matched trough this matching function

q is decreasing in 𝜃.

𝑀(𝐼,𝑉) 𝑉

1

𝜃

Existing matches are destroyed with probability 𝛿

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A match pair of worker and firm, results in aggregate productivity, z.

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=

= 𝑀( , 1)

•

–

𝐼

f is increasing in 𝜃 Probability of finding a job for an unemployed worker is s∙f(𝜃)

Job-filling probability per vacancy, q, is given by 𝑞 𝜃 = –

𝑀 (𝐼,𝑉)

In steady state 𝑧 = 𝑧ҧ

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The Model Government policy • Government cannot borrow nor lend • Balances budget each period. • Finances unemployment benefit, b, through a lump sum tax, 𝜏 on all workers (unemployed and employed). • Thus, the budget constraint is 𝜏 = 𝑢benefit ∙ 𝑏 = 𝑢1 1 − 𝑑 𝑏 • The government generally has two instruments in the UI program –

The benefit level, b

–

The expected duration, 𝑑 (d is benefit exhaustion probability)

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The Model Timing 1. The economy enters period t with state (𝑧, 𝑢, 𝑢1 ) – – –

Unemployment rate u Benefit-eligible unemployed workers 𝑢1 Productivity shock, z, is then realized

2. Government realization of policy 𝑏, 𝑑, 𝜏 3. Benefit-eligible unemployed workers, 𝑢benefit = 𝑢1 (1 − 𝑑), receive benefit •

4.

I.e. with probability d previously benefit-eligible workers lose benefit.

Employed workers produce, z, receive wages w. Unemployed workers produce, h, and if benefit-eligible, receive b. All workers pay 𝜏.

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The Model Timing 5. Given the aggregate state of the economy, o all unemployed workers choose search intensity o 𝑠 1 for benefit eligible workers o 𝑠 0 for workers without benefit. o Firms chooses the number of vacancies to post, at cost 𝜅 o Separation through job destruction 6.

The laws of motion of unemployed workers are

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The Model The Workers, three optimization problems • Unemployed workers with benefit – –

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Unemployed worker without benefit – –

• • •

consume: ℎ + 𝑏 − 𝜏 Search with intensity 𝑠 1 Consume ℎ − 𝜏 Search with intensity 𝑠 0

Workers find job with probability 𝑓 𝜃 𝑠, 𝑠 = 𝑠 1 , 𝑠 0 . 𝑉 𝑒 (𝑧, 𝑢, 𝑢1 ; 𝑔) is the value of an employed 𝑉 𝑢 (𝑧, 𝑢, 𝑢1 ; 𝑔) is the value of an unemployed

Optimization problem of an unemployed worker without benefit (superscript 0)

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The Model The Workers, three optimization problems Optimization problem of an unemployed worker with benefit (superscript 1)

Optimization problem of an employed worker

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The Model The Firms Matching happens trough vacancies posted by the firms. These are filled with probability 𝑞 𝜃 . The value of an unmatched firm posting a vacancy

Free entry condition implies that firms will post vacancies until 𝐽𝑢 = 0 The value of a matched firm posting a vacancy is

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The Model Wage determination • A match between a worker and a firm produces some economic rent, which is shared between the two parties. – Equilibrium wages are set each period through a Nash bargaining problem. – Outside value to a worker is unemployment without benefit. • Wage rigidity is implemented by letting wage grows by less than one-to-one with labor productivity, z. The Competitive equilibrium consists of all value functions and assumptions just mentioned.

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The Model Private-sector optimality 1. Optimal choice of search intensity, 𝑠 0 and 𝑠 1

2. From firm’s free-entry condition follows

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The Markov-Perfect Equilibrium Assumptions • Government is utilitarian planner • Maximizes the expected value of worker’s utility • Uses the instruments

•

–

Benefit level, b

–

Expected duration,

–

Tax, 𝜏

1 𝑑

Time-consistent government policies (Klein, Krusell and Ríos-Rull (2008)). –

Each government cannot choose future policies

Reminder, government policy chooses 𝜏 𝑢1 , 𝑏, 𝑑 : = 𝑢1 1 − 𝑑 𝑏 The Government Period Return Function is

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The Markov-Perfect Equilibrium Definition (2, page 14): ”The Markov-Perfect equilibrium consists of a value function G, government policy rules, and private decision rules, such that for all aggregate productivity, z, and unemployment states, 𝑏, 𝑑, 𝑠 0 , 𝑠 1 , 𝜃, 𝑢′ , 𝑢1′ solve:” 1 , 𝑏, 𝑑, 𝑠 0 , 𝑠 1 + 𝛽𝐸𝐺(𝑧 ′ , 𝑢 ′ , 𝑢 1′ ) max 𝑅 𝑧, 𝑢, 𝑢 0 1 ′ 1′ 𝑏,𝑑,𝑠 ,𝑠 ,𝜃,𝑢 ,𝑢

The Government value function can be shown to satisfy the functional equation

Where

′

= (𝑧 ′ , 𝑢′ , 𝑢1 ), and Γ(∙) denotes policy parameters.

Thus, the current Markov government weighs the trade-off between current and future welfare. •

•

1

Longer expected duration, 𝑑 , increases the share of unemployed workers receiving benefit today, thus raising current welfare. Also, due to moral hazard behavior of unemployed, unemployed workers on benefit choose a lower search intensity. As a results, higher duration, reduces search intensity, leading to higher future unemployment and lower future welfare.

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The Markov-Perfect Equilibrium Furthermore, they prove that in the Markov-Perfect equilibrium, the unemployment benefit policy is characterized by 𝑅𝑏 = 0 and policy d is characterized by

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Parametrization • •

Model period length is one month. They calibrate the model to match features of the U.S. labor market between 2003.I and 2007.IV.

Assumptions on functional forms • Utility:

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Search cost function:

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Matching function: –

Job-finding

–

Job-filling

𝛽 = 0.991/3 , 𝜎 = 1 and search cost curvature is 𝜙 = 1

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Parametrization Externally calibrated parameters

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Parametrization Internally calibrated parameters

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Markov equilibrium policy rules

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Markov equilibrium policy rules

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Markov equilibrium policy rules

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UI Duration Extension in Recession Since the cyclical properties of the model is consistent with U.S. policy, the model can be used to study recessions. • More specifically, the December 2007 to December 2013 recession. • They specify a path for productity, to match the observed unemployment data, and the model is the calculated based on these shock.

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UI Duration Extension in Recession Given these shock processes, they calculate the Markov equilibrium strategies.

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The effect of expectation

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Conclusion •

Uses the concept of Markov-perfect equilibrium to study time-consistent UI policy.

•

Using the theoretical framework they find – Markov policy matches well the U.S. policy • 3 percentage point gab difference • 70% of this gab difference is due to private sector expectations – Longer UI durations increases welfare (0.16% of average consumption) • Provides a new argument in the debate of UI durations

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Several simplifying assumptions were made – No saving. – Government policies take effect immediately.

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