1. Fiscal policies in the nonstohastic growth models
Ljungqvist-Sargent, Chapter 11
Presented by Pavlo Iavorskyi

2. Government. 𝑔 𝑡 , 𝜏 𝑐𝑡 , 𝜏 𝑖𝑡 , 𝜏 𝑘𝑡 , 𝜏 𝑛𝑡 , 𝜏 ℎ𝑡 , 𝑡≥0
𝑔 𝑡 , 𝜏 𝑐𝑡 , 𝜏 𝑖𝑡 , 𝜏 𝑘𝑡 , 𝜏 𝑛𝑡 , 𝜏 ℎ𝑡 , 𝑡≥0
𝑔 𝑡 - a government spending;
𝜏 𝑐𝑡 - a consumption tax rate;
𝜏 𝑖𝑡 - an investment tax rate;
𝜏 𝑘𝑡 - an “ earning from capital” tax rate;
𝜏 𝑛𝑡 - an “ earning from labor” tax rate;
𝜏 ℎ𝑡 - a lump sum tax;

3. Preferences, technology, information.
𝑡=0 ∞ 𝛽 𝑡 𝑈 𝑐 𝑡 ,1− 𝑛 𝑡 (11.2.1) 𝑈() is strictly increasing, strictly concave and twice continuously differentiable; Technology: 𝑔 𝑡 + 𝑐 𝑡 + 𝑥 𝑡 ≤𝐹( 𝑘 𝑡 , 𝑛 𝑡 ), 𝑘 𝑡+1 = 𝑥 𝑡 +(1−𝛿) 𝑘 𝑡 𝑔 𝑡 + 𝑐 𝑡 + 𝑘 𝑡+1 ≤𝐹( 𝑘 𝑡 , 𝑛 𝑡 ) + 1−𝛿 𝑘 𝑡 (11.2.3) 𝐹( 𝑘 𝑡 , 𝑛 𝑡 ) is homogeneous function with positive and decreasing marginal products.

4. Household, Government and Firm.
𝑡=0 ∞ 𝑞 𝑡 1+ 𝑡 𝑐𝑡 𝑐 𝑡 + 𝑞 𝑡 (1− 𝑡 𝑖𝑡 ) 𝑘 𝑡+1 +(1−𝛿) 𝑘 𝑡
≤ 𝑡=0 ∞ 𝑟 𝑡 1− 𝑡 𝑘𝑡 𝑘 𝑡 + 𝑤 𝑡 1− 𝑡 𝑛𝑡 𝑛 𝑡 − 𝑞 𝑡 𝜏 ℎ𝑡 (11.2.4)
𝑡=0 ∞ 𝑞 𝑡 𝑔 𝑡 ≤ 𝑡=0 ∞ 𝜏 𝑐𝑡 𝑞 𝑡 𝑐 𝑡 − 𝜏 𝑖𝑡 𝑞 𝑡 𝑘 𝑡+1 + 1−𝛿 𝑘 𝑡 + 𝑟 𝑡 𝜏 𝑘𝑡 𝑞 𝑡 + 𝑤 𝑡 𝜏 𝑛𝑡 𝑞 𝑡 + 𝑞 𝑡 𝜏 ℎ𝑡
(11.2.5a)
𝑡=0 ∞ 𝑞 𝑡 𝐹 𝑘 𝑡 , 𝑛 𝑡 − 𝑟 𝑡 𝑘 𝑡 − 𝑤 𝑡 𝑛 𝑡 (11.2.5𝑏)
𝑞 𝑡 , 𝑟 𝑡, 𝑤 𝑡 is a price system.

5. Competitive equilibrium with distorting taxes
Definition. A competitive equilibrium with distorting taxes is a budget feasible government policy 𝑔 𝑡 , 𝜏 𝑐𝑡 , 𝜏 𝑖𝑡 , 𝜏 𝑘𝑡 , 𝜏 𝑛𝑡 , 𝜏 ℎ𝑡 that satisfy (11.2.5a); a feasible allocation 𝑐 𝑡 , 𝑥 𝑡 , 𝑛 𝑡 , 𝑘 𝑡 that given the price system 𝑞 𝑡 , 𝑟 𝑡, 𝑤 𝑡 and the government policy solves the household’s(11.2.1, ) and the firm’s (11.2.3, b) problems.

6. No arbitrage condition.
𝑡=0 ∞ 𝑞 𝑡 1+ 𝑡 𝑐𝑡 𝑐 𝑡 ≤ 𝑡=0 ∞ 𝑤 𝑡 1− 𝑡 𝑛𝑡 𝑛 𝑡 − 𝑡=0 ∞ 𝑞 𝑡 𝜏ℎ 𝑡 (11.2.6) + 𝑡=1 ∞ 𝑟 𝑡 1− 𝑡 𝑘𝑡 + 𝑞 𝑡 1− 𝑡 𝑖𝑡 1−𝛿 − 𝑞 𝑡−1 1− 𝑡 𝑖𝑡−1 𝑘 𝑡 + 𝑟 0 1− 𝑡 𝑘0 + 𝑞 0 1− 𝑡 𝑖0 1−𝛿 𝑘 0 + lim 𝑇→∞ 𝑞 𝑇 1− 𝑡 𝑖𝑇 𝑘 𝑇+1 𝑞 𝑡 1− 𝑡 𝑖𝑡 = 𝑞 𝑡+1 1− 𝑡 𝑖𝑡+1 1−𝛿 +𝑟 𝑡+1 1− 𝑡 𝑘𝑡+1 (11.2.8)

7. Household’s and Firm’s Problem.
Household’s Euler equations: 𝛽 𝑡 𝑈 1𝑡 = 𝜇 𝑞 𝑡 1+ 𝜏 𝑐𝑡 ( a) 𝛽 𝑡 𝑈 2𝑡 ≤ 𝜇 𝑤 𝑡 1− 𝜏 𝑛𝑡 ( b) Firm’s zero-profit conditions: 𝑟 𝑡 = 𝑞 𝑡 𝐹 𝑘𝑡 𝑤 𝑡 = 𝑞 𝑡 𝐹 𝑛𝑡 ( )

8. Computing equilibrium. 1.
Assumptions: 𝑈 𝑐,1−𝑛 =𝑢 𝑐 , inelastic labor supply, full employment.
𝑘 𝑡+1 =𝑓 𝑘 𝑡 + 1−𝛿 𝑘 𝑡 − 𝑔 𝑡 − 𝑐 𝑡 (11.3.1)
𝑢 ′ 𝑐 𝑡 =𝛽 𝑢 ′ 𝑐 𝑡 𝜏 𝑐𝑡 𝜏 𝑐𝑡 − 𝜏 𝑖𝑡 − 𝜏 𝑖𝑡 1−𝛿 + 1− 𝜏 𝑘𝑡 − 𝜏 𝑖𝑡 𝑓 ′ ( 𝑘 𝑡+1 ) (11.3.3)
At steady state capital level and tax rates are constant:
𝑓 ′ ( 𝑘 )=(𝜌+𝛿) 1− 𝜏 𝑖 1− 𝜏 𝑘 (11.3.7)

9. Computing equilibrium. 2.
𝑐 𝑡 =𝑓 𝑘 𝑡 + 1−𝛿 𝑘 𝑡 − 𝑔 𝑡 − 𝑘 𝑡+1 (11.3.8a) 𝑞 𝑡 = 𝛽 𝑡 𝑈 1𝑡 1+ 𝜏 𝑐𝑡 (11.3.8b) 𝑟 𝑡 𝑞 𝑡 = 𝑓 ′ ( 𝑘 𝑡 ) (11.3.8c) 𝑤 𝑡 𝑞 𝑡 = 𝑓 𝑘 𝑡 − 𝑘 𝑡 𝑓 ′ ( 𝑘 𝑡 ) (11.3.8d) 𝑅 𝑡+1 = 1+ 𝜏 𝑐𝑡 1+ 𝜏 𝑐𝑡+1 1− 𝜏 𝑖𝑡+1 1− 𝜏 𝑖𝑡 1−𝛿 + 1− 𝜏 𝑘𝑡+1 1− 𝜏 𝑖𝑡 𝑓 ′ ( 𝑘 𝑡+1 ) (11.3.8e) 𝑠 𝑡 𝑞 𝑡 = (1−𝛿)−(1− 𝜏 𝑘 ) 𝑓 ′ ( 𝑘 𝑡 ) (11.3.8f) 𝑠 𝑡 is the per unit value of the capital stock at time t measured in units of time t consumption.

10. Computing equilibrium. 3.
𝑅=(1+ρ) ( ) 𝑠 𝑞 = 1+𝜌 +(𝜌+𝛿) 𝜏 𝑖 ( )

11. Remarks.
The equilibrium allocation in the “only lump sum tax” case is identical to the one that solves planning problem in which government spending is taken as an exogenous stream that is deducted from output;
When labor is inelastic, constant 𝜏 𝑐 and 𝜏 𝑛 are not distorting;
Variations in 𝜏 𝑐 over time are distorting;
Capital taxation is distorting.

12. Experiments. 1.
Assumptions: 𝑢 𝑐 = 𝑐 1−𝛾 1−𝛾 , 𝑓 𝑘 = 𝑘 𝛼 , δ=0.2, 𝛾=2, 𝛽=0.95 Responses to foreseen permanent increase in g at t = 10.

13. Experiments. 2.
Responses to foreseen permanent increase in 𝜏 𝑐 at t = 10.

14. Experiments. 3.
Responses to foreseen permanent increase in 𝜏 𝑖 at t = 10.

15. Experiments. 4.
Responses to foreseen permanent increase in 𝜏 𝑘 at t = 10.

16. Experiments. 5.
Responses to foreseen one-time increase in g at t = 10.

17. Experiments. 6.
Responses to foreseen one-time increase in 𝜏 𝑖 at t = 10.

18. Technology Growth. 1. 𝑌 𝑡 =𝐹( 𝑘 𝑡 , 𝐴 𝑡 𝑛 𝑡 ) (11.9.1)
𝑌 𝑡 =𝐹( 𝑘 𝑡 , 𝐴 𝑡 𝑛 𝑡 ) (11.9.1)
𝐴 𝑡+1 = 𝜇 𝑡+1 𝐴 𝑡 (11.9.2)
𝑘 𝑡+1 = 𝜇 𝑡+1 −1 𝑓 𝑘 𝑡 + 1−𝛿 𝑘 𝑡 − 𝑔 𝑡 − 𝑐 𝑡 (11.9.4)
𝑢 ′ 𝑐 𝑡 𝐴 𝑡 = 𝛽 𝑢 ′ 𝑐 𝑡+1 𝐴 𝑡 𝜏 𝑐𝑡 𝜏 𝑐𝑡 − 𝜏 𝑖𝑡 − 𝜏 𝑖𝑡 1−𝛿 + 1− 𝜏 𝑘𝑡 − 𝜏 𝑖𝑡 𝑓 ′ ( 𝑘 𝑡+1 )
(11.9.5)
𝑓 ′ ( 𝑘 )= 1+𝜌 𝜇 𝛾 −(1−𝛿) 1− 𝜏 𝑖 1− 𝜏 𝑘 (11.9.8)

19. Technology Growth. 2.
𝑅=(1+ρ) 𝜇 𝛾 (11.3.9) 𝑆 𝑞 = 1− 𝜏 𝑖 (1+𝜌) 𝜇 𝛾 +(1−𝛿) 𝜏 𝑖 ( )

20. Experiments. 7.
Responses to foreseen permanent increase in productivity growth at t = 10.

21. Experiments. 8.
Responses to unforeseen permanent increase in productivity growth at t = 1.

22. Questions?