Two period models: Some analytical details
ECON 331
This set of notes shows the optimization problems with the 2-period model. I will deviate
from the approach in Williamson by explicitly assuming that utility from consumption can
be separated across time and across consumption and leisure:
U (c, l, l′ , c′ ) = u(c) + ν(l) + β [u(c′ ) + ν(l′ )]
where u() and ν() are both increasing functions with negative second derivatives.
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Optimization by the household
The representative household maximizes utility subject to its lifetime budget constraint,
taking as given real interest rates, real wages in both periods, profits in both periods, and
taxes in both periods.
max
u(c) + ν(l) + β(u(c′ ) + ν(l′ )) subject to c +
′
′
c,c ,l,l
w′ (h − l) + Π′ − t′
1 ′
c ≤ w(h − l) + Π − t +
1+r
1+r
(1)
The Lagrangian for this problem is
1 ′
w′ (h − l) + Π′ − t′
−c−
c
L(c, c , l, l , λ) = u(c) + ν(l) + β(u(c ) + ν(l )) + λ w(h − l) + Π − t +
1+r
1+r
(2)
The Kuhn-Tucker conditions are (I’m not going to indicate things with stars, but these
are all evaluated at the optimal arguments to the Lagrangian)
′
′
′
′
u′ (c) − λ = 0
ν ′ (l) − wλ = 0
1
βu′ (c′ ) − λ
=0
1+r
1
βν ′ (l′ ) − λ
w′ = 0
1+r
λ≥0
w′ (h − l) + Π′ − t′
1 ′
λ w(h − l) + Π − t +
−c−
c =0
1+r
1+r
1
The chapter mentions a few assumptions; we can show those assumptions hold for this
(pretty general) setup.
Combining the first and second equation, we have that
ν ′ (l)
=w
U ′ (c)
(3)
That is, the MRS between consumption and leisure is equal to the wage. (A similar
condition holds for the second period).
Combining the first and third equations:
U ′ (c)
=1+r
βU ′ (c′ )
(4)
so the marginal rate of substitution between current and future consumption is equal to
the real interest rate (the price at which we substitute between the two).
Combining the second
and fourth equation, we get a similar condition for leisure. Since
w′ /w ′
′
′ ′
λ = ν (l)/w, βν (l ) = 1+r ν (l) or
w(1 + r)
ν ′ (l)
=
w′
βν ′ (l′ )
(5)
That is, the optimizing household also can be thought of as trading leisure off between
the current and future period, where the relevant price of leisure is related to current and
future wage rates and the real interest rate.
Relating these first-order optimality conditions to the statement to the statements in the
text from pages 382-384, we have derived equations (11-4), (11-5) (implicilty) and (11-6).
Williamson assumes that the current labor supply increases when w increases, e.g., the
substitution effect outweighs the income effect. An increase in the current labor
supply will increase ν ′ (l). The reason is the usual diminishing marginal rate of substitution
logic; as the amount of leisure falls, the marginal benefit of one additional hour is relatively
greater.
(Presumably, the increase in the wage also increases c and hence U ′ (c) falls as well).
This can be shown to be consistent with our within-period optimality condition, equation
(3). (We would have to take total differentials and do a lot of tedious algebra, or a little,
somewhat-tedious linear algebra, to show the exact conditions we need for this to hold).
Secondly, he assumes that the supply of labor increases when real interest rates increase1+
r increases. Again, this is consistent with equation (5). Holding wages constant in both
periods, an increase in (1+r) must lead to a net increase on the right hand side, which is
consistent with falling leisure in the first period (if the substitution effect outweighs the
income effect).
Finally, we need that current labor supply decreases when lifetime wealth increases. Think about a change in lifetime wealth as an increase in profits in one of the
periods (for example). If r and wages in both periods are unchanged, it must be that the
ratio on the right hand side of (5) is constant as well. Assuming that leisure is a normal
good, a change in lifetime wealth must affect leisure in both periods in the same direction.
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In other words, leisure must increase in both periods; current and future labor supply fall
when lifetime wealth increases, all else equal.
Williamson assumes that an increase in interest rates lowers current consumption. From the discussion in chapter 9, we know that, for an individual lender, this requires
the substitution effect to outweigh the income effect, and for a borrower income and substitution effects from an increase in interest rates both work to decrease consumption. So,
strictly speaking, what assumptions we need to make for the representative consumer aren’t
obvious, but we’ll stick with Williamson’s assumption that an increase in 1 + r causes a
decrease in first-period consumption.
Note that for additively separable utility (as in this case), this assumption actually also
requires that second period consumption must also fall, so that the ratio βU ′ (c′ )/U ′ (c) rises.
(whether it needs to fall more depends on us putting more structure on U ).
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Optimization by the representative firm
The firm’s problem is to maximize the present value of its profits, using a production function
Y = zF (K, N ) (with a similar production function in the second period).
Profits in the first period are Y − wN d − I and in the second period they are Y ′ −
w′ (N d )′ + (1 − δ)K ′ (we assume that after production (1 − δ)K ′ capital is left over, which is
then converted into consumption goods and given to the owners of the firm).
The future capital stock comes from the capital accumulation equation,
K ′ = (1 − δ)K + I
.
′
Π
. Substitution (for Π, Y, Y ′ , K ′ ) yields
The present value of the firm V is equal to Π + 1+r
V (N d , N d′ , I) = zF (K, N ) − I − wN d +
1 ′
z F (1 − δ)K + I, N d′ − w′ N d′ + (1 − δ)[(1 − δ)K + I]
1+r
Since we’ve gotten rid of the constraints, we can maximize this directly with respect to
the choice variables. We have that
V1 (N d , N d′ , I) = zF2 (K, N ) − w = 0
1 ′
V2 (N d , N d′ , I) =
z F2 ((1 − δ)K + I, N d′ ) − w′ = 0
1+r
′
1
V3 (N d , N d′ , I) = −1 +
z F1 ((1 − δ)K + I, N d′ ) + 1 − δ = 0
1+r
where we’ve used the chain rule in the third expression. The first two expressions show
that in both periods, the firm optimally hires until the marginal product of labor (which
depends on the available capital stock) equals the wage in that period. Notice that these
conditions implicitly tell us how much labor the firm should hire, given then other parameters
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of the problem; since z and w are exogenous to the firm, and the amount of capital is predetermined by the time the period starts, this first order condition pins down the optimal
labor demand curve. (If we picked a specific functional form for F () we could solve for it).
Notice also that the choice of N d′ is implicitly determined by the choice of I. Once we’ve
picked how much capital to build for tomorrow, we know that we’ll equate the marginal
product of labor (which depends on that capital stock and z ′ ) to w′ (which we take as given).
In particular, we know that if we invest more today, we’ll also hire more workers tomorrow
(capital and labor are complements in production). So, we can ignore the dependency of
future labor demand on current investment, although we’ll keep labor demand in there
because it will be a function of w′ . (Basically, this is a way of saying what Williamson
says, which is that the firm ignores second period labor demand when it’s choosing I. I just
wanted you to know *why* we get to do that).1
The first order condition for investment can be simplified by multiplying by 1 + r and
re-arranging terms to get
z ′ F1 ((1 − δ)K + I, N ′ ) − δ = r
The firm invests until the marginal product of capital minus the depreciation rate – the
net product of capital – is equal to the real interest rate. If the firm invested more than this
amount, then it would be sacrificing too much profits today for too little profits (in present
value terms) in the second period. This is the same as equation (11-16) in Williamson.
Another way of putting it is to bring δ to the right hand side:
z ′ F1 ((1 − δ)K + I, N ′ ) = r + δ
The right hand side is sometimes called the “user cost of capital.” Firms should invest
until the benefit equals the user cost.
How does investment demand respond to changes in variables the firm takes
as given? We would like to understand, in general, how the optimal choice of investment
varies with respect to changes in the parameters z ′ , r, K and in the intertemporal price r.
(Notice that current wages don’t matter given how we’ve set up the problem; they don’t
appear in the first order condition for investment). We’re going to assume that the firm
has a downward sloping labor demand curve, so there’s some function N(w′ ) with N′ ≤ 0.
Substitute this in: o do this, first substitute in for N d′ :
−(1 + r) + [z ′ F1 ((1 − δ)K + I, N(w′ )) + 1 − δ] = 0
Remember that for a general function F (x1 , x2 , . . . xN ) = 0, the total differential is
∂F
∂F
∂F
dx1 +
dx2 + · · · +
dxN = 0
∂x1
∂x2
∂xN
We can treat the dxj as variables that can be manipulated algebraically. In this case,
take the total differential of the expression with respect to z ′ , r, K, r, w′ and I, resulting in
this inordinately long expression:
1
Also, everything we do still “works” if we don’t make this assumption. But if you had a particular
production function, you would end up with some redundant FOCs.
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− dr + z ′ F11 (1 − δ)dK
− z ′ F11 Kdδ + z ′ F11 dI
+ F1 dz ′ + N′ (w′ )dw′
=0
It’s convenient to group things by common differential terms:
Let’s start by solving for dI/dr assuming that the other differential terms are zero (e.g.,
that dz ′ = 0, dK = 0, etc)
dIz ′ F11 − dr = 0
Solve for dI/dr
dI
1
= ′
dr
z (F11 )
Becuase the marginal product of labor is diminishing, this is negative. In other words,
the demand curve for investment is downward sloping. Good start.
in general.
Setting all the other differential terms to zero except dI and dz ′ , we can write:
dIz ′ F11 + dz ′ F1 = 0
And then solving for
dI
:
dz ′
dI
−(F1 )
= ′
>0
′
dz
z (F11
(6)
Higher future productivity increases investment demand. You want more capital tomorrow because it’s very productive, all else equal.
In a similar manner, we can solve for the implicit derivative of I with respect to the other
variables in the problem:
dI
dK
=
−[z ′ F11 (1 − δ)]
= −(1 − δ) < 0
z ′ (F11 )
(z ′ F11 K + 1)
dI
=
dδ
z ′ (F11 )
dI
dδ
(7)
(8)
has ambiguous sign. The denominator is negative. The numerator could be positive,
negative, or zero depending on whether zK ′ (F11 ) < −1. This depends a bit on magnitudes,
which is a little uncomfortable. But remembering that r + δ is the user cost, it probably
makes sense that dI
< 0 because it implies investment demand is decreasing in the user cost
dδ
as of capital.
(6) confirms the assertion on page 394 of the text that an increase in future technology
shifts the investment schedule to the right, and that a decrease in the current capital stock
also shifts the investment schedule to the right (see (7)) as asserted in the textbook. This
approach in general is useful for trying to tease out the competing forces for more complicated
derivatives or when trying to come up with quantitative measures of how much a curve shifts.
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