Why Are Savings Rate so Low and Interest Rates so High in Brazil? The Role of Unfunded Social Security and Compulsory Savings (Preliminary)

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1 Why Are Savings Rate so Low and Interest Rates so High in Brazil? The Role of Unfunded Social Security and Compulsory Savings (Preliminary) Marco Bonomo INSPER Ricardo Brito INSPER February, Marcelo R. dos Santos INSPER Abstract In this paper we examine the role of compulsory savings and unfunded social security on the macroeconomic equilibrium in an economy populated by overlapping generations individual with time-inconsistent preferences. Our model economy is informed by the Brazilian experience, where a generous social security structure with high and progressive replacement rate combined with forced savings system coexists with high historical real interest rates and low saving rates. We examine the links between those facts by simulating such model and performing counter-factual exercises. Reducing pensions replacement rates to levels comparable to US leads to substantial increase in the saving rates and reduction in interest rates. Similar results are obtained when forced savings interest rates are increased from the current below market levels. Keywords: J.E.L. codes: E, H55. Introduction In this paper we examine the role of compulsory savings and unfunded social security on the macroeconomic equilibrium in an economy populated by overlapping generations individual with time-inconsistent preferences. Our model economy is informed by the Brazilian experience, where a generous social security structure with high and progressive replacement rate combined with forced savings system coexists with high historical real interest rates and low saving rates. We examine the links between those facts by simulating such model and performing counter-factual exercises. Individuals in this economy have different abilities and face income and mortality risk and are subject to borrowing constraint. Equilibrium aggregate and distributive effects of those institutions are non-trivial. Compulsory savings earn below market returns. Those funds are channelled to investment through subsidized loans. Eventual deficits in the unfunded social security are financed by additional distortionary taxes. The progressive replacement rates have distributive effects. To further describe the social security for the economy, individuals have their wage taxed at a fixed rate and retire compulsorily at retiring age (5 years old in Brazil), when they start to receive pensions at replacement rates that are inversely related to their permanent income. This unfunded pension system could have a role of compensating undersavings resulting from time-inconsistency, and to insure savers from the death risk. Additionally, individuals have a funded system (called FGTS in Marcelo R. Santos gratefully acknowledges financial support from CNPq Proc. 337/-. All errors are our sole responsibility.

2 Brazil), which force them to save a certain fraction of their gross income (% in Brazil) which accumulates at rates below market. Those savings cannot be withdrawn before retiring ages. They constitute then another mechanism to countervail the low propensity to save of those myopic agents. We examine the joint and separate role of those two systems in the equilibrium economy, by performing counterfactual exercises. First, we compare the benchmark equilibrium with one where the the forced saving mechanism is abolished. Since in this economy forced savings help to finance subsidized loans, this reduction is compensated by an increase in the income tax rate. Voluntary savings rate increase substantially more than compensating the elimination of forced savings. Interest rates are marginally increased. In our second exercise we bring down the pension replacement rates, which are very high in Brazil, to US levels. As expected, this increases both voluntary and total saving rates. It also reduces substantially interest rates. Finally an increase in the remuneration of forced savings (another reform discussed in the Brazilian Congress) also leads to important increases in the total saving rates and reduces importantly the interest rates. Our counter-factual exercises point out that both a decrease in replacement rate and an increase in the forced savings remuneration are important reforms that could lead to substantial increase in the saving rates and reduction in the interest rates. Our results do not depend on the degree of time-inconsistency of agents, and also hold for agents with exponential utility. The plan of the paper is as follows. Section presents our model economy. The following section explains how we calibrate the model parameters in order to represent the Brazilian economy. Section displays the results of our counter-factual equilibria. The last section concludes. The Environment At each point in time, the economy is inhabited by multiple cohorts of individuals of different ages. Each cohort is comprised of a continuum of measure one of individuals who live for a finite, albeit random, number of periods.. Demography Each period, t, a new generation is born. For an individual born in period t, uncertainty regarding the time of death is captured by the fact that he or she faces a probability ψ t+ of surviving to the age t + conditional on being alive at age t. Hence, an individual born in t is alive in t + t with probability t k= ψ k. We also assume that there is T > such that ψ T + =. For most of our analysis we will focus on the steady-state allocations. Since it greatly simplifies the presentation we shall drop all time indices from aggregate variables and use t to represent age. We may map the survival probability into the time invariant age profile of the population denoted {µ t } T t=. Letting g n denote the population growth rate, the fraction of agents t years old in the population is found using the following law of motion with µ t, T t= µ t =. µ t = ψ t + g n µ t,

3 3. Households Preferences Individuals derive utility from consumption, c, and leisure, l. We use a quasi-hyperbolic preference in line with Laibson [99]. In this set-up, per period utility is discounted by, ϱβ, ϱβ, ϱβ 3,... Given that ϱ is used only to discount utility from the current period and the next and β is used to discount future utility every period, they are called shortterm and long-term discount factors, respectively. The expected life-time utility can be defined as follows: U = u(c ) + ϱe [ T t=β t ( t k= ψ k ) U t (c t, l t ) ], () where E is the expectation operator conditional on information at birth. We allow preferences over consumption-leisure bundles to vary with age by indexing the flow utility by t. More specifically, flow utility will be of the form for ρ t (, ) t, γ >, γ. U t (c t, l t ) = (c t ρt l ρt t ) γ, () γ Budget constraints Every period, individuals choose labor supply, consumption and asset accumulation to maximize their objective, (), subject to a budget constraint which we shall explain momentarily. Each person has a unit time endowment which can be directly consumed in the form of leisure, l, or used in market related activities. An individual of age t who works for l hours supplies to the market a total of l t s t e u+zt efficiency units which are paid at a rental rate w. The variable u N (, σu) is a permanent component of an individual s skills. It is realized at birth and retained throughout one s life. On the other hand, z evolves stochastically according to an AR() process, z t = ϕ z z t + ε t, with innovations ε t N(, σε). Whereas u aims at capturing the heterogeneity at birth, everyone s most relevant lottery, z is the main source of uncertainty that affects one s choices. The parameter ϕ z accommodates the empirically observed persistence of productivity shocks. s t is the age-efficiency profile, which is deterministic and intend to capture the age component of the life-cycle earnings. Labor productivity shocks are independent across agents. As a consequence, there is no uncertainty regarding the aggregate labor endowment even though there is uncertainty at the individual level. All workers in this economy pay labor income taxes (τ w, τ ss ), where the revenue from τ ss is used to finance the benefit payments to the retirees, and τ w finances overall government expenditures not related to the social security system. In addition, individuals make a compulsory contribution, τ fgts, to their FGTS account. Thus, after-tax labor income is given by: y t = ( τ w τ fgts τ ss )wlse u+z (3) At the age of 5, individuals retire and start collecting social security benefits, which we denoted by b(u). In order to capture the correlation between pensions and life-cycle earnings, we allow the benefits to depend on the permanent shock. In particular, we assume that: The reason for allowing ρ to depend on age is to better match the actual behavior of hours worked over the life-cycle. This is important in our case given the nature of the exercises we study.

4 b(u) = θ(u)y m (u) () where θ(u) the retirement replacement rate and y m (u) is the average life-cycle earnings conditioned on u. The system is progressive, so we also assume that θ (u) <. Individuals can trade a risk free asset which holdings we denote by a t. Asset holdings are subject to an exogenous lower bound. More precisely, we assume that agents are not allowed to contract debt at any age, so that the amount of assets carried over from age t to t + is such that a t+. Because no agent can hold a negative position in assets at any time, we assume without loss that asset takes the form of capital, as in Aiyagari [99]. Budget constraint for individuals aged t =,..., T r : c t + a ord,t+ = ( + r)a ord,t + y t + ɛ (5) where a ord,t+ is next period ordinary assets and ɛ is a lump sum transfer related to the involuntary bequests left by those who die before reaching age T +. At each age, individuals total wealth, a t, is simply the sum of the ordinary assets, a ord,t+, and the FGTS account balance, a fgts,t, which is accumulated through compulsory contributions. In particular, contributions to the FGTS fund accumulate according to the following law of motion: a fgts,t+ = ( + r fgts )a fgts,t + τ fgts wlse u+z () Individuals are allowed to withdraw their FGTS account balance only upon retirement. Thus, the budget constraint for individuals aged t = T r,..., T can be written as: c t + a t+ = ( + r)a t + b(u) + ɛ (7) Recursive Formulation of Households Problem Let ω = (a ord, a fgts, u, z) denote the individual state space. The choice problem of individuals aged t =,..., T r can be recursively represented as follows: V w,t (ω) = max : [ U(c, l) + ϱβψ t+ E z l,a Vw,t+ (ω ) ], () ord subject to the budget (5) and (). Let a ord = d a(ω) and l = d l (ω) denote the solution of the problem above. Thus, the value function on the right-hand side is updated as follows: where V w,t (ω) = U(c, l) + βψ t+ E z Vw,t+ (ω ) (9) c + d a (ω) = ( + r)a ord,t + ( τ w τ fgts τ ss )wd l (ω)se u+z + ɛ () Notice that, due to the time-inconsistency problem associated with the quasi-hyperbolic discounting preference, the recursive formulation is such that agents choose next period ordinary assets and hours worked applying the discount factor ϱβ, but the actual value function is evaluated with the discount factor β. The optimization problem of retired agents, t = T r,..., T, can similarly be written as follows:

5 5 subject to the budget (7) where V r,t is updated according to: V r,t (ω) = max a : [ U(c, ) + ϱβψ t+ Vr,t+ (ω ) ], () V r,t (a, u) = U(c, ) + βψ t+ Vr,t+ (a, u) () It should be stressed that we have imposed non-negativity constraints on asset holdings. We have thus taken an extreme (albeit plausible) position with regards to capital markets. Relaxing a little the assumption by allowing some exogenous limit is likely to have little effect on our conclusions, at the cost of introducing a whole new set of issues that would have to be dealt with to maintain the internal consistency of the model. Also important is the fact that we have only used individual state variables in ω. It is apparent that prices do enter the value function. Indeed, in solving the model we will need to find the equilibrium prices by explicitly taking into account how they enter the policy functions associated with () and ()..3 The Government In our economy, the government manages a pay-as-you-go social security system, wherein the pension benefits to pensioners are financed through an exogenous tax τ ss. The amount of benefit received by each retired agent was presented in the last section. The government levies proportional taxes on labor income, τ w, to finance an exogenous stream of expenditures, G. We allow τ w to adjust to ensure that government budget constraint is satisfied at equilibrium. Additionally, it also run a compulsory savings program represented by the Service Time Guarantee Fund (FGTS). Under this program, employers deposit % of the payroll in an account in the name of the employee. The interest rate on these deposits, r fgts, is determined by the government and thus is a policy parameter in our model. Finally, we assume that the government collects the accidental bequests and transfers it to all agents in the economy on a lump-sum basis.. Technology Technology is standard. The production side of the economy aggregates and the technology for producing the consumption good is summarized by a Cobb-Douglass production function with constant returns to scale, Y = BK α N α, where K is aggregate capital, N is aggregate efficient units of labor, and B is a scale parameter. Every period, the standing representative firm solves the static optimization problem { max BK α N α δk wn r( φ)k rφk }, K,N where r is rental rate of capital, w, wage. where r is the market rental rate of physical capital, r is a government subsidy to cost of capital, w is the wage rate and φ is the share of the aggregate capital that is subsided. Note that we assume that the rental rate of capital is net of depreciation costs which are born directly by the firm. We consider others treatment for the accidental bequests and the present the results as a robustness exercise.

6 The first order conditions for the firm s profit maximization problem are, w = ( α)bk α N α, (3) and r = αbkα N α δ rφ φ ().5 Recursive competitive equilibrium In all that follows we describe the recursive equilibrium in a steady state. This greatly simplifies the presentation. Moreover it dispenses with the distinction between age and time thus significantly reducing the notational burden. At each point in time, agents differ from one another with respect to age t and to state ω = (a ord, a fgts, u, z) Ω. nts of age t identified by their individual states ω, are distributed according to a probability measure λ t defined on Ω, as follows. Let (Ω, Ϝ(Ω), λ t ) be a space of probability, where Ϝ(Ω) is the Borel σ-algebra on Ω: for each η Ϝ(Ω), λ t (η) denotes the fraction of agents aged t that are in η. Given the age t distribution, λ t, Q t (ω, η) induces the age t + distribution λ t+ as follows. The function Q t (ω, η) determines the probability of an agent at age t and state ω to transit to the set η at age t +. Q t (ω, η), in turn, depends on the policy functions in (), and on the exogenous stochastic process for z. A recursive competitive equilibrium for the economy is as follows. Definition. Given the policy parameters, a recursive competitive equilibrium for the economy is a collection of value functions {V w,t (ω), V r,t (ω)}, policy functions for individual asset holdings d a,t (ω), for consumption d c,t (ω), for labor supply d lw,t(ω), prices {w, r}, age dependent but time-invariant measures of agents λ t (ω), transfers ɛ and a labor income tax τ w such that: (i) a t (ω), l w (ω) solve the dynamic problems in () and (); (ii) individual and aggregate behaviors are consistent, that is: K = N = T µ tˆ t= Ω T µ tˆ t= Ω a t (ω)dλ t l t (ω) exp(u + z t )dλ t (iii) {w, r} are such that they satisfy the optimum conditions () and (3); (iv) given the decision rules, λ t (ω) follows the law of motion: ˆ λ t+ (η) = Q t (ω, η)dλ t η Ϝ(Ω); Ω

7 7 (v) the distribution of accidental bequests is: T ɛ = µ tˆ t= Ω ( ψ t+ )d a,t (ω)dλ t (vi) τ w is such that the government s budget constraints, T G + G ss + r fgts K fgts = (τ w + τ ss ) µ tˆ t= Ω d nw,t(ω)s t (ω) exp(u + z t )dλ t + rφk are satisfied every period. 3 Calibration To carry out our quantitative analysis, we need first to find values for all the parameters of the model. We accomplish this by calibrating the model to the Brazilian economy. The population age profile {µ t } T t= depends on the population growth rate, g n, the survival probabilities, ψ t, and the maximum age, T, that an agent can live. nts enter the economy at age and may live for 7 years, T = 7, so that the real maximum age is 9 years. Data on survival probability by age are from Brazilian Institute of Geography and Statistics (IBGE). Given the survival probabilities, the population growth rate is chosen so that the age distribution in the model replicates the dependency ratio observed in the data. By setting g n =., the model generates a dependency ratio of 3%, which is close to the dependency ratio observed in the data for. To calibrate the preference parameters we proceed as follows. First, we set ϱ = and choose the discount factor β in such a way that the equilibrium of our benchmark economy implies a capital-output ratio around of., which is the value observed in the data. Then we fix the parameter γ to., from micro evidence, and choose the share of leisure in the utility function, ρ t, to match average ours for different age groups. In particular, we assume that ρ t = ρ + ρ t. To calibrate ρ, we use the average working hours for ages and for ρ the average between. The first group works on average 37. while the second.37 of their time endowment. For the last 5 years we specify a new profile ρ t = ρ + ρ t. We calibrate ρ to match the average hours during those last five years equal to To explore the implications of hyperbolic discounting, we also study economies in which ϱ <. In particular, we consider the cases where ϱ =.5 and ϱ =.. For each value of ϱ, we only adjust β to generate the same capital to output ratio of.. The parameters that characterized the stochastic component of individuals productivity are (σ u, ϕ z, σ ɛ ). Several authors have estimated similar stochastic process for labor productivity using US data. Controlling for the presence of measurement errors and/or effects of some observable characteristics such as education and age, the literature provides a range of [.9,.97] for ϕ z and of [.,.5] for σ ɛ. In Brazil, due to the lack of a household panel data survey, such as the Panel Study of Income Dynamics in the U.S., we cannot estimate ϕ properly. Thus, we set ϕ =.9 based on the evidence for the U.S. economy and then calibrate σ ɛ 3 The data for hours worked are from Brazilian National Household Survey - PNAD (Pesquisa Nacional por Amostra de DomicÃlios).

8 Table : Parameter Values - Calibration Parameter Value Source/Target β.979 K/Y =. γ. Micro evidence ρ t - life cycle profile of mean hours σu.7 Gini index of.33 at age ϕ z.9 Kaplan [] σɛ. Gini index of.5 at age δ.55 see text α.3 B.9 w = to match the income Gini at age, which was.5 in according to the PNAD. This procedure provides a value for σɛ equal to.. Finally, σu is chosen in such a way that the labor income Gini at age in the model matches its counterpart in the data, which is nearly.3. We obtain a value of.7 for σu. We discretize the two shocks in order to solve the model, using 5 states to represent the permanent shock and nine states for the persistent shock. For expositional convenience, we refer to the two extremes of the grid for the permanent shock as low and high ability/type. The values of the technological parameters (α, δ) are also in Table. We chose a value for the capital, α, of.3 based on Paes and Bugarin []. The depreciation rate, in turn, is obtained by δ = I/Y K/Y g. We set the investment-product ratio I/Y equal to.9 and the capitalproduct ratio K/Y equal to.. The economic growth rate, g, is constant and consistent with the average growth rate of GDP over the second half of the last century. Based on data from Penn-World Table, we set g equal to.%, which yields a depreciation rate of nearly 5.%. The values for the actual age-efficiency profile are constructed similarly to Huggett [99] and McGrattan and Rogerson [99]. We use annual earnings and annual hours worked for the age groups 5-, 5-3,..., 75- from PNAD. First, we construct hourly wages by dividing annual earnings by annual hours for each age group. Afterwards, we use a second order polynomial to interpolate the points to obtain the age-efficiency profile by exact age. Finally, we specify the others parameters related to government activity. First, we set government consumption, G, to % of the output of the economy under the baseline calibration. The interest rate on FGTS assets is 3% plus the reference interest rate (TR), which was nearly %. Thus, thus considering an inflation rate of.5%, we have that the real return on FGTS deposits is.5%, which is the value we use for r fgts in the benchmark economy. As for Social Security, contributions are of the form T ss (y) = τ ss min {y, y max }. The tax rate for social security is % for the employee and % for the employer, so we set τ ss =.3. To calibrate the contribution ceiling, we use the fact that in the data, y max =.3y m, where y m is the average labor income. We assume that the retirement replacement rate is given by θ(u) = θ + θ u. We choose θ and θ in such a way that the replacement rate of the lowest ability individual is. and the average replacement rate is.7, as is observed in the data. Figure 3 compares the average life-cycle profiles induced by the benchmark policies for both the exponential and the hyperbolic discounting models. The top left graphs displays the ageefficiency profile for each of model, whereas the graph in the left displays the corresponding hours. The three figures in the bottom show how taxes vary with income, how consumption For the sake of comparison, using the PSID data, Kaplan [] finds ϕ z =.9, σ ɛ =. and σ u =.5.

9 9 and assets accumulation change with age. 3.5 Average Assets. Average Consumption 3 % = : % = :5 Total assets.7.5 % = : % = : % = :5.5 FGTS assets. % = :.. Average Hours.5 Average Savings Total savings % = : Private savings.35 % = :5.3 % = : % = : % = :5 % = : Figure : Average life-cycle profiles for the benchmark economy: Exponential vs Hyperbolic discounting Averages may, of course, hide a rich diversity in life-cycle patterns. We split the individuals in our economy in five different ability groups. We group the agents in the high extreme of the grid of the distribution of innate ability, u, and label them the high ability group. The agents on the lowest extreme of the grid are labeled low ability. In Figure??, we plot the same variables considered in Figure?? for each of these groups to get a sense of how heterogeneity plays in our model. Life-cycle patterns are qualitatively similar for all groups and all models. High ability individuals do, however, work more hours and accumulate more assets than lower ability individuals for all different specifications. Results In Figure, we see that hyperbolic accumulate less assets, have less wealth to spend when retired, and end up consuming less in the elderly age. Regarding difference between the high and low type in Figure, the humps in assets and consumption are more pronounced for the high type, which saves more during the working age and then deplete the his accumulated wealth. Table presents the steady state values of the benchmark scenario in column and some counter factual exercises in the following columns. In column, we see that without the compulsory FGTS savings, the total savings of the economy coincide with the private (voluntary) savings. In the first panel, without hyperbolic discounting, the higher average return on savings provides the incentive for higher savings, more than compensating the compulsory FGTS. That was expected. However surprising is that total savings also increase in the quasi-

10 % = : % = :5 % = :.... % = : % = :5 % = : Average life cycle hours worked by type.3 Average life cycle total savings by type % = : % = :5 % = :..3. % = : % = :5 % = : Figure : Average profiles for the benchmark economy by type: Exponential vs Hyperbolic discounting hyperbolic cases of the second and third panels. To pay a higher average return on capital, the marginal product of capital has to be higher, which implies lower levels of K/Y and lower wages. By receiving lower wages, worker supply less hours, an effect common to the three panels. And without the revenue from the FGTS, the government has to raise taxes. In column 3 of Table, we see that a lower replacement rate (from.7 to.35) lowers the taxes needed to finance pensions and considerably increases the needed private savings for retirement. By accumulating more wealth, the ratio of capital to output increases and the interest rate decreases. Labor is now more productive, wages are higher and agents choose to work more in the three panels. Column of Table shows the joint effect of no compulsory FGTS savings and a lower replacement rate, where the effects of the latter seem to dominate. Finally, in column 5 of Table, we analyze the effects of a higher remuneration to the FGTS (from -.5% to 3.% per year). In this case, the government needs to raise taxes to pay for such higher interest. Because of the cumulative effect of higher interest on the FGTS, the agent needs to save less voluntary to finance retirement. The greater weight of FGTS on total savings also lowers the average interest rate and increases capital stock. Again, with more productive labor and higher wages, workers work more, though less than in the reduction of the replacement rate exercise (in column 3). References S Rao Aiyagari. Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics, 9(3):59, 99.. M. Huggett. Wealth distribution in life-cycle economies. Journal of Monetary Economics, 3(3): 9 9, 99. 3

11 % = : % = :5 % = :.... % = : % = :5 % = : Average life cycle hours worked by type.3 Average life cycle total savings by type % = : % = :5 % = :..3. % = : % = :5 % = : Figure 3: Average profiles for the benchmark economy by type: Exponential vs Hyperbolic discounting Greg Kaplan. Inequality and the lifecycle. Quantitative Economics, 3(-):7 55,., D. Laibson. Hyperbolic discount functions, undersaving and savings policy. National Bureau of Economic Research Working Paper 535, 99.. E. R. McGrattan and R. Rogerson. Changes in hours worked since 95. Federal Reserve Bank of Minneapolis Quarterly Review, (): 9, N. L. Paes and M.N.S. Bugarin. Parametros tibutários da economia brasileira. Estudos Econômicos, 3,. 3

12 ϱ =. and β =.979 ϱ =.5 and β =.993 ϱ =. and β =. avg. µ = :35 avg. µ = :35 avg. µ = : avg. µ = :35. avg. µ = :35. avg. µ = : avg. µ = :35 avg. µ = :35 avg. µ = : avg. µ = :35. avg. µ = :35. avg. µ = : Figure : Benchmark versus reducing the retirement replacement rate: Exponential and Hyperbolic discounting

13 3 Table : Results The table displays the values for the relevant variables GDP (Y ), capital-output ratio (K/Y ), average hours (Avg hours), wages (w), real interest rates(r), and avg. savings Exponential discounting: ϱ =. and β =.979 Bench τ fgts θ all r fgts Y K/Y Avg hours w r.%.9% 7.% 7.% 7.39% Private savings rate.9%.%.5%.75% 3.5% Total savings rate.%.%.%.75%.% τ w 9.5%.9% 3.% 5.99% 5.% CEV Quasi-hyperbolic discounting: ϱ =.5 and β =.993 Bench τ fgts θ all r fgts Y K/Y Avg hours w r.%.7% 7.3% 7.7% 7.% Private savings rate.%.7%.5%.7% 3.5% Total savings rate.3%.7%.%.7%.% τ w 9.5%.9%.% 5.7% 5.5% CEV Quasi-hyperbolic discounting: ϱ =. and β =. Bench τ fgts θ all r fgts Y K/Y Avg hours w r.%.% 7.% 7.% 7.% Private savings rate.3%.%.%.7%.3% Total savings rate.%.%.%.7%.% τ w 9.5%.95%.5% 5.% 5.3% CEV

14 ϱ =. and β =.979 ϱ =.5 and β =.993 ϱ =. and β =. 9 øfgts = : 9 øfgts = : 9 øfgts = : øfgts = :. øfgts = :. øfgts = : øfgts = : øfgts = : øfgts = : øfgts = :.5 øfgts = :.5 øfgts = :... Figure 5: Benchmark versus eliminating the FGTS: Exponential and Hyperbolic discounting

15 5 ϱ =. and β =.979 ϱ =.5 and β =.993 ϱ =. and β =. øfgts = : and avg. µ = :35 øfgts = : and avg. µ = :35 øfgts = : and avg. µ = : øfgts = : and avg. µ = :35. øfgts = : and avg. µ = :35. øfgts = : and avg. µ = : øfgts = : and avg. µ = :35 øfgts = : and avg. µ = :35 øfgts = : and avg. µ = : øfgts = : and avg. µ = :35. øfgts = : and avg. µ = :35. øfgts = : and avg. µ = : Figure : Benchmark versus changing both τ fgts =. and avg. θ =.35: Exponential and Hyperbolic discounting

16 ϱ =. and β =.979 ϱ =.5 and β =.993 ϱ =. and β =. rfgts = 3:% rfgts = 3:% rfgts = 3:%.... rfgts = 3:%. rfgts = 3:%. rfgts = 3:% rfgts = 3:% rfgts = 3:%.5.5 rfgts = 3:% rfgts = 3:%.35.. rfgts = 3:% rfgts = 3:% Figure 7: Benchmark versus increasing r fgts to 3%: Exponential and Hyperbolic discounting