Derivation of optimal labor supply ¶ Consider a worker with quasilinear isoelastic preferences:
U ( C , h ) = C − ψ h 1 + 1 / ε 1 + 1 / ε U(C, h) = C - \frac{\psi \, h^{1+1/\varepsilon}}{1+1/\varepsilon} U ( C , h ) = C − 1 + 1/ ε ψ h 1 + 1/ ε subject to the budget constraint C = w ( 1 − τ ) h + v C = w(1-\tau)h + v C = w ( 1 − τ ) h + v w w w τ \tau τ h h h v v v
Substituting the budget constraint:
U = w ( 1 − τ ) h + v − ψ h 1 + 1 / ε 1 + 1 / ε U = w(1-\tau)h + v - \frac{\psi \, h^{1+1/\varepsilon}}{1+1/\varepsilon} U = w ( 1 − τ ) h + v − 1 + 1/ ε ψ h 1 + 1/ ε The first-order condition with respect to h h h
U h = w ( 1 − τ ) − ψ h 1 / ε = 0 U_h = w(1-\tau) - \psi \, h^{1/\varepsilon} = 0 U h = w ( 1 − τ ) − ψ h 1/ ε = 0 Solving for h h h
h ∗ ( τ ) = ( w ( 1 − τ ) ψ ) ε h^*(\tau) = \left(\frac{w(1-\tau)}{\psi}\right)^\varepsilon h ∗ ( τ ) = ( ψ w ( 1 − τ ) ) ε This confirms that labor supply is increasing in the net-of-tax wage w ( 1 − τ ) w(1-\tau) w ( 1 − τ ) ε \varepsilon ε
Setup ¶ A worker perceives τ ^ = τ + δ \hat{\tau} = \tau + \delta τ ^ = τ + δ δ \delta δ
h ( τ ^ ) = ( w ( 1 − τ ^ ) ψ ) ε h(\hat{\tau}) = \left(\frac{w(1-\hat{\tau})}{\psi}\right)^\varepsilon h ( τ ^ ) = ( ψ w ( 1 − τ ^ ) ) ε But utility is realized at the true rate τ \tau τ
U ( τ , h ( τ ^ ) ) = w ( 1 − τ ) h ( τ ^ ) + v − ψ h ( τ ^ ) 1 + 1 / ε 1 + 1 / ε U(\tau, h(\hat{\tau})) = w(1-\tau)h(\hat{\tau}) + v - \frac{\psi \, h(\hat{\tau})^{1+1/\varepsilon}}{1+1/\varepsilon} U ( τ , h ( τ ^ )) = w ( 1 − τ ) h ( τ ^ ) + v − 1 + 1/ ε ψ h ( τ ^ ) 1 + 1/ ε Second-order Taylor expansion ¶ Define DWL ( δ ) = U ( τ , h ∗ ( τ ) ) − U ( τ , h ( τ ^ ) ) \text{DWL}(\delta) = U(\tau, h^*(\tau)) - U(\tau, h(\hat{\tau})) DWL ( δ ) = U ( τ , h ∗ ( τ )) − U ( τ , h ( τ ^ )) h ∗ h^* h ∗ U ( τ , ⋅ ) U(\tau, \cdot) U ( τ , ⋅ )
DWL ( δ ) ≈ − 1 2 U h h ∣ h = h ∗ ⋅ ( Δ h ) 2 \text{DWL}(\delta) \approx -\frac{1}{2} U_{hh}\big|_{h=h^*} \cdot (\Delta h)^2 DWL ( δ ) ≈ − 2 1 U hh ∣ ∣ h = h ∗ ⋅ ( Δ h ) 2 Computing U h h U_{hh} U hh
U h h = − ψ ε h 1 / ε − 1 U_{hh} = -\frac{\psi}{\varepsilon} h^{1/\varepsilon - 1} U hh = − ε ψ h 1/ ε − 1 At the optimum, ψ ( h ∗ ) 1 / ε = w ( 1 − τ ) \psi (h^*)^{1/\varepsilon} = w(1-\tau) ψ ( h ∗ ) 1/ ε = w ( 1 − τ ) ψ = w ( 1 − τ ) ⋅ ( h ∗ ) − 1 / ε \psi = w(1-\tau) \cdot (h^*)^{-1/\varepsilon} ψ = w ( 1 − τ ) ⋅ ( h ∗ ) − 1/ ε
U h h ∣ h ∗ = − w ( 1 − τ ) ε h ∗ U_{hh}\big|_{h^*} = -\frac{w(1-\tau)}{\varepsilon \, h^*} U hh ∣ ∣ h ∗ = − ε h ∗ w ( 1 − τ ) Computing Δ h \Delta h Δ h
Δ h = h ( τ ^ ) − h ∗ ( τ ) = ( w ( 1 − τ − δ ) ψ ) ε − ( w ( 1 − τ ) ψ ) ε \Delta h = h(\hat{\tau}) - h^*(\tau) = \left(\frac{w(1-\tau-\delta)}{\psi}\right)^\varepsilon - \left(\frac{w(1-\tau)}{\psi}\right)^\varepsilon Δ h = h ( τ ^ ) − h ∗ ( τ ) = ( ψ w ( 1 − τ − δ ) ) ε − ( ψ w ( 1 − τ ) ) ε For small δ \delta δ
Δ h ≈ − ε h ∗ 1 − τ ⋅ δ \Delta h \approx -\frac{\varepsilon \, h^*}{1-\tau} \cdot \delta Δ h ≈ − 1 − τ ε h ∗ ⋅ δ Combining:
DWL ( δ ) ≈ 1 2 ⋅ w ( 1 − τ ) ε h ∗ ⋅ ( ε h ∗ 1 − τ ) 2 δ 2 = 1 2 ε w h ∗ ⋅ δ 2 1 − τ \text{DWL}(\delta) \approx \frac{1}{2} \cdot \frac{w(1-\tau)}{\varepsilon \, h^*} \cdot \left(\frac{\varepsilon \, h^*}{1-\tau}\right)^2 \delta^2 = \frac{1}{2} \varepsilon \, w \, h^* \cdot \frac{\delta^2}{1-\tau} DWL ( δ ) ≈ 2 1 ⋅ ε h ∗ w ( 1 − τ ) ⋅ ( 1 − τ ε h ∗ ) 2 δ 2 = 2 1 ε w h ∗ ⋅ 1 − τ δ 2 Expected DWL ¶ If δ ∼ N ( 0 , σ 2 ) \delta \sim N(0, \sigma^2) δ ∼ N ( 0 , σ 2 ) E [ δ 2 ] = σ 2 E[\delta^2] = \sigma^2 E [ δ 2 ] = σ 2
E [ DWL ] ≈ 1 2 ε w h ∗ ⋅ σ 2 1 − τ \boxed{E[\text{DWL}] \approx \frac{1}{2} \varepsilon \, w \, h^* \cdot \frac{\sigma^2}{1-\tau}} E [ DWL ] ≈ 2 1 ε w h ∗ ⋅ 1 − τ σ 2 Dividing by earnings w h ∗ w h^* w h ∗
E [ DWL ] earnings ≈ 1 2 ε ⋅ σ 2 1 − τ \frac{E[\text{DWL}]}{\text{earnings}} \approx \frac{1}{2} \varepsilon \cdot \frac{\sigma^2}{1-\tau} earnings E [ DWL ] ≈ 2 1 ε ⋅ 1 − τ σ 2 Comparative statics ¶ The formula E [ DWL ] / earnings = 1 2 ε σ 2 / ( 1 − τ ) E[\text{DWL}]/\text{earnings} = \frac{1}{2}\varepsilon\sigma^2/(1-\tau) E [ DWL ] / earnings = 2 1 ε σ 2 / ( 1 − τ )
Parameter Effect on DWL Intuition ε \varepsilon ε Linear, positive More elastic workers make larger errors σ \sigma σ Quadratic, positive Larger errors cause disproportionately larger losses τ \tau τ Positive via 1 / ( 1 − τ ) 1/(1-\tau) 1/ ( 1 − τ ) Higher rates amplify errors in the net-of-tax wage w h ∗ w h^* w h ∗ Linear, positive Higher earners lose more in absolute terms
Optimal tax under misperception ¶ Consider a utilitarian planner choosing τ \tau τ v = τ w ˉ h ˉ / N v = \tau \bar{w} \bar{h} / N v = τ w ˉ h ˉ / N τ \tau τ
∂ E [ DWL ] ∂ τ > 0 \frac{\partial \, E[\text{DWL}]}{\partial \tau} > 0 ∂ τ ∂ E [ DWL ] > 0 The planner internalizes that a marginal increase in τ \tau τ 1 / ( 1 − τ ) 1/(1-\tau) 1/ ( 1 − τ )
∂ τ ∗ ∂ σ < 0 \frac{\partial \tau^*}{\partial \sigma} < 0 ∂ σ ∂ τ ∗ < 0 The optimal tax rate decreases with misperception noise. This is confirmed numerically: the calibration shows τ ∗ \tau^* τ ∗ σ \sigma σ
Cobb-Douglas knife-edge result ¶ For Cobb-Douglas utility U ( L , C ) = L α C β U(L, C) = L^\alpha C^\beta U ( L , C ) = L α C β L = T − h L = T - h L = T − h v = 0 v = 0 v = 0
L ∗ = α T α + β L^* = \frac{\alpha T}{\alpha + \beta} L ∗ = α + β α T This is independent of the tax rate. The income and substitution effects of a tax change exactly cancel, so misperceiving τ \tau τ