Overlapping Generations Model: Dynamic Efficiency and Social Security

Transcription

1 Overlapping Generations Model: Dynamic Efficiency and Social Security Prof. Lutz Hendricks Econ720 August 23, / 28

2 Issues The OLG model can have inefficient equilibria. We solve the problem of a ficticious social planner This yields a Pareto optimal allocation by construction. We learn from this: 1. Solving the planning problem may be an easy way of characterizing CE (if it is optimal). 2. Comparing it with the CE points to sources of inefficiency. 2 / 28

3 The Social Planner s Problem

4 Planner s problem Imagine an omnipotent social planner. She can assign actions to all agents (consumption, hours worked,...). She maximizes some average of individual utilities. She only faces resource constraints. Solving this problem yields one Pareto optimal allocation. No economy that faces the same technological constraints can do better. A benchmark against which equilibria can be assessed. 4 / 28

5 Welfare function The planner s objective function is assumed to be a weighted average of individual utilities: ω 0 βu(c o 1 ) + t=1 ω t[u(c y t ) + βu(c o t+1 )] By varying the weights (ω t ) we can obtain all Pareto optimal allocations. It makes sense even if comparing utilities across agents does not. To ensure that the objective function is finite, assume that t ω t <. 5 / 28

6 Planner s problem The planner only faces feasibility constraints. In this model: K t+1 + N t c y t + N t 1 c o t = F(K t,n t ) + (1 δ)k t (1) Or, in per capita young terms (k t = K t /N t ): c y t + c o t /(1 + n) + (1 + n)k t+1 = (1 δ)k t + f (k t ) 6 / 28

7 Planner s Lagrangian Γ = ω 0 βu(c o 1 ) + t=1 ω t[u(c y t ) + βu(c o t+1 [ )] ] + t=1 λ (1 δ)k t + f (k t ) t c y t c o t /(1 + n) (1 + n)k t+1 Planner s FOCs: ω t u (c y t ) = λ t ω t 1 βu (c o t ) = λ t /(1 + n) λ t [1 δ + f (k t )] = λ t 1 (1 + n) 7 / 28

8 Planner s problem Static optimality: ω t u (c y t ) = ω t 1 (1 + n)βu (c o t ) Euler equation: ω t u (c y t )[1 δ + f (k t )] = ω t 1 u (c y t 1 )(1 + n) Using the static condition, the Euler equation becomes u (c y t ) = βu (c o t+1 )[1 δ + f (k t+1 )] (2) which looks like the Euler equation of the household. This is not surprising: the planner should respect the individual FOCs unless there are externalities. 8 / 28

9 Planner s Solution Sequences { c y t,c o t,k t+1 } t=1 that satisfy: Static and Euler equation. Feasibility. A transversality condition or k t+1 0. We talk about those later. 9 / 28

10 Interpretation of the Euler equation A feasible perturbation does not change welfare. In t 1: In t: c y t 1 by (1 + n) k t by 1 (per capita of the date t young) output by f (k t ) (per capita t young) raise c y t by 1 δ + f (k t ) or raise c o t by (1 + n)(1 δ + f (k t )) From t + 1 onwards: nothing changes especially not kt+1 10 / 28

11 Planner s Steady State For a steady state to exist, weights must be of the form ω t = ω t, ω < 1 Otherwise the ratios ω t+1 /ω t in the FOCs are not constant. Then the Euler equation becomes ω (1 δ + f (k MGR )) = (1 + n) This is the Modified Golden Rule. (ω = 1 is the Golden Rule). Because ω < 1: k MGR < k GR and the MGR is dynamically efficient. 11 / 28

12 How does the planner avoid dynamic inefficiency? If the planner desires lots of old age consumption, he can implement a "transfer scheme" of the following kind: Take a unit of consumption from each young and give (1 + n) units to each old at the same date. There is no need to save more than the GR. Of course, there aren t really any transfers in the planner s world. 12 / 28

13 Social Security

14 Social Security A transfer scheme akin to Social Security can replicate the Planner s allocation and avoid dynamic inefficiency. Social Security consists of a payroll tax on workers; a transfer payment to the retired. 14 / 28

15 Two flavors of Social Security Fully funded: For each worker, the government invests the tax payments. This is equivalent to a forced saving plan. A system that is gaining popularity around the world. Pay-as-you-go: Current transfers are paid from current tax revenues. The U.S. system. 15 / 28

16 Household with Social Security The household maximizes u ( c y ) ( ) t + βu c o t+1 subject to the present value budget constraint w t τ y t τo t r t+1 = c y t + co t r t+1 (3) Lump-sum taxes do not change the Euler equation (prove this): β(1 + r t+1 )u ([1 + r t+1 ]s t+1 τ o t+1 ) = u (w t s t+1 τ y t ) 16 / 28

17 Household with Social Security The saving function remains the same s t+1 = s ( w t τt y, τt+1 o,r ) t+1 (4) For given prices, Social Security reduces saving for two reasons: Higher income when old. Lower income when young. 17 / 28

18 Household with Social Security If a tax change does not alter the present value of taxes, dτ y + dτo 1 + r t+1 = 0 then the optimal consumption path does not change. Reason: present value budget constraint and first-order condition unchanged. This is the Permanent Income Hypothesis. 18 / 28

19 Fully funded Social Security Young: pay tax τ y t. Old pay: τ o t+1 = (1 + r t+1) τ y t < 0. Government supplies revenues as capital to firms. For the household: Forced saving at rate of return r. No change to the present value budget constraint. Therefore, if prices remain fixed: No change to optimal consumption plan. Private saving (of the young) drops by the Social Security tax amount. 19 / 28

20 Fully Funded Social Security We prove that unchanged (w t,r t ) clear the markets with Social Security. Household: By PIH: no change in consumption plan. Household fully dissaves the tax: s t+1 = τt y. Government saves: s G t+1 = N tτ y t. Capital market clearing: K t+1 = N t s t+1 + s G t+1 = 0 (5) Fully funded SS is neutral. Essentially, the government just relabels some private savings as public. 20 / 28

21 Exercise Write out the equilibrium definition for the model with Fully Funded Social Security. 21 / 28

22 Pay-as-you-go Social Security Assume population growth at rate n: N t = (1 + n)n t 1. Tax collection from the current young: N t τt y. Transfer payments to the current old: N t 1 τt o. The budget balances in each period: τt o = τt y (1 + n) (6) From the household s perspective: Forced saving with return n. Saving drops by an amount different from τ y t. 22 / 28

23 Pay as you go Social Security We prove that unchanged r t+1 imply excess demand for K t+1. Household: s t+1 < 0. Government: Balanced budget. Capital market: K t+1 = N t s t+1 < / 28

24 Illustration Capital market clearing: k t+1 (1 + n) = s ( w(k t ) τ y t, w(k t+1 ) τ o t+1, r(k t+1) ) (7) Assume that the saving function is well-behaved (e.g. log utility and Cobb-Douglas). 24 / 28

25 Complications Since prices change, we cannot guarantee that Pay-as-you-go SS reduces steady state k. Totally differentiate the saving function: [1 + n s 3 f (k t+1 )]dk t+1 = s 1 dτ y s 2 dτ o < 0 A sufficient condition for dk t+1 < 0 is that s 3 > 0. Then the law of motion unambiguously shifts down. 25 / 28

26 Dynamic efficiency If SS reduces the steady state capital stock, it can alleviate dynamic inefficiency. Note that the argument is not reversible: in a dynamically efficient economy, reverse social security is not a Pareto improvement. why not? 26 / 28

27 Reading Acemoglu (2009), ch. 9. Krusell (2014), ch / 28

28 References I Acemoglu, D. (2009): Introduction to modern economic growth, MIT Press. Krusell, P. (2014): Real Macroeconomic Theory, Unpublished. 28 / 28