Chapter II: Labour Market Policy

Transcription

1 Chapter II: Labour Market Policy Section 2: Unemployment insurance Literature: Peter Fredriksson and Bertil Holmlund (2001), Optimal unemployment insurance in search equilibrium, Journal of Labor Economics 19 (2), pp Prof. Dr. Christian Holzner Page 335

2 Unemployment insurance: 1. Unemployment benefits need to be finance through taxes. This creates a budget externality (firms do not take into account that creating a vacancy reduces the budget cost and increase tax income). 2. Unemployment insurance creates like any insurance a moral hazard problem (workers have no incentive to search, if they are fully insured). The optimal unemployment insurance system needs to provide incentives for workers to search actively for a job. Research question: How does the budget constraint affect the equilibrium outcome? What is the optimal shape of the unemployment insurance scheme? Prof. Dr. Christian Holzner Page 336

3 Unemployment insurance in Germany: Unemployment insurance Workers receive 60 percent of their previous earnings (65 percent for parents), The entitlement period depends on the contribution period and on the age of a worker (6 months (age <25), 18 months (age >60)), Unemployment insurance is financed through unemployment insurance contributions by employers and employees (currently 3,0 percent of gross earnings for employers and employees). Social assistance (Hartz IV) Currently 364 Euro per month plus cost of housing and heating. Initial earnings are not taxed and social assistance is not completely withdrawn in order to increase the incentives to take up work. If a social assistance recipient rejects job offers up to three times, then they are sanctioned and loose up to 360 Euros of social assistance. Prof. Dr. Christian Holzner Page 337

4 2.1 Unemployment insurance financed by social security contributions Idea: The unemployment insurance payments are financed through contributions (taxes). Budget externality: A firm does not take into account that creating a vacancy decreases the unemployment insurance payments and increases the tax revenue. Firms create not enough vacancies compared to the social optimum. Prof. Dr. Christian Holzner Page 338

5 Framework: Like in the Mortensen-Pissarides model, except Workers finance the unemployment insurance scheme through contributions c. The net wage is given by w n = w c. (since wages are unique, we can write contributions as lump sum contributions). Government budget equation: All employed workers have to pay contributions c in order to finance the unemployment insurance benefits z for unemployed workers, i.e., u c (1 u) = zu c = z 1 u. Note, that the contributions increase with the number of unemployed workers. Prof. Dr. Christian Holzner Page 339

6 Worker s surplus: Value of being unemployed is given by rv u = z + θm (θ) (V e V u ). The value of being employed is given by rv e = w c + q (V u V e ). The surplus of becoming employed is given by S L = V e V u = w c rv u. r + q Prof. Dr. Christian Holzner Page 340

7 Firm s vacancy creation condition and firm s surplus: Free entry implies that the value of a vacancy is equal to zero, i.e., Π v = 0. The value of employing a worker (firm s match surplus) equals, S F = Π e Π v = y w r + q. Thus, firms create vacancies until the cost of recruiting a worker equals the expected discounted profit of employing a worker, i.e. h m (θ) = y w r + q. (1) Prof. Dr. Christian Holzner Page 341

8 Wage determination: Total surplus of a match: S = Π e Π v + V e V u = y c rv u r + q Nash-Bargaining outcome (surplus splitting rule): Π e Π v = (1 γ) S and V e V u = γs Wage equation: w = γy + (1 γ) rv u + (1 γ) c. Prof. Dr. Christian Holzner Page 342

9 Gross wage curves: The gross wage can be obtained by substituting the value of being unemployed, rv u = z + and the unemployment insurance contributions, γ 1 γ hθ. u c = z 1 u = z q θm (θ) into the gross wage equation, i.e. [ w = γy + (1 γ) z + γ ] 1 γ hθ + (1 γ) z ( = (1 γ) z 1 + q ) + γ [y + hθ]. θm (θ) q θm (θ) Prof. Dr. Christian Holzner Page 343

10 Slope and intuition for the gross wage curve: The gross wage is a non-linear function of the market tightness, i.e., ) w [θm (θ)] = (1 γ) z ( q θ [θm (θ)] 2 + γh. Intuition: 1. The first term is negative, because a higher the market tightness decreases the unemployment rate. This reduces the cost of financing the unemployment insurance system and therefore reduces the contribution c. The gross wage (wage cost to the employer) decreases by (1 γ) z, i.e., decreases the wage according to the fraction (1 γ) (firm s bargaining power) of unemployment insurance payments z. 2. The second term is positive, because a higher market tightness increases the worker s outside option (value of being unemployed). This increases the wage that the worker gets. Prof. Dr. Christian Holzner Page 344

11 Net wage curve: w n = z ( ) q (1 γ) γ θm (θ) + γ [y + hθ] The net wage increases with the market tightness for two reasons: 1. The first term is positive, because a higher the market tightness decreases the unemployment rate. This reduces the cost of financing the unemployment insurance system and therefore reduces the contribution c. The gross wage (wage cost to the employer) increases by γz. 2. The second term is positive, because a higher market tightness increases the worker s outside option (value of being unemployed). This increases the wage that the worker gets. Prof. Dr. Christian Holzner Page 345

12 Figure 2.1: Gross and net wage curves Prof. Dr. Christian Holzner Page 346

13 Gross wage and labour market tightness in equilibrium: Job creation curve: w = y h m (θ) (r + q) Gross wage curve: w = (1 γ) z ( 1 + q ) θm (θ) + γ [y + hθ] The non-monotonicity of the gross wage curve implies that multiple equilibria can exists. Prof. Dr. Christian Holzner Page 347

14 Figure 2.2: Multiple equilibria (in market tightness and gross wages) due to the budget externality Prof. Dr. Christian Holzner Page 348

15 Figure 2.3: Multiple equilibria (unemployment rates) due to the budget externality Prof. Dr. Christian Holzner Page 349

16 Intuition for the existence of multiple equilibria: The reason for the existence of multiple equilibria is in the budget externality, i.e., the fact that firms do not consider the effect vacancy creation has on the number of unemployed and employed workers and therefore on tax revenues and the cost of financing unemployment insurance. If firms were to take the budget externality into account, they would create more vacancies in order to reduce the cost of the unemployment insurance system. Prof. Dr. Christian Holzner Page 350

17 A good and a bad equilibrium exists: 1. In the good equilibrium firms create many vacancies. This decreases unemployment insurance payments and therefore leads to lower contribution rates. The reduced contributions increase the surplus of a match, lower the gross wage and therefore increase the profit of firms. The high profits triggers more vacancy creation such that a good equilibrium can exist. 2. In the bad equilibrium firms create few vacancies. This increases unemployment insurance payments and therefore leads to higher contribution rates. The high contributions decrease the surplus of a match, increases the gross wage and therefore decrease the profit of firms. The low profits prevents firms from creating more vacancy creation such that a bad equilibrium prevails. Prof. Dr. Christian Holzner Page 351

18 Deriving the social optimum: The social planner maximizes aggregate welfare, i.e. max θ 0 [(y c) (1 u) + zu hθu]e rt dt subject to the budget constraint zu = c (1 u) and the constraint implied by matching frictions, i.e. u = q (1 u) θm (θ) u Prof. Dr. Christian Holzner Page 352

19 Hamiltonian (after substituting the budget constraint): H = [y (1 u) hθu]e rt + µ [q (1 u) θm (θ) u] FOC: [ ] H θ = 0 he rt = µm (θ) 1 + θm (θ) m (θ) (2) Transversality condition: H u = µ [ y hθ]e rt µ [q + θm (θ)] = µ (3) lim µ u = 0 t Prof. Dr. Christian Holzner Page 353

20 Differentiating equation (2) with respect to time t implies: µ = rµ Substituting µ using equations (2) and (3) implies the following condition for the optimal labour market tightness θ, i.e., or h m (θ) h m (θ) = = [y + hθ] 1 η (θ) r + q + θm (θ) (1 η (θ)) y r + q + η (θ) θm (θ) where η (θ) equals the elasticity of the matching function with respect to the unemployment rate u, i.e., η (θ) = θm (θ) m (θ) Prof. Dr. Christian Holzner Page 354

21 Comparing the social planner s solution h m (θ) = (1 η (θ)) y r + q + η (θ) θm (θ) with the decentralized market solution [ ( (1 γ) y z 1 + q )] h m (θ) = θm (θ) r + q + γθm (θ) implies that the government can implement the social planner s solution by choosing unemployment benefits proportional to productivity, i.e., z = by, such that the social planner s solution is identical to the decentralized market solution, i.e., [ b = [1 u] 1 1 η (θ) ] r + q + γθm (θ), 1 γ r + q + η (θ) θm (θ) which is only feasible, if η (θ) > γ. Prof. Dr. Christian Holzner Page 355

22 2.2 Unemployment insurance system and moral hazard Idea: Unemployment insurance is intended to insure risk averse worker against the risk of being laid off. Full insurance, i.e., unemployment benefits equal to the net wage, would imply that no worker is willing to look for a job. Thus, only partial insurance is possible. What is the optimal shape of the unemployment insurance scheme? Prof. Dr. Christian Holzner Page 356

23 Framework: Consider a two-tier unemployment insurance system. In the first tier short-term unemployed receive unemployment insurance payments b 1 w n, which are proportional to net income. In the second tier long-term unemployed receive unemployment insurance payments b 2 w n, where b 2 b 1. Short term unemployed workers become long-term unemployed workers at rate λ. Using the framework by Fredriksson and Holmlund (2001) we want to show that it is optimal to have a two-tier system, i.e., b 2 < b 1, instead of a uniform benefit system with b 2 = b 1. The number of unemployed workers in the first and second tier are denoted by u 1 and u 2. The number of employed workers is given by l = 1 u 1 u 2. Prof. Dr. Christian Holzner Page 357

24 Labor market flows: Prof. Dr. Christian Holzner Page 358

25 Labor market flows: The number of employed workers is given by equating the inflows (s 1 θm (θ) u 1 from short-term unemployment and s 2 θm (θ) u 2 from long-term unemployment) with the outflow ql, i.e., s 1 θm (θ) u 1 + s 2 θm (θ) u 2 = ql. (4) The number of short-term unemployed is similarly given by ql = λu 1 + s 1 θm (θ) u 1. (5) The number of long-term unemployed is given by λu 1 = s 2 θm (θ) u 2. (6) Prof. Dr. Christian Holzner Page 359

26 The Beveridge curve: Using the equations (5) and (6) to substitute u 1 and u 2 implies, [ ] u θm (θ) s 1 u 1 u 1 +u 2 + s 2 2 u 1 +u 2 θm (θ) (s 1 + λ) s 2 λ+s l = [ ] = 2 u q + θm (θ) s 1 u 1 u 1 +u 2 + s 2 q + θm (θ) (s 1 + λ) s 2 2 λ+s u 1 +u 2 2 The respective unemployment rate (Beveridge curve) is given by u 1 l = q q + θm (θ) (s 1 + λ) s. 2 λ+s 2 Note that the unemployment rate falls, if short- and long-term unemployed workers search more, i.e., u/ s 1 < 0 and u/ s 2 < 0. Prof. Dr. Christian Holzner Page 360

27 Framework (continued): To model the moral hazard problem associated with unemployment insurance, the Mortensen-Pissarides model needs to be adjusted to allow for risk averse workers and an endogenous search intensity, i.e., Workers are risk averse, i.e., their instantaneous utility is given by a concave utility function u (x) = lnx. The matching probability of an unemployed worker is given by her individual search intensity s i and aggregate matching probability, i.e., where the average search intensity s i θm (θ), s = s 1u 1 + s 2 u 2 u 1 + u 2 enters the market tightness, i.e., θ = v/su. Individuals choose their search intensity according to a convex cost function c (s i ) = ln (1 s i ). Prof. Dr. Christian Holzner Page 361

28 Bellman equations for workers: Unemployed workers chose their search intensity s i in order to maximize their life-time utility Value of being short-term unemployed, rv u,1 = max s 1 [lnb 1 w n (1 s 1 ) + s 1 θm (θ) (V e V u,1 ) + λ (V u,2 V u,1 )] Value of being long-term unemployed, rv u,2 = max s 2 [lnb 2 w n (1 s 2 ) + s 2 θm (θ) (V e V u,2 )] Value of being employed, rv e = lnw n + q (V u,1 V e ). Assumption: Employed workers earn the same wage regardless their previous labor market status (short- or long-term unemployed). Prof. Dr. Christian Holzner Page 362

29 Search intensity of short- and long-term unemployed workers: Worker trade off the marginal cost of search 1/ [1 s i ] with the additional expected gains from searching θm (θ) [V e V u,i ], i.e., 1 1 s i = θm (θ) [V e V u,i ] for i {1, 2}. (7) If workers would be fully insured against layoffs, i.e., V e = V u,i, unemployed workers would not search for a job. = The government can only partially insure unemployed workers, i.e., b i < 1. Prof. Dr. Christian Holzner Page 363

30 Match surplus: Short-term unemployed, V e V u,1 = (r + s 2θm (θ)) (ln w n lnb 1 w n (1 s 1 )) + λ (lnw n lnb 2 w n (1 s 2 )) (r + s 2 θm (θ)) (r + q + s 1 θm (θ)) + λ (r + q + s 2 θm (θ)) (8) Long-term unemployed, V e V u,2 = (r + λ + s 1 θm (θ)) (ln w n lnb 2 w n (1 s 2 )) (r + s 2 θm (θ)) (r + q + s 1 θm (θ)) + λ (r + q + s 2 θm (θ)) q (ln b 1 w n (1 s 1 ) lnb 2 w n (1 s 2 )) + (r + s 2 θm (θ)) (r + q + s 1 θm (θ)) + λ (r + q + s 2 θm (θ)) (9) Workers take the net wage w n and the market tightness θ as given, then they decide on the search intensity. Prof. Dr. Christian Holzner Page 364

31 Comparative statics for unemployment benefits: Given the match surplus for short- and long-term unemployed one can show that the search intensity of short-term unemployed increases, if unemployment benefits are reduced, i.e., V e V u,1 b 1 < 0 = ds 1 db 1 < 0 and V e V u,1 b 2 < 0 = ds 1 db 2 < 0. the search intensity of long-term unemployed increases, if unemployment benefits b 2 are reduced, i.e., V e V u,2 < 0 = ds 2 < 0 and V e V u,2 > 0 = ds 2 > 0. b 2 db 2 b 1 db 1 the search intensity of long-term unemployed decreases, if unemployment benefits b 2 are reduced. This last effect is called entitlement effect. If a long-term unemployed workers become employed, they are again entitled to unemployment benefits b 1, if they become unemployed again. This additional expected income increases the gains from searching. Prof. Dr. Christian Holzner Page 365

32 Comparative static for entitlement period: Given the match surplus for short- and long-term unemployed one can show that the search intensity of short-term unemployed decreases, if the expected entitlement period (1/λ) increases, since the value of being short-term unemployed increases, i.e., V e V u,1 [1/λ] < 0 = ds 1 d [1/λ] < 0. the search intensity of long-term unemployed increases, if the expected entitlement period (1/λ) increases, since the entitlement effect increase the value of being employed, i.e., V e V u,2 [1/λ] > 0 = ds 2 d [1/λ] > 0. Prof. Dr. Christian Holzner Page 366

33 Firm s vacancy creation condition: Free entry implies that the value of a vacancy is equal to zero, i.e., Π v = 0. Thus, firms create vacancies until the cost of recruiting a worker equals the expected discounted profit of employing a worker, i.e., h m (θ) = y w r + q. (10) Prof. Dr. Christian Holzner Page 367

34 Budget equation: Employed workers pay taxes in order to finance short- and long-term unemployment benefits. The net wage is given by w n = w/ (1 + τ), which turns out to be a convenient way to write the tax. The budget equation can then be written in the following way, i.e., (w w n ) l = b 1 w n u 1 + b 2 w n u 2 τ = b 1u 1 + b 2 u 2 1 (u 1 + u 2 ) (11) The proportional unemployment benefits b 1 w n and b 2 w n guarantee that the tax is independent of the level of the wage. Prof. Dr. Christian Holzner Page 368

35 Wage bargaining: Assumption: All unemployed workers have the same outside option while bargaining, i.e., it is assumed that they are entitled to short-term unemployed benefits, if workers and firms cannot agree on a wage. w = arg max (V e V u,1 ) γ (Π e Π v ) (1 γ) FOC: γ V e/ w + (1 γ) Π e/ w = 0 V e V u,1 Π e Π v where due to the log-utility function,. V e w = τ r + q w τ and Π e w = 1 r + q. Prof. Dr. Christian Holzner Page 369

36 Wage equation: The bargaining outcome implies the following gross wage, i.e., γy w = (1 γ) [V e V u,1 ] (r + q) + γ (12) The gross wage is independent of the tax rate τ for the following two reasons: 1. The log-utility function, i.e., lnw n, ensures that V e / w is independent of the tax. 2. The proportional unemployment benefit payments b 1 w n and b 2 w n (together with log-utility) ensure that the surplus from becoming employed, i.e., V e V u,1, is independent of the net wage (see equation 8). Intuition: The gross wage w increases with the market tightness θ, since the value of being unemployed increases with θ, which increases the value of being unemployed and thus decreases the worker s surplus of becoming employed, i.e., (V e V u,1 ) / θ < 0. Prof. Dr. Christian Holzner Page 370

37 Comparative statics of the gross wage curve: Unemployment benefits: The surplus of becoming employed decreases, if unemployment benefits b 1 and b 2 increase, i.e., [V e V u,1 ]/ b i < 0. This implies that the gross wage increases with unemployment benefits. The intuition is the same as in the original MP, i.e., an increase in unemployment benefits increases the value of being unemployed, which increases the worker s outside option during bargaining. Entitlement duration: An increase in the entitlement period for short-term unemployment benefits, i.e., an increase in 1/λ, leads to a higher value of being short-term unemployed V u,1. This increases the worker s outside option and therefore the gross wage paid to workers. Prof. Dr. Christian Holzner Page 371

38 Figure 2.4: Comparative statics of the gross wage curve Prof. Dr. Christian Holzner Page 372

39 Equilibrium: Part 1: The vacancy creation curve (10), the wage curve (12) and the search intensity condition for short-term unemployed (7) determine the market tightness θ, the gross wage w and the search intensity for short-term unemployed s 1 in equilibrium. Part 2: Given {θ, w, s 1 } the search intensity condition for long-term unemployed (7) determines s 2, the market tightness θ and the search intensities s 1 and s 2 determine the unemployment rates u 1,u 2 and u, which in turn determine the tax rate τ and the net wage w n in equilibrium. Prof. Dr. Christian Holzner Page 373

40 Equilibrium market tightness: The equilibrium market tightness can be obtained by substituting the wage curve (12) into the vacancy creation curve (10) h m (θ) = y ( ) γ 1. (13) r + q (1 γ) (r + q) (V e V u,1 ) + γ Comparative statics: Using the implicit function theorem implies that the market tightness decreases, if unemployment benefits increase, i.e., θ b i = dθ db i = RHS [V e V u,1 ] [V e V u,1 ] b i RHS [V e V u,1 ] [V e V u,1 ] θ + hm (θ) m(θ) 2 < 0, because the direct effect of an increase in the gross wage associated with higher benefits, which decreases profits, dominates the indirect effect of a lower market tightness, which reduces wages and profits. Prof. Dr. Christian Holzner Page 374

41 Comparative statics: An increase in the entitlement period 1/λ increases the value of of being shortterm unemployed and implies [V e V u,1 ]/ [1/λ] < 0. This associated higher outside option while bargaining leads to a higher gross wage. The higher wage in turn reduces profits and vacancy creation, i.e., dθ d [1/λ] = RHS [V e V u,1 ] [V e V u,1 ] [1/λ] RHS [V e V u,1 ] [V e V u,1 ] θ + hm (θ) m(θ) 2 < 0. If unemployment benefits are equal, i.e., b 1 = b 2, which also implies that workers search with equal intensity, i.e., s 1 = s 2, then the implicit function theorem implies, due to θ b 1 θ = r + s 2θm (θ) b 2 λ (14) V e V u,1 = (r + s 2θm (θ)) ( lnb 1 (1 s 1 )) + λ ( lnb 2 (1 s 2 )) (r + s 2 θm (θ)) (r + q + s 1 θm (θ)) + λ (r + q + s 2 θm (θ)). (15) Prof. Dr. Christian Holzner Page 375

42 Figure 2.5: Equilibrium comparative statics: market tightness and gross wage Prof. Dr. Christian Holzner Page 376

43 Equilibrium search intensity for short-term unemployed: Using the wage equation (12) to substitute [V e V u,1 ] and the vacancy creation condition (10) to substitute the gross wage implies the following equation for the search intensity of short-term unemployed, i.e., 1 1 s 1 = θm (θ) [ γ 1 γ h ym (θ) h (r + q) ] (16) This search intensity curve describes an increasing relationship between the market tightness θ and the search intensity of a short-term unemployed s 1, i.e., ds 1 dθ > 0. Prof. Dr. Christian Holzner Page 377

44 Intuition: A higher market tightness increases the matching probability of a worker, i.e., θm (θ), and therefore the return to search. Comparative statics: The search intensity s 1 is independent of unemployment benefits b 1 and b 2 and the entitlement period 1/λ, i.e., s 1 b 1 = 0, s 1 b 2 = 0 and s 1 [1/λ] = 0. only the market tightness influences the search intensity. Thus, the search intensity is only indirectly influences by unemployment benefits b 1 and b 2 and the entitlement period 1/λ. Prof. Dr. Christian Holzner Page 378

45 Equilibrium search intensity for long-term unemployed: The search intensity of long-term unemployed workers is given by 1 = θm (θ) lnb 1 (1 s 1 ) lnb 2 (1 s 2 ) 1 s 2 r + λ + s 2 θm (θ) + r + λ + s 1θm (θ) 1 r + λ + s 2 θm (θ) 1 s 1 s 2 increases with b 1 and with the entitlement period 1/λ due to the entitlement effect (the value of becoming employed increases) and decreases with b 2, since the value of being long-term unemployed rv u,2 increases, i.e., s 2 b 1 > 0, s 2 b 2 < 0 and s 2 [1/λ] > 0. If we start with equal unemployment benefits, i.e., b 1 = b 2, then all workers search with the same intensity, i.e., s 1 = s 2. In such a situation, a marginal increase in b 1 and a marginal decrease in b 2 has the same effect on the search intensity, i.e., s 2 b 1 = s 2 b 2. (17) Prof. Dr. Christian Holzner Page 379

46 Equilibrium budget: τ = b 1u 1 + b 2 u 2 1 (u 1 + u 2 ) The tax rate τ decreases, if the unemployment rate u i decreases, i.e., τ/ u i < 0. In the following welfare analysis we will consider a marginal decrease in b 2 and a marginal increase in b 1 starting from the situation where b 1 = b 2, such that the tax rate stays constant, i.e., db 2 = u 1 = s 2θm (θ), (18) db 1 u 2 λ where the last equality follows from the definition of u 2 in equation (21) below. Prof. Dr. Christian Holzner Page 380

47 Equilibrium unemployment: The total unemployment rate is given by u 1 l = q q + θm (θ) s 2 s 1 + λ s 2 + λ. (19) The unemployment rates for short- and long-term unemployed is given by u 1 = u 2 = q λ + s 1 θm (θ) l (20) λ s 2 θm (θ) u 1 (21) Prof. Dr. Christian Holzner Page 381

48 Employment effects of a budget neutral change in benefits: A marginal increase in short-term unemployment benefits b 1 and a simultaneous, budget neutral decrease in long-term unemployment benefits b 2 has the following effects on employment, i.e., dl db 1 = ( l θ + l s s 1 1 θ + s 2 l s 2 θ +l s 1 ( s1 b 1 + s 1 b 2 db 2 db 1 ) )( θ b 1 + θ db 2 b 2 db 1 ) + l s 2 ( s2 b 1 + s 2 b 2 db 2 db 1 ). Using equation (14) to substitute θ b 2 and equation (18) to substitute db 2 /db 1 as well as noting that s 1 / b i = 0 and s 2 / b 1 = s 2 / b 2 implies, ( ) dl = l θ + l s 1 s db 1 1 θ + s 2 r l s 2 θ r + s 2 θm (θ) θ b 1 + l s λ + s 2 θm (θ) s 2 2. (22) λ b 1 Prof. Dr. Christian Holzner Page 382

49 Intuition for the employment effect: 1. The first term is associated with a change in the market tightness. (a) A higher market tightness increases the number of employed workers, i.e., l θ > 0. (b) A higher market tightness also increases the probability to find a job and therefore increases the search intensity, i.e., l s i [ s 1 / θ] > 0. (c) An increase in unemployment benefits, however, reduces the market tightness, i.e., θ b 1 < 0, since the increased outside option of workers increases the wage and reduces profits. Thus, an increase in unemployment benefits b 1 and the associated decrease in the market tightness has a negative impact on employment. 2. The second term denotes the effect of an increase in short-term unemployment benefits on the search intensity of long-term unemployed. Due to the entitlement effect, i.e., s 2 / b 1 > 0, long-term unemployed increase their search intensity and therefore enter employment at a higher rate, i.e., l s 2 > 0. Prof. Dr. Christian Holzner Page 383

50 Welfare analysis: In the following analysis, we follow Fredriksson and Holmlund (2001) and concentrate on a budget neutral increase in short-term unemployment benefits and associated decrease in long-term unemployment benefits starting from a uniform unemployment benefit level b 1 = b 2. We also focus on the special case where the interest rate is equal to zero, i.e., r = 0. This implies that we do not consider the negative impact of this policy change on employment, i.e., the first term in equation (22) is set to zero. Using simulations Fredriksson and Holmlund (2001) show that the second term dominates the first term for realistic parameter values. The assumption r = 0 is identical to assuming that the market tightness (and the wage) does not change according to the budget neutral policy change, i.e., θ b 1 + θ b 2 db 2 db 1 = 0 θ b 1 θ b 2 = db 2 db 1. (23) Prof. Dr. Christian Holzner Page 384

51 Welfare function: The government maximizes the life-time utility of all agents in the economy, i.e., Ω = max b 1,b 2 [lrv e + u 1 rv u,1 + u 2 rv u,2 + lrπ e + vrπ v ]. For r = 0, this is equivalent of maximizing the sum of instantaneous utilities for employed and unemployed workers, i.e., Ω = max b 1,b 2 [l lnw n + u 1 lnb 1 w n (1 s 1 ) + u 2 lnb 2 w n (1 s 2 )] = max b 1,b 2 [lnw n + u 1 lnb 1 (1 s 1 ) + u 2 lnb 2 (1 s 2 )], (24) where the government has to take into account that net wages are given by w n = w/ (1 + τ). Prof. Dr. Christian Holzner Page 385

52 The budget externality: Using welfare function (24) one can show that unemployed workers do not search enough, since they do not take into account the positive effect that their search intensity has on the reduction of the tax rate for all other agents, i.e., Ω s i = lnwn 1 u i + u 1 lnb 1 (1 s 1 ) + u 2 lnb 2 (1 s 2 ) b1 =b 2 s i 1 s i s i s i = lnwn 1 u i + u i lnb i (1 s i ) s i 1 s i s i since b 1 = b 2 and s 1 = s 2 implies we get V e V u,i = lnb i (1 s i ) q + s i θm (θ) Ω s i = 1 b1 =b τ u θm (θ) q and = s i [q + s i θm (θ)] 2 ( ) τ 1 u i θm (θ) [V e V u,i ]. s i 1 s i Prof. Dr. Christian Holzner Page 386

53 The optimal search intensity of unemployed workers, i.e., implies 1 1 s i = θm (θ) [V e V u,i ] Ω s i = 1 b1 =b τ τ s i. Using the budget equation (11) implies that a higher search intensity reduces unemployment and hence, the tax rate, i.e., τ/ s i < 0. Thus,welfare increases, if the search intensity of unemployed workers rises, i.e., Ω s i > 0. b1 =b 2 Prof. Dr. Christian Holzner Page 387

54 First Order Conditions: We start at b 1 = b 2. Since unemployment benefits are uniform, the entitlement period 1/λ does not influence the optimal outcome. Thus, the results are valid for any entitlement period. The government chooses the long-term unemployment benefits such that unemployed workers are best insured, i.e., it trades off, dω db 2 = Ω + Ω s 1 + Ω s 2 + dω b1 =b 2 b 2 s 1 b 2 s 2 b 2 dθ θ b 2 = 0. Prof. Dr. Christian Holzner Page 388

55 The government trades off, - the direct income effect of b 2 (first term), which is positive, i.e., Ω/ b 2 > 0, - the effect that long-term unemployment benefits have on the search intensity of short-term unemployed (second term), which we know is zero, since s 1 / b 2 = 0 as explained on page 379, - the effect that long-term unemployment benefits have on the search intensity of long-term unemployed (third term). This effect is negative, since the gain from searching decreases, i.e., s 2 / b 2 < 0, as explained on page 402. A higher search intensity increases welfare, i.e., Ω/ s 2 > 0, since worker do not take their positive effect on the budget into account (budget externality), - the effect that long-term unemployment benefits decrease the market tightness, i.e., θ b 2 < 0, which increases or decreases welfare depending on Ω/ θ 0. Prof. Dr. Christian Holzner Page 389

56 The first order condition for short-term unemployment benefits at b 1 = b 2 is given by dω db 1 = Ω + Ω s 1 + Ω s 2 + dω b1 =b 2 b 1 s 1 b 1 s 2 b 1 dθ θ b 1. Using the fact that short-term unemployed s search intensity is not affected by unemployment benefits, i.e., s 1 / b 1 = 0, and using the FOC for b 2 to substitute dω/dθ and equation (23) to substitute θ b 1 /θ b 2 implies dω db 1 = Ω + Ω s 2 + b ( 2 Ω + Ω ) s 2. b1 =b 2 b 1 s 2 b 1 b 1 b 2 b 2 b 2 This can be further simplified by noting that the welfare function (24) and budget neutrality, i.e., b 2 / b 1 = u 1/ u 2, implies, Ω b 1 + Ω b 2 b 2 b 1 = u 1 1 b 1 + u 2 1 b 2 b 2 b 1 = 0. (25) Prof. Dr. Christian Holzner Page 390

57 Optimal unemployment insurance system: Using equation (25) to simplify the FOC implies dω db 1 = Ω s 2 + Ω s 2 b 2 > 0. b1 =b 2 s 2 b 1 s 2 b 2 b 1 Short-term unemployment benefits should be increased above the uniform benefit, since long-term unemployed workers do not search enough, because they do not take the budget externality into account, i.e., Ω/ s 2 > 0, - the search intensity of long-term unemployed workers increases with short-term unemployment benefits due to the entitlement effect, i.e., s 2 / b 1 > 0, - the search intensity of long-term unemployed workers decreases with long-term unemployment benefits ( s 2 / b 2 < 0), which have to fall according to the budget neutrality, i.e., b 2 / b 1 = u 1/ u 2. Thus, a two-tier unemployment benefit system is optimal in order to increase the search intensity of long-term unemployed workers. Prof. Dr. Christian Holzner Page 391