ramsey con externo

7/29/2019 Ramsey Con Externo

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11.1. Market economy with a public sector 381

General equilibrium

The increase in the capital stock, per time unit equals aggregate grosssaving:

= = ( T) 0 0 given(11.3)

We assume is proportional to the work force measured in efficiency units,that is = T where 0 is decided by the government. In growth-corrected form the dynamic aggregate resource constraint (11.3) then is

= () (+ + ) 0 0 given, (11.4)

where (T) T, T and is the production functionon intensive form, 0 0 00 0 In view of the assumption that satisfiesthe Inada conditions, we have

lim0

0() = lim

0() = 0

Since marginal utility of private consumption is by assumption not af-fected by the Keynes-Ramsey rule of the household will be as if there

were no government sector:

=1

( )

In equilibrium the real interest rate, equals 0() In terms of technology-corrected consumption the Keynes-Ramsey rule thus becomes

=1

h0()

i (11.5)

In terms of the transversality condition of the household can be written

lim

0(0()) = 0 (11.6)

The phase diagram of the dynamic system (11.4) - (11.5) is shown in Fig.11.1. It is assumed that is of moderate size compared to the productivecapacity of the economy so as to not rule out the existence of a steady state.

Apart from a vertical downward shift of the

= 0 locus, due to 0 thephase diagram is similar to that of the Ramsey model without government.The lump-sum taxes have no effects on resource allocation at all. Indeed,

C. Gr ot h, Lect ure not es i n macr oeconomi cs, ( mi meo) 2011


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11.1. Market economy with a public sector 385

budget restriction. In this setting, whether income takes the form of dispos-able income or transfers does not matter and so there are no real effects onthe economy. If the model were extended with endogenous labor supply, theresult would be different.

A capital income tax

It is different when it comes to a tax on capital income because saving in theRamsey model responds to incentives. Consider a capital income tax at therate , 0 1 The households dynamic budget identity now reads

= [(1 ) ] + + 0 given.As above, is the per capita lump-sum transfer. In view of a balancedbudget, we have at the aggregate level = So = TheNo-Ponzi-Game condition is changed to

lim

0 [(1)] 0

and the Keynes-Ramsey rule becomes

=1

[(1 ) ]

In general equilibrium we get

=1

h(1 )(

0() ) i

(11.11)

The differential equation for is again (11.7).In steady state we get (0() )(1 ) = + , that is,

0() = +

1 + +

where the last inequality comes from the parameter condition (A1). Because00 0, the new is lower than if = 0. Consequently, consumption inthe long term becomes lower as well.3 The resulting resource allocation is

not Pareto optimal. There exist an alternative technically feasible resourceallocation that makes everyone in society better off. This is because thecapital income tax implies a wedge between the marginal transformationrate in production over time, 0() , and the marginal transformationrate over time, (1 )(0() ), which consumers adapt to.

3 In the Solow growth model a capital income tax, which finances lump-sum transfers,will have no effect. This is because saving does not respond to incentives in that model.In the Diamond OLG model a capital income tax, which finances lump-sum transfers tothe old generation, has an ambiguous effect on capital accumulation, cf. Exercise 5.?? inChapter 5.



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390 CHAPTER 11. APPLICATIONS OF THE RAMSEY MODEL

*k

GRk k k

'E

c

*'c

0new c

0k

B *'k

E

A *c

0

F G

Figure 11.4: Phase portrait of an unanticipated temporary rise

This analysis illustrates that when economic agents behavior depend on

forward-looking expectations, a credible announcement of a future changein policy has an effect already before the new policy is implemented. Sucheffects are known as announcement effects or anticipation effects.

An unanticipated temporary rise in

Once again we change the scenario. The economy with low taxation hasbeen in steady state up until time 0. Then the new tax-transfer scheme isunexpectedly introduced. At the same time it is credibly announced that thehigh taxation of capital income and the corresponding transfers will cease at

time 1 0. The phase diagram in Fig. 11.4 illustrates the evolution ofthe economy for 0 For 1 the dynamics are governed by (11.7) and(11.11) with the old again starting from whatever value obtained by attime 1

In the time interval [0 1) the new, temporary dynamics with the high0 and high transfers rule. Yet the path that the economy takes immediatelyafter time 0 is different from what it would be without the information thatthe new tax-transfers scheme is only temporary. Indeed, the expectation ofthe future shift to a higher after-tax rate of return and cease of high trans-fers implies lower present value of expected future labor and transfer earnings



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11.2. Learning by investing andfirst-best policy 395

where is aggregate net investment and =

P

6

The idea is that investment the production of capital goods as anunintended by-product results in experience or what we may call on-the-joblearning. This adds to the knowledge about how to produce the capital goodsin a cost-efficient way and how to design them so that in combination withlabor they are more productive and better satisfy the needs of the users.Moreover, as emphasized by Arrow, each new machine produced and putinto use is capable of changing the environment in which production takesplace, so that learning is taking place with continually new stimuli (Arrow,1962).

The learning is assumed to benefi

t essentially allfi

rms in the economy.There are knowledge spillovers across firms and these spillovers are reason-ably fast relative to the time horizon relevant for growth theory. In ourmacroeconomic approach both and are in fact assumed to be exactlythe same for all firms in the economy. That is, in this specification the firmsproducing consumption-goods benefit from the learning just as much as thefirms producing capital-goods.

The parameter indicates the elasticity of the general technology level, with respect to cumulative aggregate net investment and is named thelearning parameter. Whereas Arrow assumes 1 Romer focuses on thecase = 1 The case of 1 is ruled out since it would lead to explosivegrowth (infinite output in finite time) and is therefore not plausible.7

The individual firm

From now we suppress the time index when not needed for clarity. Considerfirm Its maximization of profits, = ( ) ( + ) leadsto the first-order conditions

= 1( ) ( + ) = 0 (11.21)

= 2( ) = 0

Behind (11.21) is the presumption that eachfi

rm is small relative to theeconomy as a whole, so that each firms investment has a negligible effecton the economy-wide technology level . Since is homogeneous of degreeone, by Eulers theorem, the first-order partial derivatives, 1 and 2 arehomogeneous of degree zero. Thus, we can write (11.21) as

1( ) = + (11.22)

6 For arbitrary units of measurement for labor and output the hypothesis is =

0 In (11.20) measurement units are chosen such that = 1 .7 Empirical evidence of learning-by-doing and learning-by-investing is briefly discussed

in Bibliographic notes.



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Log-differentiating w.r.t. in (11.48) and combining with (11.49) givesthe social planners Keynes-Ramsey rule,

=1

((1 ) ) (11.51)

We see that This is because the social planner internalizes theeconomy-wide learning effect associated with capital investment, that is, thesocial planner takes into account that the social marginal product of capitalis = (1 ) 1(1 ) To ensure bounded intertemporal utility wesharpen (A1) to

(1

) (A1)To find the time path of, note that the dynamic resource constraint (11.46)can be written

= ((1 ) ) 0

in view of (11.51). By the general solution formula (11.43) this has thesolution

= (0 0

(1 ) )((1)) +

0(1 )

(11.52)

In view of (11.49), in an interior optimal solution the time path of the adjoint

variable is = 0

[((1)]

where 0 = 0 0 by (11.48) Thus, the conjectured transversality condi-

tion (11.50) implieslim

((1)) = 0 (11.53)

where we have eliminated 0 To ensure that this is satisfied, we multiply from (11.52) by ((1)) to get

((1)) = 0

0(1 )

+0

(1 ) [((1))]

0 0(1 )

for

since, by (A1), + = (1 ) in view of (11.51). Thus,(11.53) is only satisfied if

0 = ((1 ) )0 (11.54)

Inserting this solution for 0 into (11.52), we get

=0

(1 ) = 0



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11.3. Concluding remarks 409

11.3 Concluding remarks

11.4 Exercises



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