Lecture 15: Comparative Statics Using The Ramsey Model

Spring Semester ’12-’13 Akila Weerapana

Lecture 15: Comparative Statics Using The Ramsey Model

I. INTRODUCTION

• In the last class, we derived the neo-classical growth model, also known as the Ramsey model, which endogenized the consumption decision. We derived a de-centralized version of the model with households and individuals making consumption decisions and firms making production decisions separately and then combined these to do a phase diagram analysis, where we compared the insights to the insights we got from the Solow model.

• We showed that the dynamics of the model indicated that the economy is saddle-path stable and that the transversality conditions will ensure that the economy gravitates towards the finite steady state of the model.

• In today’s class, we focus on comparative statics using the Ramsey model. In particular we will look at how changes in the various key parameters of the model affect the short-run and long-run path of consumption and capital in the economy.

II. COMPARISON TO THE SOLOW MODEL

• Before doing the comparative statics, it is a good idea to compare the predictions of the Ramsey model to the predictions of the Solow model. In the Solow model, we had a steady state of k

=

(

)

1−α

. In that model consumption was just a fixed fraction of income so steady state consumption which is c

= (1 − s)y

can be written as k

= (1 − s) (

s n+δ

)

1−α

• In the Ramsey model, the ˙k = 0 locus is given by the equation c

= k

− (n + δ)k

and the

˙c = 0 locus is given by the equation αk

= (ρ + δ). Putting these together, we can see that the Ramsey model’s steady state is at k

=

(

)

1−α

and c

= k

− (n + δ)k

, which simplifies to

c

=

( α

ρ + δ )

1−α

(

1 − (n + δ)α ρ + δ

)

• In the Solow model s is exogenous and could be higher or lower than α. But we can get some insight into the comparison between the two model by considering the optimal choice of s (in terms of maximizing consumption), which is known as the Golden Rule. As you were asked to show on the problem set, the Golden Rule saving rate is s = α.

• In that case, the comparison of steady state capital per worker becomes straightforward:

k

≡

( α

ρ + δ )

< k

≡

( α

n + δ

)

• The comparison of steady state consumption per worker is more complicated because (

1 −

)

>

(1 − α) but y

< y

. So we need to look closer at the consumption value for the Ramsey equation, and how it changes when ρ changes.

c

=

( α ρ + δ

)

(

1 − (n + δ)α ρ + δ

)

∂c

∂ρ =

( α ρ + δ

)

[

(n + δ)α (ρ + δ)

] +

(

1 − (n + δ)α ρ + δ

) [( α 1 − α

) ( α ρ + δ

)

(

−α (ρ + δ)

)]

=

( α ρ + δ

)

1−α

( α (ρ + δ)

) [

(n + δ) − (

1 − (n + δ)α ρ + δ

) ( α 1 − α

) ( ρ + δ α

)]

=

( α ρ + δ

)

1−α

( α (ρ + δ)

) [

(n + δ) − (

1 − (n + δ)α ρ + δ

) ( ρ + δ 1 − α

)]

=

( α ρ + δ

)

1−α

( α (ρ + δ)

) [

(n + δ) −

( ρ + δ 1 − α

)

+ (n + δ)

( α

1 − α )]

=

( α ρ + δ

)

( α (ρ + δ)

) [( n + δ 1 − α

)

−

( ρ + δ 1 − α

)]

=

( α ρ + δ

)

1−α

( α (ρ + δ)

) ( n − ρ 1 − α

)

• This derivative is zero at n = ρ and the second derivative is clearly negative since an increase in ρ will raise the denominator of the first term and the second term and reduce the numerator in the third term. Therefore, we can conclude that the expression c

reaches its maximum value at ρ = n, where we already showed that c

= c

. For ρ > n, the Ramsey model generates lower consumption, capital and output (per-capita) in steady state than the Solow model.

III. COMPARATIVE STATICS

• Let’s begin by considering what happens when there is an increase in ρ. This affects the

˙c locus but not the ˙k locus, shifting it to the left. To get on the new saddle-path towards

equilibrium, the consumption level has to sharply rise to point B before continuing on to the

new steady state at k

.

- 6

c

k

˙k

= 0

˙c

= 0

k

k

c

c

B

 6

?



? -



6-

• Intuitively, this makes sense. An increase in impatience is like a reduction in the saving rate because we consume more up front. The result is that consumption jumps up initially but then falls down to an even lower level than it was before. The resulting time-path for consumption looks like

- 6

ln c

t ln c

ln c

t

t

q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q

An increase in n

• Now consider what happens when there is an increase in the growth rate of the population (household) n. This affects the ˙k = 0 locus and moves it down but has no impact on the

˙c = 0 locus. The only way to get on the new saddle-path towards equilibrium is for the

consumption level to drop sharply to point B. In other words, we move immediately to a new

steady state with the level of capital per worker but a lower level of consumption.

- 6

c

k

˙k

= 0

˙c

= 0

k

c

c

B

 6

?



? -

? 6-

• Intuitively, how do we make sense of this? An increase in n raises the size of the household and increases household utility. So the household is willing to immediately accept lower consumption per individual in the household (i.e. the other individuals save more to allow for the new entrants to consume. The time path for consumption looks as follows.

- 6

ln c

t ln c

ln c

t

t

q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q

An increase in δ

• Now consider what happens when there is an increase in the depreciation rate δ. This affects BOTH the ˙k = 0 locus (moving it down as in the case of a decrease in n) but ALSO moves the ˙c = 0 locus to the left. We know for sure that the end result is a lower steady state capital stock and a lower steady state consumption level but the transition dynamics are less clear.

Depending on whether the new steady is to the left, the right or even on the old stable arm the dynamics will vary.

• For example, if the intersection is on the same stable arm, then consumption does not have to

sharply rise or fall and can smoothly decrease down to the new steady state. If the intersection

is to the right of the old stable arm, then the new stable arm is lower and consumption will initially fall to get onto the new stable arm. Conversely, if the intersection is to the left of the old stable arm, then the new stable arm is higher and consumption will initially rise to get onto that new stable arm. The latter case is shown below

- 6

c

k

˙k

= 0

˙c

= 0

k

c

c

B

 6

?



? -

? 6-

• Intuitively, the ambiguous nature of this result comes because an increase in δ has both an income and substitution effect. A higher δ, means that machines are breaking down at a faster rate and they need to be replaced at a faster rate leaving less resources available for consumption (the income effect). On the other hand, a higher δ reduces the real interest rate (the effective return to saving) αk

− δ and thus makes it less attractive to save (more attractive to spend) the substitution effect. When I moved the ˙c line by more, I was basically saying that the substitution effect dominates the income effect.

• The time path for consumption looks as follows.

- 6

ln c

t ln c

ln c

t

t

q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q