Equilibrium Unemployment in a General Equilibrium Model 1

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1 Equilibrium Unemployment in a General Equilibrium Model 1 Application to the UK Economy Keshab Bhattarai 2 Hull University Bus. School July 19, Based on the paper for the 4th World Congress of the Game Theory Society, Istanbul, Turkey. July 21-26, Business School, University of Hull, HU6 7RX, Hull, UK. K.R.Bhattarai@hull.ac.uk K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

2 Abstract Ratio of unemployed to vacancies has risen sharply after the recession of 2008/09. How harmful is it for the long run growth of the economy? An attempt is made to develop an applied dynamic general equilibrium model with Pissarides (1985,1986, 1987) and Mortensen and Pissarides (1994, 1999) type equilibrium unemployment to assess impacts of such unemployment and accompanying tax-transfer programmes in UK. Improvements in the matching technology lowers the equilibrium unemployment and raises the long-run growth rate and life time utilities of households. Matching could be made more e cient by in uencing the relative price system by optimal set of tax and transfer instruments. Better matching techniques can make transition of job-seekers to employment more e cient; the intertemporal labour-leisure and consumption-saving decisions more e cient to have greater impacts on growth and redistribution. K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

3 Empirical Evidence -1 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

4 Empirical Evidence -2 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

5 Empirical Evidence -3 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

6 Empirical Evidence -4 Figure 3 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

7 Empirical Evidence -5 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

8 Evidence for Beveridge curve Table: Change in vacancy on change in unememployment rate in UK Coe cient Standard Error t-value t-prob Intercept u R 2 = 0.20, F (1, 128) = 32.4 (0.00), N = 130 Table: Change in vacancy on change in unememployment rate in UK Coe cient Standard Error t-value t-prob Intercept u R 2 = 0.20, F (1, 128) = 32.4 (0.00), N = 130 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

9 Empirical Evidence -6 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

10 Objectives of the papers To integrate the basic results on equilibrium unemployment in the job search model expressed in terms of the Beveridge and wage curves phenomenon contained in Pissarides (1985, 1986, 1987), Mortensen and Pissarides (1994, 1999) and Pissarides (2000) into a standard Walrasian dynamic general equilibrium model to study its implications on accumulations, e ciency and growth. To nd the level of equilibrium matches consistent to a general equilibrium model. To assess impacts of matching with various scal policy measures including VAT, income and production taxes and transfers in the system of relative prices and allocations of resources as well as on the evolution of the economy. To nd the full impacts of the equilibrium rate of unemployment on labour supply, consumption and saving behaviors of households, investment and capital accumulation behavior of rms and relative prices of commodities and factors of production in the broader economy. K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

11 Selected literature Game theory: Hicks (1939), Nash (1953) Bargaining model; Arrow and Debreu (1954),Diamond and Maskin (1979),Dri ll and Schultz (1992), Cripps M. (1993) Unemployment: Meade (1936), Blanchard, Diamond, Hall and Yellen (1989),Layard and Nickell (1990) Manning(1990), Ball and Mankiw (2002), Hall (2003), Saint-Paul (2004), Lockwood, Miller and Zhang (1998), Ljungqvist and Sargent (2007). Wage curve:beveridge (1942), Phelps, Edmund S. (1968), Blanch ower and Oswald (1994), Lockwood and Manning (1989). Estimation: Lancaster (1979), Chesher and Lancaster (1983) Equilibrium unemployment: Nickel(1982), Pissarides (1979, 1985, 1986, 1987), Mortensen and Pissarides (1970, 1994, 1999), Calvo (1979), Shapiro and Stiglitz (1984), Lindebeck and Snower (1988), Petrongolo and Pissarides (2001), Shimer (2005) General equilibrium and growth: Stone (1961), Whalley J (1977), Auerbach and Kotliko (1987),Mercenier and Philippe (1994), Dixon and Rankin (1994), Rutherford (1998), Hutton and Roucco (1999),Dixon and Rimmer (2002), Ngai and Pissarides (2007), Bhattarai (2007), Basu and Bhattarai (2011). Fiscal policy: Shoven and Whalley (1973), Whalley (1977), Whalley (1975), Blundell (2001), Böhringer, Boeters, and Feil (2005), Bhattarai and Whalley (2009), Mirrlees and editors (2010) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

12 Household Preferences: Demand subject to budget and max U h 0 = t=0 β t,h U h t Ct h, lt h (1) t=0 P i,t 1 + tc h i and time constraint: Ci,t h h = wt h t=0 1 tw h i LSt h + r t (1 tk) Kt h + Rt h (2) LS h t = L h t l h t ; L h t = L h 0e nt (3) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

13 Production and Trade max Π j,t = A i,t = Ψ t=0 (1 δ i ) PD θ i PY j,t σy 1 σy σy 1 σy j,t θ d i t=0 + δ i PE a i,j P i,t δ d i Di,t + δ m i MM σm 1 σm i,t PE i,t E i,t = t=0 σy 1 σy j,t 1 σy 1 θ m i t=0 σm σy 1 a m i,jp i,t (4) (5) PM i,t MM i,t (6) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

14 Drivers of Dynamics Y j,t = Ψ i hγ i L ρ i i,t + (1 γ i ) K ρ i i,t i 1 ρi (7) I i,t = K i,t (1 δ) K i,t 1 (8) LS h t + L h t = L h t ; L h 0 = L h t e nt (9) E t + U t = H h=1 LS h t (10) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

15 Equilibrium Unemployment of Pissarides (2000) and Mortensen and : Matchning, Search and Bargaining (Nobel Prize Laureates 2010) Matching function aggregates vacancies and unemployment with job creation as: M t = M (V t, U t ) = V γ (1 γ) t U t ; t Π M i,t = Π t M (V i,t, U i,t ) (11) i=1 i=1 M denote the number of matching of vacancies and job seekers, V is number of vacancies and U the number of unemployed, γ is the parameter between zero and one Nash-product of the bargaining game over the di erence between the earnings from work (W ) rather than in being unemployed (U) and earnings to rms from lled and vacant jobs. NP i,t = (W i,t U t ) β (J i,t V i,t ) 1 β (12) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

16 Nash Bargaining Solutions Symmetric solution of this satis es joint pro t maximisation condition for worker as: and for employers as: (W i,t U t ) = β (J i,t + W i,t V i,t U t ) (13) (J i,t V ) = (1 β) (J i,t + W i,t V i,t U) (14) Lagrangian used for solving Nash (1954) bargaining L i,t = (W i U) β (J i V i,t ) 1 β + ψ [J i,t + W i,t V i,t U] (15) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

17 Matching Model of Unemployment Parameter θ which is the ratio of vacancy to job seeking workers θ = V U. The probability lling a vacancy is given then by f (θ) and not lling it by 1 f (θ). probability of nding a job by an unemployed worker is q (θ) t and the not nding is 1 q (θ) t; job creation occurs when matching takes place between rms with vacancies and workers seeking the job. With labour force L and the unemployment rate u, the number of workers who enter unemployment is λ (1 u) L t. There is a balance between job creation, ml t = θq (θ) L t; and job destruction, λ (1 u) L t. in the steady state. The term θq (θ) measures the transition probability from unemployed to employed. Normalising L to 1 the dynamics of unemployment is explained by transition dynamics between the job destruction and job creation u = λ (1 u) θq (θ) u and in equilibrium u = λ λ+θq(θ). K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

18 Matching Model of Unemployment 1) Dynamics of unemployment depends on the rate of job destruction,λ (1 u), and the rate of job creation, θq (θ) u. and in equilibrium u = λ (1 u) θq (θ) u (16) u = λ λ + θq (θ) (17) where λ is the rate of idiosyncratic shock of job destruction and θ is the ratio of vacancy to the unemployment and q (θ) is the probability of lling a job with a suitable candidate through the matching process. K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

19 Diagram 1: Equilibrium wage and mark Wage, w w i W ( 1 β ) + βp( + θc) = z 1 p w ( r + ) = 0 ( θ ) λ q pc K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

20 Diagram 2: Beveridge and job creation curve Vacancies, v Job creation curve r + λ + βθ ( θ ) ( 1 β )( p z) pc = 0 q( θ ) q v* λ Beveridge u = curve λ + θq( θ ) u* u: equilibrium unemployment K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

21 Matching model of unemployment 2) Optimal job creation or (demand for labour curve) shows how rms balance the marginal revenue product of labour to wage and hiring and ring costs in (θ, w) space p w (r + λ) pc q (θ) = 0 (18) where p is the price of product, w the wage rate, and (r + λ) pc is the q(θ) cost of hiring and ring 3) With 0 < β < 1 the wage curve shows positive links between the reservation wage (z) the price of product p and costing of hiring (θc) w i = z (1 β) + βp (1 + θc) (19) This equation is derived below return functions for work, unemployment, vacancy and hiring in the job K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

22 Public sector G t = N i=1 It collects revenue from direct and indirect taxes as: RV t = H h=1 P i,t tc h i C h H i,t + h=1 g i,t (20) r t tkk h t + Revenue is balanced over the model horizon: t=1 G t = t=1 RV t + H Rt h h=1 H h=1! w h t tw h LS h t (21) (22) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

23 Optimal conditions for the dynamic equilibrium T N t=0i=1 Y i,t = P i,t 1 + t h ci T t=0 H C h h=1 P i,t Y i,t = C h i,t = T t=1 i,t + I i,t + E i,t + g i,t + θ i M i,t (23) T t=0 G t = T t=0 " h r t (1 t k ) K h t + R h t + w h L hi (24) r t (1 t k ) K i,t + T t=1 RV t + H Rt h h=1 H h=i! w h t L h i,t # (25) (26) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

24 Optimal conditions for the dynamic equilibrium N i=1 PE i,t E i,t N PM i,t MM i,t = F (27) i=1 u T = λ T λ T + θ T q (θ T ) (28) [Ut h0 Y j,t L h j,t C h t, l h t = Y j,t+1 L h j,t+1 I i,t = (g i + δ i ) K i,t 1 (29) = β h Ut+1 h0 C t+1 h, l t+1 h Y and j,t 1 ]. 1+wt h K j,t = Y j,t+1 K j,t r t or K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

25 Utility and Income Distribution R1 R2 R3 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38 R4

26 Drivers of Economic Growth R5 R6 R7 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38 R8

27 Consequences of tax and transfer programme R9 R10 R11 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

28 Consequences of reforms R5_1 R6_1 R7_1 R8_1 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

29 Matching and equilibrium unemployment R13 R14 R15 R16 K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

30 Conclusion-1 An attempt is made to incorporate the Pissarides (1985, 1986), Mortensen and Pissarides (1994, 1999) type equilibrium unemployment in the general equilibrium model to evaluate the impact of matching technology in the long-run growth and level of utilities and lifetime income of households in the UK economy. Dynamic interactions among heterogenous consumers and producers generates interesting results. Utility of all households increase over time as levels of output, capital stock and labour supply rise over time. Model reproduces income distribution pattern and the Gini coe cient as one would get from the analysis of the real data. K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

31 Conclusion-1 Taxes create distortions and raise prices of commodities in all sectors when taxes and transfers rise relative to the benchmark. Price index rises steadily, economy becomes more expensive, cost of production rise and output falls steadily relative to that in the benchmark economy in almost all sectors of the economy. Level of equilibrium unemployment rises but the job matching increases at a faster space in the policy reform scenario causing equilibrium unemployment to decline. Better matching technology lowers the rate of equilibrium unemployment rate as vacancies are lled more e ciently. Long run growth and redistribution are more sensitive to exibility of markets as re ected in the intertemporal elasticity of substitution between leisure and consumption and substitutability of commodities in consumption and between capital and labour in production across sectors. More e cient matching technology improves e ciency in consumption and production, enhances growth and raises K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

32 Returns from vacancy and job occupancy Return from vacancy rv = pc + q (θ) (J V ) (30) where V denotes the value of vacancy and J the expected value for occupied jobs, pc the cost of vacancy. In equilibrium V = 0 and thus J = pc q(θ). Returns from an occupied job is given by rj = p w λj, (31) where a job generates revenue p against the cost of wage rate w and loss due to the stochastic job termination λj, λ being the ratio of idiosyncratic shocks. K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

33 Returns from vacancy and job occupancy Thus the optimal condition for employment is given by equality between price of the product, wage rate and the hiring cost of the job, p w (r + λ) pc = 0. (32) q (θ) Firms take price and interest rate (r) as given in the market, parameters λ and θ are set exogenously. Return from unemployment includes reservation wage (z) and expected wage if employed ru = z + θq (θ) (W i U) (33) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

34 Return from Unemployment and Employment Return for employed worker is or (r + λ) W = w + λu or ru + θq (θ) U = z + θq (θ) W i. (34) rw = w + λ (U W ). (35) W = Putting this in unemployment equation ru + θq (θ) U = z + θq (θ) w (r + λ) + λ (r + λ) U (36) = z + θq (θ) (r + λ) U w (r + λ) + λ w (r + λ) + θq (θ) λ (r + λ) U (37) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

35 Return from Unemployment ru ru (r + λ) ru [(r + λ) θq (θ) λ (r + λ) U + θq (θ) U = z + θq (θ) w (r + λ) θq (θ) λu + θq (θ) (r + λ) U = z (r + λ) + θq (θ) w θq (θ)] = z (r + λ) + θq (θ) w (38) ru = Similarly (r + λ) W = w + λu or (r + λ) W = w + λ r z(r +λ)+θq(θ)w [(r +λ)+θq(θ)] z (r + λ) + θq (θ) w [(r + λ) + θq (θ)] = wr [(r +λ)+θq(θ)]+λ(z(r +λ)+θq(θ)w ) r [(r +λ)+θq(θ)] (39) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

36 Return from Employment and Bargaining rw = = wr (r + λ) + λ wr (r + λ) + λ (r + λ) (r + λ) ru Wage bargaining between rms and workers rj = p w λj or J = p w (r +λ) rw = w + λ (U W ) or W i = w (r +λ) + λ (r +λ) U Nash-product of the bargaining game z (r + λ) + θq (θ) w [(r + λ) + θq (θ)] (40) (W i U) β (J i V ) 1 β (41) Symmetric solution of this satis es value maximisation jointly by rms and workers (W i U) = β (J i + W i V U) with V = 0 W i (1 β) = βj i + (1 β) U K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38

37 w i = z (1 β) + βp (1 + θc) (45) K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38 Outcome of bargaining between workers and employees wi r + λ + λ p r + λ U (1 β) = β wi r + λ (1 β) w i + λu (1 β) = β (p w i ) + (1 β) (r + λ) U (1 β) w i = β (p w i ) + (1 β) ru w i = βp + (1 β) ru + (1 β) U (42) w i = βp + (1 β) fz + θq (θ) (W i U)g (43) From (W i U) = β (J i + W i V U) ; (W i U) = β 1 β J i ; (W i U) = β pc 1 β q(θ) Therefore β pc w i = βp + (1 β) (z + θq (θ)) 1 β q(θ) w i = βp + (1 β) z + θβpc (44)

38 Matching Model of Unemployment Thus wage rate includes reservation wage (z) and average hiring costs. Putting the wage curve in job creation curve p w (r + λ) pc q(θ) = 0 or p z (1 β) βp (1 + θc) (r + λ) pc q (θ) = 0 (46) βθq (θ) + r + λ (1 β) ( p z) pc = 0 (47) q (θ) This analysis is based on constant labour supply assumption though could be extended to a growing economy. Adding sectoral and structural features of the economy makes equilibrium unemployment theory even closer to the real economy as presented in this paper. K. Bhattarai (Hull University Bus. School) EU in GE with taxes July 19, / 38