Consider an OLG economy. People receive y > 0 when young and nothing when old. T
ID: 1196181 • Letter: C
Question
Consider an OLG economy. People receive y > 0 when young and nothing when old. The only asset
available in the economy is capital. One unit of consumption today could be transformed to x > 0 units of
consumption tomorrow through a given linear technology. Preferences are given by the following lifetime
utility function:u ( c 1 ) + u ( c 2 )
1. Set up the consumer’s problem. This is going to be considered as the standard problem
2. Identify the givens and the choice variables of the problem
3. Characterize the solution of the standard problem. That is, find the first-order conditions of the
maximization problem
4. Discuss the intuition of the Euler equation and of the lifetime budget constraint
5. Findalgebraicformulasfortheoptimalconsumptionwhenyoung,whenold,andforcapitalholdings.
Discuss the intuition behind each algebraic formula. That is, think on how each variable that is part
of the formula affects consumption when young, when old, and capital holdings. Also think on why
some key variables ( y , or x ), if it is the case, are missing in each formula
6. Draw the graph introduced in class (see Quiz 4 for further references) and label the curves
7. Consider x = 1.2 , y = 5 , = 0.9 , and u () = ln () , to find numerical values for the optimal consump-
tion when young, when old, and for capital holdings
8. In what follows you will be asked to modified the standard problem. Treat each case separately and
always considering the standard problem as the starting point. Your job here is thinking about the
implications of each modification in parts 1-5.
(a) The implicit assumption in the standard problem is that capital fully depreciates once it is used
in the production of second-period consumption. Now, suppose that capital depreciates at rate
( 0,1 ) after production takes place
(b) Suppose the government steps in and now charges a proportional tax to capital holdings
(c) Suppose the government charges, instead, a lump-sum tax T
(d) Use your graph in (6), that is, move the curves if you consider it as necessary) to discuss the
modification (a). Explain intuitively
(e) Use your graph in (6), that is, move the curves if you consider it as necessary) to discuss the
modification (b). Explain intuitively
(f) Use your graph in (6), that is, move the curves if you consider it as necessary) to discuss the
modification (c). Explain intuitively
Explanation / Answer
Time is discrete, t = 1, 2, 3, . . . and the economy (but not its people) lives forever. In each period there is a single, nonstorable consumption good. In each time period a new generation (of measure 1) is born, which we index by its date of birth. People live for two periods and then die.
(e t t , e t t+1 ): generation tís endowment of the consumption good in the Örst and second period of their live. (c t t , c t t+1 ): consumption allocation of generation t. In time t there are two generations alive: 1 One old generation t 1 that has endowment e t1 t and consumption c t1 t . 2 One young generation t that has endowment e t t and consumption c t t . In period 1 there is an initial old generation 0 that has endowment e 0 1 and consumes c 0 1 .
Outside money: money that is, on net, an asset of the private economy. This includes Öat currency issued by the government. Inside money (such as bank deposits) is both an asset as well as a liability of the private sector (in the case of deposits an asset of the deposit holder, a liability to the bank). If m 0, then m can be interpreted as Öat money. If m < 0, one should envision the initial old people having borrowed from some institution (outside the model) and m is the amount to be repaid.
Preferences of individuals are representable by: ut(c) = U(c t t ) + U(c t t+1 ) Preferences of the initial old generation is representable by: u0(c) = U(c 0 1 ) We shall assume that U is strictly increasing, strictly concave, and twice continuously di§erentiable.
In the presence of money (m 6= 0), we will take money to be the numeraire. This is important since we can only normalize the price of one commodity to 1. With money, no further normalizations are admissible. Let pt be the price of one unit of the consumption good at period t.
Trade takes place sequentially in spot markets for consumption goods that open in each period. In addition, there is an asset market through which individuals do their saving. Let rt+1 be the interest rate from period t to period t + 1 and s t t be the savings of generation t from period t to period t + 1. We will consider assets that cost one unit of consumption in period t and deliver 1 + rt+1 units tomorrow. Those assets are easier to handle than zero-coupon bonds if the asset at hand is Öat money. However, both assets have identical implications.
Given m, an Arrow-Debreu equilibrium is an allocation cˆ 0 1 , f(cˆ t t , cˆ t t+1 )g t=1 and prices fptg t=1 such that 1 Given fptg t=1 , for each t 1, (cˆ t t , cˆ t t+1 ) solves: max (c t t ,c t t+1 )0 ut(c t t , c t t+1 ) s.t. pt c t t + pt+1c t t+1 pt e t t + pt+1e t t+1 2 Given p1, cˆ 0 1 solves: max c 0 1 u0(c 0 1 ) s.t. p1c 0 1 p1e 0 1 + m 3 For all t 1 (resource balance or goods market clearing): c t1 t + c t t = e t1 t + e t t for all t 1
Given m, a sequential markets equilibrium is an allocation cˆ 0 1 , f(cˆ t t , cˆ t t+1 ,sˆ t t )g t=1 and interest rates frtg t=1 such that: 1 Given frtg t=1 for each t 1, (cˆ t t , cˆ t t+1 ,sˆ t t ) solves: max (c t t ,c t t+1 )0,s t t ut(c t t , c t t+1 ) s.t. c t t + s t t e t t c t t+1 e t t+1 + (1 + rt+1)s t t 2 Given r1, cˆ 0 1 solves: max c 0 1 u0(c 0 1 ) s.t. c 0 1 e 0 1 + (1 + r1)m 3 For all t 1 (resource balance or goods market clearing): cˆ t1 t + cˆ t t = e t1 t + e t t for all t 1
Given that the period utility function U is strictly increasing, the budget constraints hold with equality. Summing the budget constraints of agents: c t t+1 + c t+1 t+1 + s t+1 t+1 = e t t+1 + e t+1 t+1 + (1 + rt+1)s t t By resource balance: s t+1 t+1 = (1 + rt+1)s t t Doing the same manipulations for generation 0 and 1: s 1 1 = (1 + r1)m
By repeated substitution: s t t = t =1 (1 + r)m The amount of saving (in terms of the period t consumption good) has to equal the value of the outside supply of assets, t =1 (1 + r)m. Interpretation. This condition should appear in the deÖnition of equilibrium. By Walrasílaw however, either the asset market or the good market equilibrium condition is redundant.
For rt+1 > 1, we combine both budget constraints into: c t t + 1 1 + rt+1 c t t+1 = e t t + 1 1 + rt+1 e t t+1 Divide by pt > 0: c t t + pt+1 pt c t t+1 = e t t + pt+1 pt e t t+1 Divide initial old generation by p1 > 0 to obtain: c 0 1 e 0 1 + m p1 Hence, it looks that 1 + rt+1 = pt pt+1 must play a key role.
Given equilibrium Arrow-Debreu prices fptg t=1 , deÖne interest rates: 1 + rt+1 = pt pt+1 1 + r1 = 1 p1 These interest rates induce a sequential markets equilibrium with the same allocation than the Arrow-Debreu equilibrium. Conversely, given equilibrium sequential markets interest rates interest rates frtg t=1 , deÖne Arrow-Debreu prices by p1 = 1 1 + r1 pt+1 = pt 1 + rt+1 These prices induce allocations that are equivalent to the sequential markets equilibrium.
From the equivalence, the return on the asset equals: 1 + rt+1 = pt pt+1 = 1 1 + t+1 (1 + rt+1)(1 + t+1) = 1 rt+1 t+1 where t+1 is the ináation rate from period t to t + 1. The real return on money equals the negative of the ináation rate.
Using: p1 = 1 1 + r1 pt+1 = pt 1 + rt+1 with repeated substitution delivers: pt = 1 t =1 (1 + r) ) t =1 (1 + r) = 1 pt Interpretation.
Now, note that we argued before that s t t = t =1 (1 + r)m Hence: s t t = m pt You can think about this last condition both as: 1 An equilibrium condition. 2 A money demand function.
DeÖne the excess demand functions: y (pt , pt+1) = c t t (pt , pt+1) w1 z(pt , pt+1) = c t t+1 (pt , pt+1) w2 These functions summarize, for given prices, consumer optimization. y and z only depend on pt+1 pt , but not on pt and pt+1 separately (the excess demand functions are homogeneous of degree zero in prices). Varying pt+1 pt between 0 and (not inclusive), we obtain the o§er curve: a locus of optimal excess demands in (y, z) space: (y, f (y )) f can be a multi-valued correspondence. A point on the o§er curve is an optimal excess demand function for some pt+1 pt 2 (0, ).
Pick an initial price p1 (this is NOT a normalization since p1 determines the real value of money m p1 the initial old generation is endowed with; we have already normalized the price of money). Hence, we know z0(p1, m). This determines y (p1, p2). 2 From the o§er curve, we determine z(p1, p2) 2 f (y (p1, p2)). Note that if f is a correspondence then there are multiple choices for z. 3 Once we know z(p1, p2), we can Önd y (p2, p3) and so forth. In this way we determine the entire equilibrium consumption allocation: c 0 1 = z0(p1, m) + w2 c t t = y (pt , pt+1) + w1 c t t+1 = z(pt , pt+1) + w2 4 Equilibrium prices can then be found, given p1.
Any initial p1 that induces sequences c 0 1 , f(c t t , c t t+1 ), ptg t=1 such that the consumption sequence satisÖes c t1 t , c t t 0 is an equilibrium for given money stock. This already indicates the possibility of a lot of equilibria for this model. In general, the price ratio supporting the autarkic equilibrium satisÖes: pt pt+1 = U 0 (e t t ) U0(e t t+1 ) = U 0 (w1) U0(w2) and this ratio represents the slope of the o§er curve at the origin.
Intitution
Take the autarkic allocation and try to construct a Pareto improvement. In particular, give additional 0 > 0 units of consumption to the initial old generation. This obviously improves this generationís life. From resource feasibility this requires taking away 0 from generation 1 in their Örst period of life. To make them not worse o§, they have to receive 1 in additional consumption in their second period of life, with 1 satisfying 0U 0 (e 1 1 ) = 1U 0 (e 1 2 ) or 1 = 0 U 0 (e 1 1 ) U0(e 1 2 ) = 0(1 + r2) 1 > 0
In general, the required transfers in the second period of generation tís life to compensate for the reduction of Örst period consumption: t = 0 t =1 (1 + r+1) 1 Such a scheme does not work if the economy ends at Öne time T since the last generation (that lives only through youth) is worse o§. But as our economy extends forever, such an intergenerational transfer scheme is feasible provided that the t do not grow too fast, that is, if interest rates are su¢ ciently small. But if such a transfer scheme is feasible, then we found a Pareto improvement over the original autarkic allocation, and hence the autarkic equilibrium allocation is not Pareto e¢ cient.