Note Tvc (Transversality condition)

7/27/2019 Note Tvc (Transversality condition)

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Fall 2013. UdeM. Example:

Suppose that individuals choose fctg1t=0 in order to maximize

1Pt=0

11 + t

(ect

) (1)

where ; > 0 subject to the budget constraint

at+1 at = wt + rtat ct (2)

and the no-Ponzi game condition

limn!1

at

t1Qs=1

1

(1 + rs)

0

We let at denote the stock of accumulated asset (debt if negative) at time t; while ct is consumption.

Initial a0 is given; wt denotes the wage, the net interest rate is rt. The consumer inelastically suppliesone unit of labor. Per-period utility in (1) is known as CARA (constant absolute risk aversion) utility.

1. Write the Euler condition. Write the transversality condition of the problem.

The Euler Equation of each individual is

ect =(1 + rt+1)

1 + ect+1

Taking log, you can write

ct+1ct= 1 + 1ct

1 ln( 1 + rt+1

1 + )

Then the consumption of the rich (high ct) grows at a slower pace if rt+1 > : To explain why therich have dierent saving behavior, rst note that the absolute coecient of risk aversion is constantand equal to :1 Then, CARA implies increasing relative risk aversion (hence, decreasing elasticity ofintertemporal substitution). The higher incentive to keep a smooth consumption prole for the richexplains why consumption growth of the rich is lower when rt+1 > : The higher elasticity of substitutionof the poor implies that they consume less to enjoy higher consumption tomorrow. Things are oppositewhen rt+1 < : the consumption of the rich (high ct) grows at a higher pace

The transversality condition takes the form2

limt!1

t(1 + rt)u0(ct)at = 0

The transversality condition (TVC) requires that the product between (1) the derivative of the per-period utility u((1+ rt)at + wt at+1) with respect to the current state variable and (2) the state variableshould not increase as t goes to 1 at a rate faster or equal than 1=:

Why do we need the transversality condition? Wouldnt it be enough to impose that the EulerEquations are satised at all t in order to nd the optimal consumption path? The answer is no. Whenthe horizon is nite (see Problem 1) in addition to the Euler equations, we also need a terminal condition:

1 The coecient of absolute risk aversion is

u

00

(c)u0(c) :

2 See Appendix at the end.

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aT+1 = 0: That is, the individual does not save in the last period. When T = 1 the TVC substitutesfor the missing terminal condition.

The transversality condition states that the value of at, when measured in terms of discounted utility,goes to zero asymptotically. It is important to note that we do not require that at itself converges tozero, only the (shadow) value of a

thas to converge to zero. The TVC ensures that individuals are not

accumulating too many assets and postponing consumption for ever.There is no unique representation of the TVC condition. The TCV can be written in many forms. A

second formulation, using the EE u0(ct1) = u0(ct)(1 + rt) is

limt!1

t1u0(ct1)at = 0 (3)

Changing the timing, write as:limt!1

tu0(ct)at+1 = 0 (4)

The TVC is sometimes written this way because tu0(ct) is equal to the multiplier associated to the budgetconstraint.3 Then, we can write

limt!1tat+1 = 0 (5)

which says, once again, that the limiting value of assets from the perspective of time zero (this is givenby the price at time zero in utility terms multiplied by the quantity at+1) converges to 0.

Lets see a third way of writing the TCV. Using the Euler equation to substitute for u0(ct1) in (3)we obtain

limt!1

t11

(1 + rt1) u0(ct2)at = 0 (6)

Using the EE at t 2; we substitute for u0(ct2). We keep substituting backwards and, after droppingthe constant term u0(c0) > 0; one can write the transversality condition as:

limt!1att1Qs=1

1

(1 + rs) = 0 (7)

Notice how (7) relates to the No-Ponzi game constraint. There is, however, an important dierence.Expression (7) is an optimality condition: it is the innite horizon analogue to the terminal conditionaT+1 = 0. The No-Ponzi game is not an optimality condition: it is instead a constraint imposed by themarkets. It says that the present value of assets must be asymptotically nonnegative: in other terms,individuals should pay the debt asymptotically and should not be able to increase consumption by takingloans to nance the payments of the interests. The debt cannot grow as fast as rt: Acemoglu, Section8.1.2 shows that by using the no-Ponzi-game condition with equality, we can write the intertemporalbudget constraint, which states that the present value of consumption and present value of incomes areequal. Financial markets impose a proper lifetime budget constraint on consumers. If not (for instance,

if consumers could run a Ponzi scheme), consumers would consume more than their incomes and nancialmarkets would eventually loose money. A nal remark: in Appendix A we used the no-Ponzi constraintwith equality. If we used the no-Ponzi constraint with inequality, you would obtain that the present valueof consumption is weakly less than the present value of incomes. This is also ne: nancial markets wouldnot loose money in this case either. In Appendix A we used the no-Ponzi constraint with equality becausewe integrated the fact (by using the transversality condition) that in equilibrium consumers would notaccumulate positive assets for ever.

3 Dene the Lagrangian

L =1X

t=0

tu(ct) + t[(1 + rt)at + wt ct)]

From the rst order condition, tu0(ct) = t. That is, the Lagrange multiplier is the marginal utility of consumption (ameasure of how much another unit of consumption at time t is worth from the perspective of time zero)

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Appendix: Given a general sequential problem

maxfxt+1g

1

t=0

1Xt=0

tF(xt; xt+1) ()

subject to xt+1 2 (xt); x0 is given

:X ! X is the correspondence describing the feasibility constraints. (xt) is the set of feasiblevalues for the state variable next period if today the state is xt 2 X: F : A ! X is the one-period returnfunction. xt is the current state, while xt+1 is the state next period. The state (can be a vector) is acomplete description of the current position of the system. We just need to know xt to know what theagent will do. Knowing past values of xj , j < t does not add information.

In the neoclassical growth model we have that xt = kt and xt+1 = kt+1; while F(xt; xt+1) = u(f(kt) kt+1) and (xt) = [0; f(kt)].

Denition The path fxt+1g1t=0 satises the Transversality Condition if

limt!1

tFx(xt; xt+1)xt = 0

F1 is the vector of marginal returns from increases in the current state variable x: we will see, forexample, that F1 is related to the vector of capital goods prices. Then Fx(:; :)xt is the value of capitalstock at time t: ). Therefore, with the TVC we require that the present discounted value of capital stockgoes to zero as t ! 1. Clearly, in general we do not demand that the capital itself goes to zero. Basically,we do not want to postpone too much utility in the distant future.

We write down the TVC for the growth model.

limt!1

tu0(f(kt) kt+1)f0(kt)| {z }

V a l u e o f o n e e x t r a u n i t

o f c a p i t a l i n d i s c . u t i l . t e r m s .

kt|{z}Total cap. stock

= 0

In other words, the TVC condition requires that the value of the capital stock, when it is measured indiscounted utility terms, goes to zero as t ! 1:

For more on this, see equation (6.32) at page 203 of the book by Acemoglu for a general formulation.

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Note Tvc (Transversality condition)