Housing with overlapping generations Chiara Forlati, Michael Hatcher, Alessandro Mennuni University of Southampton Preliminary and Incomplete May 16, 2015 Abstract We study the distributional and efficiency implications of housing in an overlapping generations model with rental markets. We show three main results. First, housing drastically alters the competitive equilibrium relative to the case where fixed-supply fiat money (or bonds) is the only asset. This is because housing delivers a relatively high rate of return and so induces a great deal or redistribution of consumption from the young to the old. Second, if rental markets are closed down, the competitive equilibrium is Pareto inefficient due to the presence of a wedge in the intratemporal trade-off between consumption and housing in the last period of life. Lastly, when we add in productive capital, we find that the stationary competitive equilibrium is dynamically efficient because an arbitrage condition between housing and capital investment pushes the return on capital above 1. This is in stark contrast to the model without housing. JEL Classification Codes: R31 Keywords: Housing, OLG economy, rental markets, Golden Rule
1 Introduction There is a widespread feeling amongst academics, politicians, and the wider society, that the housing market is inequitable. It is not clear, however, what this really means. To clarify this issue, we introduce housing in an overlapping generations economy (OLG) with two-period lives. In the first period of life households are young; in the second period they are old. Housing can be either purchased or rented. Purchased housing is both a store of value and a consumption good that delivers utility. Rented housing, by contrast, only delivers utility. Using this simple life-cycle setting, we are able to shed light on distributional and efficiency effects of housing markets and draw some implications for government policy. We start with an OLG exchange economy as in Samuelson (1958) and Gale (1973) and introduce housing and rental markets. We show that housing drastically changes the competitive equilibrium relative to fixed-supply fiat money (or bonds). The stationary competitive equilibrium is Pareto efficient but far from the Golden Rule allocation, which corresponds to the competitive equilibrium with fiat money (or bonds). This is because housing delivers a relatively high rate of return and so induces a great deal or redistribution of consumption from the young to the old. A similar result has been noted by Geanakoplos (2008) in a model with productive land and a different timing assumption, but in his model the economy is at a Pareto efficient allocation even if rental markets are absent. 1 By contrast, our analysis highlights the the role of rental markets. In particular, if these markets are shut down, the competitive equilibrium is no longer Pareto efficient. The reason is that the old must die owning housing in order to enjoy the housing service in the last period of life; if instead the old could rent, this would enable them to save less when young yet consume just as much in old age. A common feature of the case with and without rental markets is that the young save more than in a model with fiat money or government debt but without housing. In this latter case, the competitive equilibrium coincides with the Golden Rule: the allocation that maximizes the welfare of future generations but gives to the initial old the same level of welfare attained by future old (see Champ et al., 2011). With housing, the competitive equilibrium does not satisfy the Golden Rule because consumption is too backloaded to old age. This is because housing serves the dual role of store of value and also gives utility. This tendency to overaccumulate housing relative to the Golden Rule is akin to the well-known result of dynamic inefficiency in the OLG model with capital (Diamond, 1965). Notice, however, that over- 1 We assume that the possession of a house during the period is necessary to enjoy the housing consumption service, which delivers utility. In the model of Geanokoplos, by contrast, the agent can sell land at the beginning of a period but receives a dividend nonetheless. 2
accumulation of capital results in Pareto inefficiency because too much capital is accumulated in the competitive equilibrium. With a constant housing stock instead, the overaccumulation of saving results in a redistribution of consumption from the young to the old. In the case where rental markets are absent, the competitive equilibrium is Pareto inefficient due to the presence of an intratemporal wedge - a static inefficiency. The wedge arises because in the last period of life agents want housing for its utility, but can gain this only at the expense of having to die owning housing with positive resale value. Avoiding this unintentional bequest to future generations would lead to a Pareto improvement. The interatemporal wedge thus highlights the role of rental markets: they eliminate a source of Pareto inefficiency. It is also, to the best of our knowledge, a novel contribution of our paper. When productive capital is added to the model, there is an interesting interaction between housing and capital. Since the return on housing exceeds the Golden Rule return of 1, the arbitrage condition between housing and capital investment ensures that capital accumulation is dynamically efficient. This result stands in stark contrast to the model without housing, where Diamond (1965) and others have shown that capital overaccumulation is possible. It is also an interesting result because it suggests that government interventions which cure supposed overaccumulation of capital (money, government debt, social security, pensions) might be unnecessary. It may also help to explain why real-world economies do not appear to overaccumulate capital. The paper proceeds as follows. In Section 2 we set out the model, characterize the social planner solution and discuss the distrbutional and efficiency properties of the competitive equilibrium. Section 3 considers the case where rental markets are shut down. In Section 4 we introduce capital into the model and study how this affects efficiency. Finally, Section 5 concludes. 2 Model Consider a two-period OLG model in which population is constant and normalized to 1. Agents receive utility from consumption and the total amount of housing services consumed in each period (purchase + rental). When young, agents receive an endowment income y > 0 which they can use for consumption c 1, housing purchases h p 1 or to rent housing h r 1. In old age, agents choose new house purchases h p 2, rent housing h r 2 and sell their old home h p 1 for consumption goods. They also receive consumption from the sale of any bequest of housing h p 2 left to them as a result of the previous old dying with positive housing assets (as we shall see, this bequest is equal to zero in the presence of rental markets). The problem of a young 3
individual born in time t is: max U(c 1,t, h p [c 1,t,c 2,t+1,h p 1,t,hr 1,t,hp 2,t+1,hr 2,t+1 ] 1,t + h r 1,t) + βu(c 2,t+1, h p 2,t+1 + h r 2,t+1) (1) s.t. p t h p 1,t + r t h r 1,t + c 1,t y (2) c 2,t+1 + p t+1 h p 2,t+1 + r t+1 h r 2,t+1 p t+1 (h p 1,t + h p 2,t) (3) h p 1,t 0 (4) h p 2,t+1 0. (5) where p denotes the purchase price of housing and r is the rental price of housing. These prices are taken as given by the agents. The utility function U is increasing in the two arguments, with decreasing marginal utilities and Inada conditions are satisfied. Market-clearing conditions There is a fixed stock of housing H > 0, so in equilibrium it is the case that for all t h p 1,t + h r 1,t + h p 2,t + h r 2,t = H. (6) Furthermore, the rental market has to clear: h r 1,t + h r 2,t = 0, (7) That is, if one generation is a tenant, the other generation has to be a landlord. Finally, the goods market has to clear: c 1,t + c 2,t = y. (8) First order conditions c 1,t : = λ 1,t (9) c 2,t+1 : β+1 = λ 2,t+1 (10) h p 1,t : = λ 1,t p t λ 2,t+1 p t+1 λ h,1,t (11) h r 1,t : = λ 1,t r t (12) h p 2,t+1 : βu h2,t+1 = λ 2,t+1 p t+1 λ h,2,t+1 (13) 4
h r 2,t+1 : βu h2,t+1 = λ 2,t+1 r t+1 (14) where λ 1,t and λ 2,t+1 are the Lagrange multipliers on the budget constraints of the young and old, respectively, while λ h,1,t and λ h,2,t+1 are the Lagrange multipliers on the nonnegativity constraints on house purchases. They are all positive and the complementary slackness conditions on the constraints hold (for instance: λ h,1,t ( h p 1,t) 0 and λ h,1,t 0). These first order conditions, together with the market clearing conditions, define a competitive equilibrium. 2.1 Characterization We now present three propositions that help characterize the equilibrium of this model. The first two propositions are instrumental in reaching our Proposition 3: the young are landlords and the old tenants. This result helps to simplify the solution and makes comparison with the social planner solution straightforward. Proposition 1 The purchase price of housing exceeds the rental price: p t > r t for all t. Proof. The first order conditions (9) (12) imply that (p t r t )U c,1,t = β+1 p t+1 + λ h,1,t (15) With p t+1 > 0 the right hand side is positive and with U c,1,t > 0 the result follows. That p t+1 > 0 is immediate from Equation (13). Proposition 2 h p 2,t+1 = 0 and λ h,2,t+1 > 0 for all t. Proof. From equations (13) and (14) it follows that h p 2,t+1 : λ 2,t+1 (p t+1 r t+1 ) = λ h,2,t+1 (16) By Proposition 1 we know that the left hand side is positive, so λ h,2,t+1 > 0. This means that the constraint (5) is binding: h p 2,t+1 = 0. Proposition 3 h p 1,t = H and λ h,1,t = 0 for all t. Proof. From the previous proposition and stationarity we have h p 2,t = 0. Then the market clearing conditions (6) and (7) imply h p 1,t = H. Since h p 1,t > 0, the complementary slackness condition implies that λ h,1,t = 0. 5
A corollary of these propositions is that there are no housing bequests: h p 2,t = 0. Thus, in any competitive equilibrium of this model, the housing property market is stationary, with the young holding the entire stock of housing. Intuitively, the old will not purchase any housing because dying with property is strictly dominated by a strategy of renting from the current young - the landlords in our setting. 2 Due to the above propositions, the first order conditions (9) (14) simplify to = p t β+1 p t+1, (17) = r t, (18) βu h2,t+1 = β+1 p t+1 λ h,2,t+1, (19) βu h2,t+1 = β+1 r t+1. (20) These expressions are useful because they are readily comparable to the social planner solution discussed in the next section. 2.2 Social planner problem The social planner maximises a weighted sum, W, of the lifetime utilities of all generations, subject to the resource constraint and the available supply of housing in each period. It takes the past consumption and housing of the initial old, c 1, 1 and h 1, 1, as given. There is no need to distinguish between property and rented houses here, so we define housing at time t as h 1,t for the young and h 2,t for the old. The social planner problem is as follows: max W = [c 1,t,c 2,t,h 1,t,h 2,t ] t=0 = ω 1 [U(c 1, 1, h 1, 1 ) + βu(c 2,0, h 2,0 )] + [U(c 1,t, h 1,t ) + βu(c 2,t+1, h 2,t+1 )] t= 1 [U(c 1,t, h 1,t ) + βu(c 2,t+1, h 2,t+1 )] t=0 s.t. c 1,t + c 2,t = y (21) h 1,t + h 2,t = H (22) 2 As we show later on, if rental markets were absent the old generation would purchase housing for its utility and leave its purchase as a positive bequest to the next old generation despite not having any altruistic motives. 6
for t = {0, 1, 2,..., }. The weights [ ] t= 1 are known finite numbers. The first-order conditions for t = {0, 1, 2,..., } are: = Λ 1,t (23) β 1 = Λ 1,t (24) = Λ 2,t (25) β 1 U h2,t = Λ 2,t (26) where Λ 1,t and Λ 2,t are Lagrange multipliers on the resource and housing constraints. The first-order conditions imply that = β 1 (27) for t = {0, 1, 2,..., }. 3 = β 1 U h2,t (28) The intuition is straightforward. The social planner has two fixed pies that it wants to allocate optimally across ages: output and the housing stock. To do so, it must ensure that both young and old receive the same marginal utility from each pie (taking into account the weights on each generation), because this means that there is no reallocation of the pies that will benefit one age more than it hurts the other. It is useful to rearrange the two conditions above as follows: = U c 2,t U h2,t (29) = β 1 (30) A case that deserves special attention is the Golden Rule as defined below: Definition 1 The Golden Rule is the stationary allocation that maximizes the lifetime utility of new born generations at birth and gives to the initial old the same allocation given to future old. 3 Notice that there is no conflict between the utility of the initial old and the utility of future generations because the past consumption and housing of the intial old are given so do not enter into the social planner s first-order conditions. 7
The Golden Rule is the case where the weights are constant, as summarized in the following proposition. 4 Proposition 4 With constant weights, the solution to the social planner problem is the Golden Rule allocation. Proof. With constant weights, Equation (30) becomes = β. (31) This condition and (29) are the first order conditions to U(c 1,t, h 1,t ) + βu(c 2,t, h 2,t ) subject to the feasibility constraints (64) and (65) at any t. Furthermore (31) and (29) imply a stationary allocation, so the initial old has the same allocation as the old in future generations. Under stationarity, these conditions also maximize the lifetime utility of a new generation born at t: U(c 1,t, h 1,t ) + βu(c 2,t+1, h 2,t+1 ). 2.3 Competitive equilibrium: efficiency and distribution We show that the competitive equilibrium is Pareto efficient but redistributes consumption from the young to the old as compared to the Golden Rule allocation. We focus on the Golden Rule because besides being widely used, it is the equilibrium of our model without housing but with an alternative means of intergenerational transfers (e.g. money, bonds). 2.3.1 Efficiency of the competitive equilibrium Proposition 5 The stationary decentralized equilibrium is Pareto efficient. Proof. We need to show that (i) there exist weights [ ] t= 1 such that the planner first-order conditions (29) and (30) are satisfied in a stationary competitive equilibrium, and (ii) that these weights are such that (29) is satisfied: 1 < 1. 5. To show this, first note that (18) and (20) imply that = r t = U h2,t U c 1,t = U c 2,t U h2,t (32) 4 Trivially, constant weights need to be such that the problem is bounded. To this end, let the sum end at T and take the limit for T to infinity. Let the constant wight be a positive number devided by T. 5 Condition (ii) is necessary to ensure that the planner objective function is finite for any feasible allocation. It is equivalent to the Balasko and Shell (1980) criterion for Pareto optimality: t=1 where r ce is the interest rate in the competitive allocation; in our case r ce = U h 1,t β p t 8 t τ=1 (1 + rce τ ) = +,
To show that there exist weights such that (30) is also satisfied, note that under stationarity (17) and (27) imply that = β + U h 1,t p t = β 1 (33) It is then possible to pick Pareto weights such that the competitive equilibrium matches the planner allocation: 1 1 + U h 1,t β p t > 1. (34) Since the Pareto weights are decreasing, the planner objective function remains finite. The intuition for this result is that an allocation with decreasing weights is one in which future generations cannot be made better offer without lowering the utility of the initial old. Gale (1973) refers to this as the classical case and to the case of increasing weights as the Samuelson case. The latter case cannot coincide with a Pareto efficient allocation because increasing weights make the planner objective function infinite, so that the maximization problem is ill defined. 2.3.2 Distribution in the competitive equilibrium Proposition 6 The stationary decentralized equilibrium violates the Golden Rule. Proof. The Golden Rule is the case of constant weights = 1, but we have shown above that only a decreasing set of weights are consistent with the competitive equilibrium. The intuition for this result is that in the competitive equilibrium, the young invests too much relative to the Golden Rule and has too little left for the consumption good: too high. Intuitively, too much of income when young is invested in the housing market. The ratio of Pareto weights exceeds the Golden Rule ratio of 1 by the amount is β p t. Since this term need not be close to zero, the competitive equilibrium can imply substantial redistribution relative to the Golden Rule. This result is notable because, without housing, this economy would not violate the Golden Rule. Indeed, it is trivial to replace housing with another means for executing intergenerational transfers which does not give utility, such as a bond or fiat money, and show that the Golden Rule would be satisfied; see Champ et al. (2011). The key for the inefficiency is that housing is both a means of saving and a good that gives utility. This implies that in the second period of life, people demand housing while they would not want to purchase a bond or money. Hence the return on housing is high : it exceeds the Golden Rule return of 9
1. This result is true even though we have rental markets. A further source of inefficiency arises when rental markets are shut down, as we show in the next section. 3 The role of rental markets To highlight the role of rental markets, we shut them down in this section: h r 1,t = h r 2,t = 0. As an implication, the old will have to buy housing and leave their property as a bequest, h p 2,t, even though they are not altruistic. 6 With rental markets closed, the problem of a young individual born in time t is: max U(c 1,t, h p 1,t) + βu(c 2,t+1, h p 2,t+1) (35) [c 1,t,c 2,t+1,h p 1,t,,hp 2,t+1 ] s.t. p t h p 1,t + c 1,t y (36) c 2,t+1 + p t+1 h p 2,t+1 p t+1 (h p 1,t + h p 2,t) (37) h p 1,t 0 (38) h p 2,t+1 0. (39) The first-order conditions are now = U h 1,t p t + β+1 p t+1 p t (40) U h2,t+1 = +1 p t+1 (41) This last equation shows the trade-off faced by the old: to increase housing, he needs to buy it and give up +1 p t+1 consumption. This was not the trade-off in the model with renting; there it was not necessary to buy housing when old in order to enjoy its service. As the next proposition shows, this intra-temporal condition is not consistent with any Pareto efficient allocation. Proposition 7 The decentralized equilibrium is inconsistent with any interior solution to the planner problem. It is therefore Pareto inefficient. Proof. We rearrange (40) as = 1 p t + β U c 2,t+1 p t+1 p t. 6 The amount of housing bequests observed in practice might be due to altruism, but it may also reflect imperfect rental markets. This motivates us to shut down rental markets in this section. 10
Then, to satisfy the efficiency condition (29) it has to be that Then, from quation (41) we have that 1 p t + β U c 2,t+1 p t+1 p t = U c 2,t U h2,t. 1 p t + β U c 2,t+1 p t+1 p t = 1 p t. This last condition is only satisfied with Uc 2,t+1 = 0. Since housing and consumption are U p h 1,t bounded by the feasibility constraint, this last condition requires either =, p t+1 = 0 or p t =. These price conditions violate condition (41) for a competitive equilibrium. = implies h p 1,t = 0 which is not an interior solution to the planner problem. p t+1 p t+1 This proposition rules out all Pareto efficient allocations other than the extreme one where h p 1,t = 0. Notice that for this to be efficient, from condition (29), we require c 1,t = 0. So the only Pareto efficient solution not ruled out by this proposition is the corner one where the old consume all consumption goods and housing and the young has nothing. Pareto inefficiency is driven by the presence of an intratemporal wedge. This wedge arises because in the last period of life households want to purchase both consumption goods and housing (which provides utility), but the latter has the undesirable consequence that they will be forced to die with positive housing assets because renting is not an option. Avoiding this unintentional bequest through the introduction of rental markets would lead to a Pareto improvement. Rental markets thus eliminate a Pareto inefficiency. Interestingly, this intratemporal wedge does not seem to have any counterpart in an infinitely-lived representative agent models and differs from other inefficiencies discussed in OLG models. Notice that the Pareto efficient allocations that we ruled out also include the Golden Rule case where the weights are constant. 4 Introducing capital In this section, we investigate the implications of housing and rental markets for dynamic inefficiency of capital. It is shown that capital investment is necessarily dynamically efficient because it must match the return on housing, which is greater than 1. We first set out the competitive equilibrium and the social planner problem before proving dynamic efficiency and discussing some policy implications. 11
4.1 Competitive equilibrium with capital The model contains two sectors: a household sector and a sector devoted to production of a single output good. Each household inelastically supplies a unit of labour when young. Firms combine labour with capital to produce output using a constant-returns-to-scale production function y t = f(k t ), where k t is capital per person, y t is output per person (aggregates if population is normalized to 1) and f(.) is twice continuously differentiable, strictly concave and satisfies the Inada conditions. Capital depreciates at rate 0 < δ < 1 per period. Households Households receive wage income w t when young. This income can be used to purchase housing, rent, or to buy capital k t+1 which pays a return 1 + rt+1 k δ next period. The budget constraints of a young individual born at time are now: 7 k t+1 + p t h p 1,t + r t h r 1,t + c 1,t w t (42) c 2,t+1 + p t+1 h p 2,t+1 + r t+1 h r 2,t+1 (1 + rt+1 k δ)k t+1 + p t+1 (h p 1,t + h p 2,t) (43) h p 1,t 0 (44) h p 2,t+1 0. (45) First order conditions c 1,t : = λ 1,t (46) c 2,t+1 : β+1 = λ 2,t+1 (47) h p 1,t : = λ 1,t p t λ 2,t+1 p t+1 λ h,1,t (48) h r 1,t : = λ 1,t r t (49) h p 2,t+1 : βu h2,t+1 = λ 2,t+1 p t+1 λ h,2,t+1 (50) h r 2,t+1 : βu h2,t+1 = λ 2,t+1 r t+1 (51) k t+1 : λ 1,t = λ 2,t+1 (1 + r k t+1 δ) (52) 7 As with the endowment economy above, housing bequests h p 2,t will be zero in a model with renting. 12
where λ 1,t and λ 2,t+1 are the Lagrange multipliers on the budget constraints of the young and old, respectively, while λ h,1,t and λ h,2,t+1 are the Lagrange multipliers on the nonnegativity constraints on house purchases. They are all positive and the complementary slackness conditions on the constraints hold. These first order conditions, together with the market clearing conditions, define a competitive equilibrium. Firms A representative firm hires capital and labour in competitive markets to maximise current period profits. The real wage and return on capital are given by w t = f(k t ) f (k t )k t (53) rt k = f (k t ). (54) Market-clearing conditions Clearing in the housing and rental markets is unchanged, but goods market clearing now depends on investment in productive capital: h p 1,t + h r 1,t + h p 2,t + h r 2,t = H (55) h r 1,t + h r 2,t = 0 (56) c 1,t + c 2,t + k t+1 = f(k t ) + (1 δ)k t. (57) Characterization Since Propositions 1 to 3 remain true in the model with capital, the first order conditions can be simplified to = p t β+1 p t+1, (58) = r t, (59) βu h2,t+1 = β+1 p t+1 λ h,2,t+1, (60) βu h2,t+1 = β+1 r t+1, (61) = β+1 (1 + rt+1 k δ). (62) The final expression (62) is the Euler equation for capital; all other first-order conditions are unchanged relative to the exchange economy setting. Comparing (62) and (58) shows that an arbitrage condition between housing and capital investment places the following restriction on the equilibrium return on capital: 13
1 + rt+1 k δ = p ( t+1 1 + p t β+1 p t+1 ) (63) This arbitrage condition shows that the gross return on capital (net of depreciation) will exceed 1 in a stationary equilibrium where house prices are bounded. Cass (1972) has shown that a return greater than or equal to 1 is sufficient to ensure that capital accumulation is dynamically efficient in our economy. This provides the intuition for our result that capital accumulation is dynamically efficient and the competitive equilibrium Pareto efficient. We provide a formal proof in the next section after setting out the social planner problem. 4.2 Social planner problem The social planner problem with capital is as follows: s.t. max W = [c 1,t,c 2,t,h 1,t,h 2,t ] t=0 = ω 1 [U(c 1, 1, h 1, 1 ) + βu(c 2,0, h 2,0 )] + [U(c 1,t, h 1,t ) + βu(c 2,t+1, h 2,t+1 )] t= 1 [U(c 1,t, h 1,t ) + βu(c 2,t+1, h 2,t+1 )] t=0 c 1,t + c 2,t + k t+1 = f(k t ) + (1 δ)k t (64) h 1,t + h 2,t = H (65) The first-order conditions for t = {0, 1, 2,..., } are: = Λ 1,t (66) β 1 = Λ 1,t (67) = Λ 2,t (68) β 1 U h2,t = Λ 2,t (69) Λ 1,t = Λ 1,t+1 (1 + rt+1 k δ) (70) where Λ 1,t and Λ 2,t are Lagrange multipliers on the resource and housing constraints. The first-order conditions imply that = β 1 (71) 14
= β 1 U h2,t (72) = β(1 + r k t+1 δ)+1 (73) for t = {0, 1, 2,..., }. It is useful to simplify the planner s first-order conditions as follows: = U c 2,t U h2,t (74) = β 1 (75) 1 + r k t+1 δ = 1 +1 (76) Proposition 8 The stationary decentralized equilibrium with capital is Pareto efficient. Proof. We need to show that (i) there exist weights [ ] t= 1 such that the planner first-order conditions (74) and (75) are satisfied in a stationary competitive equilibrium, and (ii) that these weights are such that (74) is satisfied: 1 < 1. 8. To show this, first note that (59) and (61) imply that = r t = U h2,t U c 1,t = U c 2,t U h2,t (77) To show that there exist weights such that (75) is also satisfied, note that under stationarity (58) and (71) imply = β + U h 1,t p t = β 1 (78) It is then possible to pick Pareto weights such that the competitive equilibrium matches the planner allocation: 1 1 + U h 1,t β p t > 1. (79) With these weights and stationarity (76) tells us that 1 + rt+1 k δ = 1 = 1 > 1. Hence, there is dynamic efficiency and the competitive equilibrium is Pareto efficient. 8 Condition (ii) is necessary to ensure that the planner objective function is finite for any feasible allocation and also ensures that capital accumulation is dynamically efficient. It is a case of the Balasko and Shell (1980) criterion for Pareto optimality in a production economy: t=1 t τ=1 (1 + rk t δ) = +, and was first shown by Cass (1972). In our case, 1 + r k δ = 1 + U h 1,t β p t 15
Intuitively, the high return on housing forces the return on capital to a similarly high level and thus makes capital accumulation dynamically efficient. In turn, this ensures that the competitive equilibrium is Pareto efficient. Crucially, the same result does not hold in an economy with no housing: capital accumulation can be dynamically inefficient. 4.3 Policy implications The above result on dynamic efficiency is interesting because it shows that government interventions to cure supposed overaccumulation of capital, such as government debt, social security and pensions, might not be necessary. It may also help to explain why economies do not appear to overaccumulate capital in practice. 5 Conclusions We have shown three main results. First, housing drastically alters the competitive equilibrium relative to the case where fixed-supply fiat money (or bonds) is the only asset. This is because the competitive equilibrium induces a great deal of redistribution of consumption from the young to the old since housing delivers a rate of return higher than the Golden Rule value of 1. Second, if rental markets are closed, the competitive equilibrium is Pareto inefficient due to the presence of a wedge in the intratemporal trade-off between consumption and housing in the last period of life. Finally, when productive capital is added to the model, we find that the stationary competitive equilibrium is dynamically efficient since the arbitrage condition between housing and capital investment pushes the return on capital above 1. This is in stark contrast to the model without housing where dynamic inefficiency is possible. References [1] Balasko, Y. and Shell, K. 1980. The overlapping generations model I: The case of pure exchange without money. Journal of Economic Theory 23, 281-306. [2] Cass, D. 1972. On capital accumulation in the aggregative neoclassical model of economic growth: a complete characterization. Journal of Economic Theory 4, pp. 200-213. [3] Champ, B., Freeman, S. and Haslag, J. 2011. Modeling monetary economies. Cambridge University Press: Cambridge, MA. 16
[4] Diamond, P. 1965. National debt in a neoclassical growth model. American Economic Review 55, pp. 1126-50. [5] Gale, D. 1973. Pure exchange equilibrium of dynamic economic models. Journal of Economic Theory 6(1), pp. 12-36. [6] Geanakoplos, J. 2008. Overlapping generations models of general equilibrium. Cowles Foundation Discussion Paper 1663. [7] Samuelson, P. 1958. An exact consumption-loan model with or without the social contrivance of money. Journal of Political Economy 66(6), pp. 467-482. 17