are determined, the key to solving the equilibrium is determining the labor market tightness, θ . Note that in equilibrium, θ equalizes labor supply and labor demand: l s ( θ , Δ v ) = l d ( θ , w ) . (9) Hence, we determine θ by solving (9). Once θ is determined, l is determined from l = l s ( θ , Δ v ) , e from e = e s ( f ( θ ) , Δ v ) , n from n = l / ( 1 + τ ( θ )) , and c e and c u from the budget constraint (4) and Δ v = v ( c e ) - v ( c u ) . The equilibrium is represented in a ( l , θ ) plane in Panel A of Figure 1. The labor supply curve is upward sloping, and it shifts inward when UI increases. The labor demand curve may be horizontal or downward sloping, and it responds to UI when the wage responds to UI. The intersection of the labor supply and labor demand curves gives the equilibrium level of labor market tightness, employment, and unemployment. III. The Social Welfare Function We express social welfare as a function of two arguments: the generosity of UI and the labor market tightness. We compute the derivatives of the social welfare function with respect to UI and tightness. These derivatives are the key building blocks of the optimal UI formula derived 9
in Section IV. Following Chetty [2006 a ], we express the derivatives in terms of statistics that can be estimated empirically. Section VII will discuss estimates of these statistics. We begin by defining the social welfare function. Consider an equilibrium parameterized by a utility gain from work Δ v and a wage w . For a given Δ v , there is a direct relationship between w and the labor market tightness θ . This relationship is given by (9). Hence, it is equivalent to parameterize the equilibrium by Δ v and θ . Social welfare is thus a function of Δ v and θ : SW ( θ , Δ v ) = e s ( θ , Δ v ) · f ( θ ) · Δ v + v ( c u ( θ , Δ v )) - k ( e s ( θ , Δ v )) , (10) where c u ( θ , Δ v ) is the equilibrium level of consumption for unemployed workers. The con- sumption c u ( θ , Δ v ) is implicitly defined by y l s ( θ , Δ v ) 1 + τ ( θ ) = l s ( θ , Δ v ) · v - 1 ( v ( c u ( θ , Δ v ))+ Δ v )+( 1 - l s ( θ , Δ v )) · c u ( θ , Δ v ) . (11) This equation ensures that the government’s budget constraint is satisfied when all the variables take their equilibrium values. Since v ( c e ) = v ( c u )+ Δ v , the term v - 1 ( v ( c u ( θ , Δ v ))+ Δ v ) gives the consumption of employed workers. Next, we define two elasticities that measure how search effort responds to UI and labor market conditions. We use these elasticities to analyze the social welfare function. D EFINITION 1. The microelasticity of unemployment with respect to UI is ε m = Δ v 1 - l · ∂ l s ∂ Δ v θ . The microelasticity measures the percentage increase in unemployment when the utility gain from work decreases by 1%, taking into account jobseekers’ reduction in search effort but ignoring the equilibrium adjustment of labor market tightness. The microelasticity can be estimated by measuring the reduction in the job-finding probability of an unemployed worker whose unemployment benefits are increased, keeping the benefits of all other workers constant.