Macroeconomic Theory

Transcription

1 Macroeconomic Theory Francesco Franco Nova SBE February 12, 218 Francesco Franco Macroeconomic Theory 1/15

2 Infinite Horizon The decentralized economy 2 factor markets labor with rental price w(t) and capital services with rental price r(t) there is a debt market in which families can borrow and lend many identical families can save in k or in bonds many identical firms with crs technology perfect foresight Francesco Franco Macroeconomic Theory 2/15

3 Infinite Horizon The decentralized economy Each family decides how much to consume-save taking as given the sequence of prices by maximizing U(s) = s e ρ(t s) u(c(t))dt subject to and a(t) = k(t) b p (t) c(t) + ȧ(t) + na(t) = w(t) + r(t)a(t) Francesco Franco Macroeconomic Theory 3/15

4 Infinite Horizon The decentralized economy Firms maximize profits, foc are f (k(t)) = r(t) f (k(t)) k(t)f (k(t)) = w(t) Francesco Franco Macroeconomic Theory 4/15

5 Infinite Horizon The decentralized economy equilibrium Equilibrium: consider an arbitrary path for wages and rental rates. This sequence lead each family to determine consumtion and wealth accumlation. Private debt is in zero net supply, therefore wealth accumulation determine the capital stock sequence. In turn it determines the prices. Francesco Franco Macroeconomic Theory 5/15

6 Infinite Horizon The No-Ponzi-Game condition There is an equilibrium we want to rule out: given the absence of borrowing constraint you can indebt to reach a level of c that commands a marginal utility of zero and let the bc determine the evolution of a. This path of consumption will lead to higher and higher debt. We need a condition that rules out this pathologial case but allows for temporary indebtness: the NPG condition [ t lim t a(t)e (r(τ) n)dτ]. Francesco Franco Macroeconomic Theory 6/15

7 Infinite Horizon The No-Ponzi-Game condition Integrate the bc from to T, using the NPG you get the intertemporal budget constraint (IBC) c(t)e t (r(τ) n)dτ dt = a + h where h w(t)e t (r(τ) n)dτ dt. Francesco Franco Macroeconomic Theory 7/15

8 Ramsey FOC We have the necesarry conditions lim t a(t)u (c(t))e ρt = u (c(t)) = λ(t) λ(t) = λ(t) [ρ + n r(t)] Francesco Franco Macroeconomic Theory 8/15

9 Ramsey Decentralized economy Using the foc of the firms and the market clearing conition a(t) = k(t), we obtain k(t) = f (k(t)) c(t) nk(t), ċ(t) c(t) = σ(c(t)) [ f (k(t)) ρ n ] which is identical to the planner allocation Francesco Franco Macroeconomic Theory 9/15

10 Ramsey Expectations Using a CRRA σ is constant and we can obtain the level of consumption explicitely from Euler: ċ(t) = c(t)σ [ f (k(t)) ρ n ] integrate [ t c(t) = c e σ(r(τ) ρ n)dτ] and insert in IBC to obtain c = β (a + h ) [ { where β } exp t [(σ 1) (r(τ) n) ρσ] dτ Discuss Income-Substitution effects. dt] 1. Francesco Franco Macroeconomic Theory 1/15

11 EIS The effect of the interest rate on consumption-saving decisions Consumption is a linear function of wealth, with a propensity β that depends on the expected path of the interest rates. An increase in the interest rates: 1 makes consumption later more attractive and threfore increase saving, this is the substitution effect 2 allow for higher consumption now and later and decreases saving: this is the income effect 3 there is also a wealth effect that changes h Francesco Franco Macroeconomic Theory 11/15

12 Ramsey Government Assume first that government is consuming resources, g(t), and paying them with taxes τ(t). To begin with let us study a budget that is balanced every period. The family bc: c(t) + ȧ(t) + na(t) = w(t) + r(t)a(t) τ(t) integrate the bc, using the NPG you get (define R t = e t (r(τ) n)dτ ) c(t)r(t)dt = k b p + h τ(t)r(t)dt and using the balanced budget G = τ(t)r(t)dt. Government expediture affects decisions, prices and therefore allocations. Phase diagram. Crowding out. Francesco Franco Macroeconomic Theory 12/15

13 Ramsey Government Let us assume debt financing ḃ(t) + nb(t) = g(t) τ(t) + r(t)b(t) Integrate the bc and impose a NPG b + g(t)r(t)dt = τ(t)r(t)dt which shows that the government need not to run a balanced budget at every moment. Francesco Franco Macroeconomic Theory 13/15

14 Ramsey Government For the families now a(t) = k(t) b p (t) + b(t). Same interest rate. Integrate the bc, using the NPG you get c(t)r(t)dt = k b p + b + h τ(t)r(t)dt by substituting the government budget we obtain the strong result that only expenditure matter for the allocations not the method of financing: lump sum or debt. Plus for a given path of g the method of finance has no effect on the allocation of resources. Francesco Franco Macroeconomic Theory 14/15

15 Readings (*) David Romer. Advanced MAcroeconomics, 4th Edition Chapter II. Francesco Franco Macroeconomic Theory 15/15