Discounting Lecture slides Bård Harstad University of Oslo 2019 Bård Harstad (University of Oslo) Discounting 2019 1 / 20
Pay C to later get B? Bård Harstad (University of Oslo) Discounting 2019 2 / 20
Public Investments with Long-term Consequences Abate, reduce emission, recycle Conserve exhaustible/renewable resources Infrastructure (windmills, roads, bridges) Technology Academic research, knowledge Bård Harstad (University of Oslo) Discounting 2019 3 / 20
The Standard Approach Rae, Jevons, Senior, Bohm-Bawerk: Multiple psychological factors Bård Harstad (University of Oslo) Discounting 2019 4 / 20
The Standard Approach Rae, Jevons, Senior, Bohm-Bawerk: Multiple psychological factors Ramsey (1928): ∞ ∑ max v 0 = d t u t . t = 0 Bård Harstad (University of Oslo) Discounting 2019 4 / 20
The Standard Approach Rae, Jevons, Senior, Bohm-Bawerk: Multiple psychological factors Ramsey (1928): ∞ ∑ max v 0 = d t u t . t = 0 Samuelson (1937): � � t 1 d t = δ t = ≈ e − ρ t . 1 + ρ Bård Harstad (University of Oslo) Discounting 2019 4 / 20
The Standard Approach Rae, Jevons, Senior, Bohm-Bawerk: Multiple psychological factors Ramsey (1928): ∞ ∑ max v 0 = d t u t . t = 0 Samuelson (1937): � � t 1 d t = δ t = ≈ e − ρ t . 1 + ρ Koopman (1960): axiomatic foundation Bård Harstad (University of Oslo) Discounting 2019 4 / 20
The Standard Approach Rae, Jevons, Senior, Bohm-Bawerk: Multiple psychological factors Ramsey (1928): ∞ ∑ max v 0 = d t u t . t = 0 Samuelson (1937): � � t 1 d t = δ t = ≈ e − ρ t . 1 + ρ Koopman (1960): axiomatic foundation "the simplicity and elegance of this formulation was irresistible" and the criterion became "dominant... largely due to its simplicity... not as a result of empirical research demonstrating its validity" (Frederick et al, ’02: 355-6;352-3) Bård Harstad (University of Oslo) Discounting 2019 4 / 20
Continuity 1 Sensitivity 2 Non-Complementarity 3 Stationarity 4 Boundedness 5 Koopmans (1960): With 1-5, v 0 = ∑ ∞ t = 0 δ t u t . Bård Harstad (University of Oslo) Discounting 2019 5 / 20
The Value of a future dollar (in cents today) interest rate \ years: 50 100 200 r = 1 % 60 37 13 r = 4 % 13 1,8 0,03 r = 8 % 1,8 0,03 0,00001 Stern-review vs Nordhaus: debate on interest rate. Bård Harstad (University of Oslo) Discounting 2019 6 / 20
Ramsey’s social discount rate for consumption A dollar at time t has the same value as a ( t ) ≡ e − rt dollars today (time 0) if: a ( t ) u � ( c 0 ) e − ρ t u � ( c t ) ⇒ = a � ( t ) − ρ + u �� ( c t ) ∂ c t / ∂ t = u � ( c t ) c t ⇒ a ( t ) c t r = ρ + η t µ t . Bård Harstad (University of Oslo) Discounting 2019 7 / 20
Ramsey’s social discount rate for consumption A dollar at time t has the same value as a ( t ) ≡ e − rt dollars today (time 0) if: a ( t ) u � ( c 0 ) e − ρ t u � ( c t ) ⇒ = a � ( t ) − ρ + u �� ( c t ) ∂ c t / ∂ t = u � ( c t ) c t ⇒ a ( t ) c t r = ρ + η t µ t . In estimates, often η t = 2 and µ t = 0 , 03 . Bård Harstad (University of Oslo) Discounting 2019 7 / 20
Ramsey’s social discount rate for consumption A dollar at time t has the same value as a ( t ) ≡ e − rt dollars today (time 0) if: a ( t ) u � ( c 0 ) e − ρ t u � ( c t ) ⇒ = a � ( t ) − ρ + u �� ( c t ) ∂ c t / ∂ t = u � ( c t ) c t ⇒ a ( t ) c t r = ρ + η t µ t . In estimates, often η t = 2 and µ t = 0 , 03 . If ρ = 0 , 01, r = 0 , 07 = 7 %. Bård Harstad (University of Oslo) Discounting 2019 7 / 20
Ramsey’s social discount rate for consumption A dollar at time t has the same value as a ( t ) ≡ e − rt dollars today (time 0) if: a ( t ) u � ( c 0 ) e − ρ t u � ( c t ) ⇒ = a � ( t ) − ρ + u �� ( c t ) ∂ c t / ∂ t = u � ( c t ) c t ⇒ a ( t ) c t r = ρ + η t µ t . In estimates, often η t = 2 and µ t = 0 , 03 . If ρ = 0 , 01, r = 0 , 07 = 7 %. This used to be the recommendation in Norwegian public cost-benefit analysis. Bård Harstad (University of Oslo) Discounting 2019 7 / 20
Ramsey’s social discount rate for consumption A dollar at time t has the same value as a ( t ) ≡ e − rt dollars today (time 0) if: a ( t ) u � ( c 0 ) e − ρ t u � ( c t ) ⇒ = a � ( t ) − ρ + u �� ( c t ) ∂ c t / ∂ t = u � ( c t ) c t ⇒ a ( t ) c t r = ρ + η t µ t . In estimates, often η t = 2 and µ t = 0 , 03 . If ρ = 0 , 01, r = 0 , 07 = 7 %. This used to be the recommendation in Norwegian public cost-benefit analysis. Note that with CRRA (constant relative risk aversion); u ( c ) = c 1 − η / ( 1 − η ) , then u �� ( c t ) c t / u � ( c t ) = − η . Bård Harstad (University of Oslo) Discounting 2019 7 / 20
The discount rate under uncertainty � � ∑ t With growth rate µ t , c t = c 0 exp τ = 1 µ τ . Bård Harstad (University of Oslo) Discounting 2019 8 / 20
The discount rate under uncertainty � � ∑ t With growth rate µ t , c t = c 0 exp τ = 1 µ τ . � − η ∑ t � With CRRA, u � ( c t ) / u � ( c 0 ) = exp τ = 1 µ τ , so � � t ∑ a ( t ) = exp − ρ t − η µ τ . τ = 1 Bård Harstad (University of Oslo) Discounting 2019 8 / 20
The discount rate under uncertainty � � ∑ t With growth rate µ t , c t = c 0 exp τ = 1 µ τ . � − η ∑ t � With CRRA, u � ( c t ) / u � ( c 0 ) = exp τ = 1 µ τ , so � � t ∑ a ( t ) = exp − ρ t − η µ τ . τ = 1 Suppose y t ≡ ∑ t τ = 1 µ τ is uncertain and distributed as f ( y t ) . Bård Harstad (University of Oslo) Discounting 2019 8 / 20
The discount rate under uncertainty � � ∑ t With growth rate µ t , c t = c 0 exp τ = 1 µ τ . � − η ∑ t � With CRRA, u � ( c t ) / u � ( c 0 ) = exp τ = 1 µ τ , so � � t ∑ a ( t ) = exp − ρ t − η µ τ . τ = 1 Suppose y t ≡ ∑ t τ = 1 µ τ is uncertain and distributed as f ( y t ) . The expected future value of a dollar is today worth: � � � t ∑ e − ρ t − η y f ( y ) dy . a ( t ) = E exp − ρ t − η µ τ = τ = 1 Bård Harstad (University of Oslo) Discounting 2019 8 / 20
The discount rate under uncertainty - continued � υ , σ 2 � If µ τ ∼ N , iid, then � e − η y f ( y ) dy = e − ρ t − ηυ t + 1 2 η 2 σ 2 t ⇒ e − ρ t a ( t ) = − a � ( t ) a ( t ) = ρ + ηυ − 1 2 η 2 σ 2 . r = Bård Harstad (University of Oslo) Discounting 2019 9 / 20
The discount rate under uncertainty - continued � υ , σ 2 � If µ τ ∼ N , iid, then � e − η y f ( y ) dy = e − ρ t − ηυ t + 1 2 η 2 σ 2 t ⇒ e − ρ t a ( t ) = − a � ( t ) a ( t ) = ρ + ηυ − 1 2 η 2 σ 2 . r = So, large uncertainty reduces the discount rate. Bård Harstad (University of Oslo) Discounting 2019 9 / 20
The discount rate under uncertainty - continued � υ , σ 2 � If µ τ ∼ N , iid, then � e − η y f ( y ) dy = e − ρ t − ηυ t + 1 2 η 2 σ 2 t ⇒ e − ρ t a ( t ) = − a � ( t ) a ( t ) = ρ + ηυ − 1 2 η 2 σ 2 . r = So, large uncertainty reduces the discount rate. If shocks µ τ are correlated over time, then uncertainty grows. Bård Harstad (University of Oslo) Discounting 2019 9 / 20
The discount rate under uncertainty - continued � υ , σ 2 � If µ τ ∼ N , iid, then � e − η y f ( y ) dy = e − ρ t − ηυ t + 1 2 η 2 σ 2 t ⇒ e − ρ t a ( t ) = − a � ( t ) a ( t ) = ρ + ηυ − 1 2 η 2 σ 2 . r = So, large uncertainty reduces the discount rate. If shocks µ τ are correlated over time, then uncertainty grows. Then, r t becomes time-dependent and decreasing over time. Bård Harstad (University of Oslo) Discounting 2019 9 / 20
The discount rate under uncertainty - continued � υ , σ 2 � If µ τ ∼ N , iid, then � e − η y f ( y ) dy = e − ρ t − ηυ t + 1 2 η 2 σ 2 t ⇒ e − ρ t a ( t ) = − a � ( t ) a ( t ) = ρ + ηυ − 1 2 η 2 σ 2 . r = So, large uncertainty reduces the discount rate. If shocks µ τ are correlated over time, then uncertainty grows. Then, r t becomes time-dependent and decreasing over time. May well be negative. Bård Harstad (University of Oslo) Discounting 2019 9 / 20
Uncertainty/Disagreement about the discount rate Suppose r = r j with probability p j (or, for that fraction of people) Bård Harstad (University of Oslo) Discounting 2019 10 / 20
Uncertainty/Disagreement about the discount rate Suppose r = r j with probability p j (or, for that fraction of people) The certainty-equivalent discount factor at time t is = ∑ p j e − r j t ⇒ A ( t ) − A � ( t ) A ( t ) = ∑ w j ( t ) r j , where R ( t ) ≡ p j e − r j t p j w j ( t ) = ∑ i p i e − r i t = ∑ i p i e − ( r i − r j ) t . Bård Harstad (University of Oslo) Discounting 2019 10 / 20
Recommend
More recommend
