Prep. Course Macroeconomics

Prep. Course Macroeconomics Intertemporal consumption and saving decision; Ramsey model Tom-Reiel Heggedal tom-reiel.heggedal@bi.no BI 2014 Heggedal (BI) Savings & Ramsey 2014 1 / 30

Overview this lecture In the Solow model and in Pikettys second fundamental law the saving rate is important but not modelled. In this lecture we ll endogenize the consumption decision. i) Utility over periods ii) Two period optimization iii) Permanent income iv) Ricardian Equivalence v) Optimal growth model - Ramsey Cass Koopmans model iv) Pikettys third fundamental law of capitalism. Heggedal (BI) Savings & Ramsey 2014 2 / 30

Piketty: saving Table 5.4. Private and public saving in rich countries, 1970 2010 National saving (private + public) (net of depreciation) (% national income) U.S. 5,2% incl. Private saving 7,6% incl. Public saving 2,4% Japan 14,6% 14,5% 0,1% Germany 10,2% 12,2% 2,0% France 9,2% U.K. 5,3% 11,1% 1,9% 7,3% 2,0% Italy 8,5% 15,0% 6,5% Canada 10,1% 12,1% 2,0% Australia 8,9% 9,8% 0,9% A large part (variable across countries) of private saving is absorved by public deficits, so that national saving (private + public) is less than private saving. Sources: voir piketty.pse.ens.fr/capital21c Heggedal (BI) Savings & Ramsey 2014 3 / 30

Intertemporal modelling: Two period model We use a representative agent (household/consumer) for the macroeconomy (model not used to analyze intratemporal effi ciency between individuals ->heterogenous agents models) analyze intertemporal problems as saving and investment. Consider a two period economy. A preference system (a (social) welfare function) is given by ( ) 1 W = U(C 0 ) + U(C 1 ), 1 + ρ where U(C t ) is instantaneous (per period) utility that is well behaved: U (C ) > 0, U (C ) < 0. ρ is utility discount rate (time preference rate). Note that labor is not modelled here (e.g. inelastic supply of labor) Heggedal (BI) Savings & Ramsey 2014 4 / 30

Two period model The problem (for a representative consumer): s.to ( 1 1+ρ max U(C 1 ) + C 1,C 2 Budget constraint ) U(C 2 ) The intertemporal budget constraint is given by C 1 + C 2 1+r = Ω, where omega is lifetime income (income+initial wealth) r is the normal discount rate for consumption and investment decisions (e.g. in cba analysis). r is the real rate of return on investments (ror on capital) (no money or prices in the model). r is an equilibrium outcome of the growth model (we keep it exogenous in the two-period example). Heggedal (BI) Savings & Ramsey 2014 5 / 30

Two period model: The budget constraint Income in the periods is what is produced Y 1 and Y 2 (this is allocated as labor or capital income -> the consumer owns all capital and receives all labor income). minus taxes (that are exogenous here and for simplicity G do not enter the utility function). Initial wealth is given by V 1 (positive or negative). {derivation of bc} The intertemporal budget constraint is C 1 + C 2 1+r = V 1 + Y 1 T 1 + Y 2 T 2 1+r. We define omega Ω V 1 + Y 1 T 1 + Y 2 T 2 1+r, since all these variables are exogenous in this simple two period model. {solve problem} ( ) max U(C 1 ) + 1 C 1,C 1+ρ U(C 2 ) 2 s.to C 1 + C 2 1+r = Ω Heggedal (BI) Savings & Ramsey 2014 6 / 30

Two period model: Solution The necessary condition for maximum is U C2 U C1 = 1 + ρ 1 + r. Since marginal utility is decreasing (U CC < 0): ρ > r C 2 < C 1 ρ = r C 2 = C 1 ρ < r C 2 > C 1. Net gain of saving is U (C 1 ) + 1+r 1+ρ U (C 2 ). The Keynes-Ramsey rule: Under a perfect capital markets assumption, save (borrow) until the net gain is zero. Heggedal (BI) Savings & Ramsey 2014 7 / 30

Two period model: Exercise 1. Solve the problem (find U C2 /U C1 ) ( ) max U(C 1 ) + 1 C 1,C 1+ρ U(C 2 ) 2 s.to C 1 + C 2 1+r = Ω using substitution instead of Lagrange. 2. Then find C 1 and C 2 when utility is given by U(C t ) = σ 1 σ σ(c t 1), σ 1 where σ is the elasticity of substitution for consumption between periods (note that this is the Constant Relative Risk Aversion (CRRA) function with the constant risk parameter θ, where θ 1/σ). Heggedal (BI) Savings & Ramsey 2014 8 / 30

Two period model: Solution to 2. ( ) C 2 C 1 = 1+r σ, 1+ρ C1 = ΩH (1+ρ)σ 1+H, where H. (1+r ) σ 1 Intertemporal elasticity of substitution: the curvature of the indifference curve, the rate of change in C 2 C 1 of a change in the interest rate. Heggedal (BI) Savings & Ramsey 2014 9 / 30

Permanent income hypothesis Heggedal (BI) Savings & Ramsey 2014 10 / 30

Ricardian Equivalence Heggedal (BI) Savings & Ramsey 2014 11 / 30

Intertemporal modelling - several periods Welfare (/representative agent) function in discrete time: ( ) ( ) 2 ( ) T W = U(C 0 ) + 1 1+ρ U(C 1 ) + 1 1+ρ U(C2 ) +... + 1 1+ρ U(CT ) = T t=0 ( 1 1+ρ ) t U(Ct ). Infinite horizon: W = t=0 ( ) 1 t U(C t ), 1 + ρ discount factor( 1 1+ρ ) t is weight on utility obtained in period t, seen from period 0. i.e. W is the present discounted value of the utility stream. Function with continuous time: W = T t=0 U(C t)e ρt dt. Infinite horizon: W = t=0 U(C t )e ρt dt discount factor e ρt is the weight on utility obtained in period t, seen from period 0. Heggedal (BI) Savings & Ramsey 2014 12 / 30

Optimal growth model - simple Ramsey model Ramsey Cass Koopmans model (Ramsey growth model) is the staple neo-classical model of economic growth (long run macro). Its the Solow model with endogenous saving. Consider the simplest closed economy Ramsey model: One good produced: used for consumption and investment; per period utility U(C t ); Firms are competitive; price P t, production technology Y (K t ); Capital accumulation K = Y (K t ) C t ; Y (K t ) > C t implies saving. Problem (planner and representative HH): s.to max W (C) = max C C t=0 U(C t)e ρt dt K = Y (K t ) C t, K 0, lim K (t) 0. t Heggedal (BI) Savings & Ramsey 2014 13 / 30

Ramsey model cont. Solution is U C U C = ρ r, (1) where LHS is the rate of change in the (instantaneous) marginal utility. If ρ < r (Y (K )) then U C U U is negative. C C U negative means C consumption grows over time. Intuition: at the margin the cost of deferring consumption is smaller than the gain from saving. So capital is accumulated and consumption increases. Until the point where decreasing marginal productivity (Y (K ) < 0) lowers r so much that ρ = r, and capital is constant. Heggedal (BI) Savings & Ramsey 2014 14 / 30

Ramsey model cont. (2) The implied expression for the consumption growth rate from eq. (1) : Ċ C = σ(y (K ) ρ), (2) which is called the Euler equation (in terms of consumption growth). Denote Ċ C = g c : g c > 0 when Y (K ) > ρ, i.e. when payoff of delaying consumption is larger than the impatience. In this case there are savings and the capital stock increases. g c < 0 when Y (K ) < ρ;... g c = 0 when Y (K ) = ρ;... -> the rate of return on capital (investment) r Y (K ) explain growth rates (typically at 2%). Reformulate eq. (2) : r = ρ + 1 σ g c. Heggedal (BI) Savings & Ramsey 2014 15 / 30

Define (competitive) equilibrium Allocation defined by the following objects: time paths of quantities [C (t), K (t)] t=0. An competitive equilibrium is an allocation of quantities with prices ([P(t)] t=0, [r(t)] t=0 ) such that: i) The representative HH max C t=0 U(C t )e ρt dt subject to the intertemporal budget constraint (includes values of assets, i.e. shares in firms and the capital stock), taking factor prices [r(t)] t=0 as given. ii) Firms max P ty (K t ) r t K t taking factor prices [r(t)] t=0 as given K (and the market price P(t)) iii) Time paths of interest rate [r(t)] t=0 is consistent with market clearing (of asset markets); iv) technology/environment constraints hold: no technology or population growth. Heggedal (BI) Savings & Ramsey 2014 16 / 30

The welfare theorems 1 The first welfare theorem is that any competitive equilibrium (Walrasian equilibrium) is Pareto effi cient. 2 The second welfare theorem is that any Pareto effi cient equilibrium can be sustained by a competitive equilibrium. Redistribution versus effi ciency Heggedal (BI) Savings & Ramsey 2014 17 / 30

What is the size of the consumption discount rate? The discount rate is important for project evaluation and all intertemporal analysis: r = ρ + 1 σ g c. Piketty reports r at 4-5%. What should the this rate be? ρ: descriptive (incl. impatience, ρ = 0.03) vs prescriptive (zero plus extinction rate, ρ = 0.01 0.001). σ: the intertemporal elasticity of substitution. σ > 1 quite willing to substitute between periods (generations); σ > 1 not so willing to substitute between periods one is rich with the periods one is poor; a σ < 1 in implies that an increase of consumption for the poorer by 1% is more valuable than for the richer. A lower σ implies less weight on gains/losses to those better off. The parameter is typically 0.5 to 1. g c : belief about growth rate (typically in long run g c = g Y = 0.02): environment-economy linkages, substitution possibilities from limited resources, technology. taken together r = 2.001 7. One of the major discussions regarding climate change (the Stern review). Heggedal (BI) Savings & Ramsey 2014 18 / 30

Piketty: diminishing returns to capital Piketty is concerned with that happens with α in the long run; will most income go to capital (the wealthy owners)? The 1th law, α = r β, states that the income share to capital is increasing in r and the capital-incom ratio β. Though, there are diminishing returns to capital, Y K < 0, so r goes down as capital accumulates.then is there a natural limit to α!? This depends on the elasticity of substitution between capital and labor in Y when capital accumulates (β go up: If elas. of subs is (0, 1), an increase in β implies that r goes down by more than β, so α goes down. If elas. of subs is [0, ), an increase in β implies that r goes down by less than β, so α goes up. If elas. of subs is 1, an increase in β implies that r goes down by the same as β, so α is constant. This is the case with the C-D production function. Heggedal (BI) Savings & Ramsey 2014 19 / 30

Piketty: diminishing returns to capital cont Piketty estimates that the historical elas. of subs is around 1.5. Thus, "...no self-corrective mechanism exists to prevents a steady increase of the capital/income ratio, β, together with a steady rise in capital s share of national income, α." Though we have learned form the Ramsey model that how fast capital accumulates depends on the savings decision: A low r implies low savings, ceteris paribus (g c < 0 when Y (K ) < ρ): r = ρ + 1 σ g c MPK = ρ + 1 σ g c Piketty have exogenous savings in his model. Really, it is endogenous savings that determines β, and puts limits this ratio. Heggedal (BI) Savings & Ramsey 2014 20 / 30

Piketty: predictions Piketty does not claim that all income will go to capital, but that from the capital s side of the economy it is in principle possible to get α = 100%. What does Piketty s numbers and laws predict? From the second law, β = s/g, it follows that the return to a low growth regime (without population growth) g = 1.5, with a savings rate of 12%, leads to the return of capital β = 8 (i.e. wealth accumulated in the past takes on considerable importance) From the first law α = r β, with a return to capital of 4 5, capital s share of income is 32 40 percent (as observed for the eighteenth and nineteenth century). Heggedal (BI) Savings & Ramsey 2014 21 / 30

Piketty: the fundamental inequality The main hypothesis of Piketty: the primary reason for the hyperconcentration of wealth in most societies before WW1 is that the growth rates are low compared to the rate of return on capital, i.e. r > g unequal wealth distribution. The third fundamental law of capitalism is that r > g. Consider a country with ninetheent century low growth rates (say 1 %) and a RoR that is much higher (5 %). This means that wealth may be recapitalized much more quickly than the economy grows: E.g with these rates, a household may save one-fifth of its capital income and still ensure that the next generation inherits the same wealth level (relative to national income) as the current enjoys. Or rather, wealthy families may easily build up the family fortune, as saving more than one 5th of income implies that the wealth grows faster than national income (output) -> "inheritance society" Heggedal (BI) Savings & Ramsey 2014 22 / 30

Piketty: r>g and unequal mechanism So the thesis following from the third law (r > g) is that people with (enough) inherited wealth need only to save a fraction of their income to get their wealth to increase faster than the economy grows -> inherited wealth will dominate wealth from lifetime labor income! This does not follow directly from r > g (combined with the 1th and 2nd law). Additional assumptions are needed: There is no theory for the initial distribution -> assume that there is initial inequality (and it is today). Savings rates are assumed to be constant, but this is not enough to give more unequalness. Sure the absolute numbers will be affected, but if everyone has the same constant savings rate, the wealth distribution will be unchanged (though capital s share of income will increase over time. Assume constant and heterogenous savings rates, and assumed that rate is higher for the more wealthy. These assumptions, together with the three laws, give a more unequal wealth distribution over time Heggedal (BI) Savings & Ramsey 2014 23 / 30

Piketty: r>g empirics Heggedal (BI) Savings & Ramsey 2014 24 / 30

Piketty: r>g empirics Heggedal (BI) Savings & Ramsey 2014 25 / 30

Piketty: r>g predictions The historical RoR on capital is calculated by using the first law and estimations of β and α. Predictions for the future are based on assumptions and calculations that are consistent with both laws. The long run growth rate (of a converged frontier country) g = 1, 5% Then e.g. assuming s = 12% gives β = s/g = 8. Then assuming α = 32%, gives r = α/β = 0.04. A g = 1, 5% and pre a tax RoR on capital net of depreciation of 4%, implies that todays unequal distribution of wealth will be even more so in the future. As r > g may give rise to a more unequal distribution of wealth (given the additional assumptions on saving rates?), policy is needed. Tax returns on capital so that r is closer to g and more tax on inheritance. Heggedal (BI) Savings & Ramsey 2014 26 / 30

Piketty criticism: theory The first law, α = r β. Is true, though on evaruable cannot be used to explain the other, e.g it is true that given α, a higher β must be associated with a higher r. Though it does not make sense neither to have one of the variables fixed nor to state that one follow from another, since they are all endogenous variable (depending on savings). The second law, β = s/g, is an outcome of any standard macro model, though it does not make sense to explain a change in β following from g without letting s change (it is silly to let β be explained by g, and vice versa, since both are outcomes (endogenous variables)). The third law, r > g : may give or may not give rise to a more unequal distribution of wealth. r > g is the outcome of many standard macro models more mechanisms are needed to explain changes in the distribution. The law (combined with the 1th & 2nd) does not imply a more unequal distribution (unless constant and heterogeneous saving rates are assumed). Heggedal (BI) Savings & Ramsey 2014 27 / 30

Piketty criticism: empirics on the rate of return That the long run RoR is fairly constant is disputed: r = i π. On government long run bonds today r may even be negative (US treasury bills are seen as almost risk-free) Short run i is today close to zero, 10 years is about 2,5, and 30 years are about 3,5. Using an inflation rate of 1% we get: r = [ 0.01; 0.015; 0.03]. Note that Pikettys average return is comparing old time government bonds with a set of assets today that is much more risky, i.e. he is comparing the ror of a more or less restless asset with more risky assets. If we adjust for risk, the picture will look different. Heggedal (BI) Savings & Ramsey 2014 28 / 30

It is a difference between an (one time) increase in the value of an asset and the long run steady return. The ror we have discussed (with diminishing marginal product) is from the firms production function; not from changing house prices. 1. it is not given that this increase will continue as deregulation and centralization is done. 2. Is it an equality problem if most people owns their house and it increases in value? Heggedal (BI) Savings & Ramsey 2014 29 / 30 Piketty criticism: empirics on the rate of return The ror is partly estimated and partly backed out using the first law. Thus the decline and the rise of the capital-income ratio is an important input for the ror: Half of the capital stock is the value of housing:

End Heggedal (BI) Savings & Ramsey 2014 30 / 30