The Ramsey Discounting Formula for a Hidden-State Stochastic Growth Process / 8

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1 The Ramsey Discounting Formula for a Hidden-State Stochastic Growth Process Martin L. Weitzman May 2012 Bergen Conference Long-Term Social Discount Rates

2 What is Approach of This Paper? Increasing fuzziness of distant-future growth rates as key issue. No one knows how to model fuzzy distant-future growth rates. Worth trying di erent approaches. Here try simplest Muth-Kalman hidden-state approach. Model here is a hybrid o spring of two parents. Mother = Ramsey formula connecting growth rates to discount rates. Father = Muth-Kalman hidden-state model of stochastic growth rates. Purposely pick speci cations and functional forms to obtain simple analytical formulas. Hope that simple understandable formulas give useful general insights.

3 Basic Growth Model With a Visible State ln C t ln C t 1 = Y t (1) Y t = X t + z t, z t iidn (0, V y ) (2) X t = X t 1 + w t, w t iidn (0, V x ) (3)! ln C t ln C 0 = z τ + X 0 + w s (4) t τ=1 τ s=1 ln C t ln C 0 N (tx 0, tv y + t 3 V x /3) (5) In (5), where is the term t 3 V x /3 coming from? Why cubic in time? What does it mean? What happens if X 0 is a random variable that is not observed directly?

4 Hidden-State Growth Model (Quick Version) X t 1 N (µ t 1, V xy ) (6) X t 1! X t =) V xy! V xy + V x (7) Y t 1! Y t =) V xy + V x! V xy =) = 1 (8) V xy + V x V y V xy ( p 4Vx V y + Vx =) V xy = 2 2 µ 0 = (1 λ) s=0 λ s Y s, λ = V x σ x σ y (9) ) V y V xy + V x + V y (10) t X 0 N (t µ 0, t 2 V xy ). (11) ln C t ln C 0 N µ 0 t, V y t + V xy t 2 + V x 3 t3. (12)

5 Ramsey Hidden-State Discount Rates W = E " e ρt U(C t ) t=0 # (13) exp( r t t) = e ρt E [U 0 (C t )] U 0 (C 0 ) (14) CRRA =) U 0 (C ) = C η (15) Combine (12), (14), (15) and make use of formula for expectation of lognormal to obtain basic hidden-state formula η r t = ρ + 2 η µ 0 V y + V xy t + V x 2 3 t2. (16) Interpretation? Note linear and quadratic time decline in (16). Note that discount rate eventually becomes negative (meaning and signi cance?). Note e ect of V y >> V x =) V y >> V xy >> V x.

6 Interpreting Ramsey Hidden-State Discounting Formula by Building up Sub-Components V y > 0, V x > 0 : r t = ρ + η µ 0 η 2 2 V y + V xy t + V x 3 t2 (17) (V y = 0, V x = 0) =) r t = ρ + η µ 0 (18) η (V y > 0, V x = 0) =) r t = ρ + 2 η µ 0 2 V y (19) η (V y = 0, V x > 0) =) r t = ρ + 2 η µ 0 6 V x t 2 (20) Thought experiment: past is same but in future all uncertainty miraculously ceases (e ectively X t frozen at X 0 ). Then η r t = ρ + 2 η µ 0 2 V xy t (21) What is interpretation of equation (21)?

7 A Numerical Example Immense subjectivity. Set ρ = 0, η = 2. Set µ 0 = 2%, σ y = 3% (per year). Most brazen calibration is σ x. I calibrate σ x by requiring that the probability of a stagnant (no growth) century due to the random walk alone is one out of a million (10 6 ) =) σ x =.08%. With above parameter values, following table gives discount rate schedule t = 0 yrs 50 yrs 100 yrs 150 yrs 200 yrs 250 yrs r t = 3.8% 3.5% 2.9% 2.1% 1.2%.4% Table 1: Discount rates r t (% per year) as function of time t (years)

8 Concluding Questions and Comments What do we learn from all of this about long-term discounting? Three source types of lower discount rates under uncertainty. Two source types of time-declining discount rates. Seemingly insightful interaction between variances and time. Hint that forces causing declining discount rates may be powerful over long term. Even small amount of random walking can have eventual discounting impact over the long term. Sense that fuzziness about future growth rates is important ingredient in analysis. As yet unresolved how best to model fuzzy distant-future growth. What about caveats, limitations? More work is needed. Are alternative models better?