Lecture notes for Macro I, page 1

ŠJohn Hassler, revised February 5, 1997.

Growth

The Samuelson-Diamond-Blanchard OLG model

Households

Live for two periods and solve

1

Firms

Hire labor and capital on competitive market and combine them in a CRS production function to produce the only good. This is sold on a competitive market to the households.

Firms solve

2

where *A**t *is a productivity index that grows geometrically at rate
*g. *We can think of *AL *as the number of *effective
*units of labor. Also *L *grows at a geometric rate *n.
*So

3

The competitive market imply that factors are
paid their marginal products. *r = F**K
*and *w=F**AL*.

Since *F *is H(1) we have

4

So *f *is production per effective labor
unit and *k *is capital per effective labor unit. Note that

5

Since we CRS and competitive markets firms make zero profits. We can write this as

6

Note that we now see that both wages and interest rates are determined by the current capital to effective labor ratio. We thus write

7

Capital market

The labor income of a young generation in *t *equals
*w**t **A**t **L**t** *. Part of this is consumed (*c*1,*t*) and the rest is saved.
Let *s**t*
denote the share of labor income that is saved. This will, in
general depend on labor income and the interest rate. The savings
in period *t *is next periods capital stock

Now assume that *s**t** *only depends on *r**t+*1. (Which class of utility functions produce this result?
What are the effects of higher *r*?) Then we can write the
last line of as

This (implicitly) defines a difference equation
for *k, *i.e., a relation between *k**t *and *k**t+*1* *that has to be
satisfied in this model. Note that if we made the Solow
assumption of a constant savings rate, the difference equation
takes an explicit form. Since in this case the RHS contains no *k**t+*1. Fortunately there is a
utility function for which the income and substitution effects of
higher interest rates cancel so that the household will choose a
constant savings rate regardless of the interest rate.

Functional specification

Let us look at a particularly simple
specification. Assume *U *is the log function and that that
production is (Wicksell-) Cobb-Douglas, .

Now we can write the consumption decision of the consumer as follows.

10

with foc

11

Now we only need to specify the wage function
to have an explicit version of . We see immediately that *. *So becomes

Can you solve this difference equation and is it stable? Here is an example of a plot of .

The steady state of is its fixed point

13

The interest rate in steady state is

Dynamic Inefficiency

In each time period the amount of resources is *K**t *+ *F*(*K**t *,*A**t**L**t*). This has to be split
into aggregate consumption and next periods capital stock. We can
than write

15

where *C**t *is total consumption of young and old in period *t. *Now
divide by *A**t **L**t *

16

*c**t *is aggregate consumption per unit of current effective
unit of labor.

Now consider an economy in a steady state. We then have

17

Let us find the value of the steady state capital stock that maximizes aggregate consumption per effective unit of labor. The FOC for this problem is

This is a variant of the Ramsey Golden Rule. A
steady state capital stock above *k**gr
*is not Pareto efficient. If we are
above *k**gr* we could increase aggregate consumption in all periods
now and in the future by reducing forcing the steady state
capital stock to be lower.

Now compare and

19

So it is possible that the economy is not on the Pareto frontier and thus dynamically inefficient. What goes wrong here and why does not the first welfare theorem hold? One way of understanding this is to realize that the only way a young generation can reduce the capital stock is by saving less but then consumption would have to be lower for them in the second period. If, however, there was a mechanism that specified that young people saved less but in return got some transfers from the young next period, everybody could be made better off. What does this remind you off?

More general savings functions

In a more general case, the difference equation given by can be less nice. It can have no or multiple steady states. With multiple steady states, some will be non-stable. Multiple steady states could occur if, for example, the savings rate increase as interest rate falls (the income effect dominates).