Macro assignment

Macroeconomic Theory

1 Chapter 8: Infinite-Period Framework

All of the dynamic models we have build thus far consist of only two periods. Here we expand those models to an infinite amount of periods. The main takeaway is that the economic intuition gained from the simplified two-period framework holds when the number of periods is expanded.

• Infinite-period Framework: an infinite sequence of overlapping two-period frameworks. → The representative household or firm has an “Infinite Planning Horizon” →We use indices t,t + 1, t + 2, ... instead of 1,2,3, ... → Nothing before period t can be changed

1.1 Households

Households choose consumption/savings and labor/leisure each period over an infinite horizon to maximize utility subject to a budget constraint.

1.1.1 Preferences

Preferences are represented by:

V (ct, lt,ct+1, lt+1,ct+2, lt+2...) = u(ct, lt) + βu(ct+1, lt+1) + β 2u(ct+2, lt+2) + ...

or more compactly:

V = ∞∑ s=0

βsu(ct+s, lt+s) (1)

where each sub-utility function u(·) has the usual properties.

1.1.2 Budget Constraint

In the absence of taxes, households have the following period-t real budget constraint:

ct + at = (1 + rt)at−1 + wtnt (2)

with the unitary time endowment:

1 = lt + nt (3)

→ One could use this period-t budget constraint to derive a lifetime budget constraint over infinite periods. We will not be doing that in course.1

1If you are curious, the infinite period LBC can be derived by expressing the budget constraint as a difference equation and solving forward to get:

∑∞ s=0

ct+s/(1 + rt+1+s) s =

∑∞ s=0

(wt+snt+s)/(1 + rt+1+s) s with the usual

terminal and initial conditions imposed.

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1.1.3 Optimal Choice

Since the model as a recursive structure where the events occur each period, we can derive a single intertemporal and a single intratemporal optimality condition that will hold for each period using a sequential Lagrangian.

max {ct+s,at+s,lt+s}∞s=0

V =

∞∑ s=0

βsu(ct+s, lt+s)

subject to:

ct + at − (1 + rt)at−1 −wt(1− lt) = 0 for all t = 1,2,3, ...

The sequential Langrangian can be written in the same form we learned for two periods, but for an infinite amount of periods as follows:

L = ∞∑ s=0

{βsu(ct+s, lt+s) + λt+s (ct+s + at+s − (1 + rt+s)at−1+s −wt+s(1− lt+s))} (4)

Equation (4) might look intimidating, and you might be thinking ‘How do do I take for the first order conditions for this?’ It could be helpful to first write out the summation for a few periods:

L = u(ct, lt) + λt (ct + at − (1 + rt)at−1 −wt(1− lt))

+ βu(ct+1, lt+1) + λt+1 (ct+1 + at+1 − (1 + rt+1)at −wt+1(1− lt+1))

+ β2u(ct+2, lt+2) + λt+2 (ct+2 + at+2 − (1 + rt+2)at+1 −wt+2(1− lt+2)) + ...

The summation would go on forever in this pattern as s approaches ∞, but writing it out until s = 2 as above is sufficient for our purposes. So now what? The key thing to keep in mind is that since the model has a recursive structure, the optimality conditions hold for any t period. This means that we could take the FOCs as usual for any t period:

→ For intertemporal optimality condition we need to optimize over ct and ct+1. → For intratemporal optimality condition we need to optimize over ct and lt. → Note that we can take these FOCs while ignoring the expanded Lagrangian for s > 1.

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Taking FOCs:

∂L ∂ct

= 0 −→ ∂u

∂ct + λt = 0 −→

∂u

∂ct = −λt (5)

∂L ∂ct+1

= 0 −→ β ∂u

∂ct+1 + λt+1 = 0 −→ β

∂u

∂ct+1 = −λt+1 (6)

∂L ∂lt

= 0 −→ ∂u

∂lt + λtwt = 0 −→

∂u

∂lt

1

wt = −λt (7)

∂L ∂at

= 0 −→ λt − (1 + rt+1)λt+1 = 0 −→ λt = (1 + rt+1)λt+1 (8)

Remember that with the sequential Lagrangian we need to optimize over at to get an expres- sion that links together the Lagrange multipliers across periods.

Combining the FOCs:

→ Intertemporal Optimality Condition: Using Equations (5) and (6) into (8):

−λt = −(1 + rt+1)λt+1

∂u

∂ct = β

∂u

∂ct+1 (1 + rt+1)

⇒ ∂u/∂ct

β∂u/∂ct+1 = (1 + rt+1) (9)

→ Intratemporal Optimality Condition: Using Equations (5) and (7):

∂u

∂ct = ∂u

∂lt

1

wt

⇒ ∂u/∂lt ∂u/∂ct

= wt (10)

→ The solution to the model, {ct+s∗, lt+s∗}∞s=0, is fully characterized with an intertemporal and intratemporal optimality condition for each period and a LBC with initial and terminal conditions on wealth. → We will not be solving the infinite period models in this class. With the exception of a few

special cases, computational methods are typically needed to obtain the solution.

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2 Firms

The representative firm chooses labor and capital inputs to produce output at profit maximizing levels over an infinite horizon.

→ The firm’s real profit function over an infinite horizon:

Profit = ∞∑ s=0

(1 + rt+s) −s{f(kt+s,nt+s)− (kt+1+s − (1− δ)kt+s)−wt+snt+s} (11)

As with the households’ optimization problem, it will help to write out the summation for a few periods. We’ll again expand to s = 2.

Profit =f(kt,nt)− (kt+1 − (1−δ)kt)−wtnt

+

( 1

1 + rt+1

) (f(kt+1,nt+1)− (kt+2 − (1−δ)kt+1)−wt+1nt+1)

+

( 1

1 + rt+2

)2 (f(kt+2,nt+2)− (kt+3 − (1−δ)kt+2)−wt+2nt+2) + ...

→ For intertemporal optimality condition we need to optimize over kt+1. → For intratemporal optimality condition we need to optimize over nt. → Note that we can take these FOCs while ignoring the expanded profit function for s > 1.

Taking FOCs:

nt : ∂f

∂nt −wt = 0 ⇒

∂f

∂nt = wt (12)

kt+1 : −1 + (

1

1 + rt+1

)( ∂f

∂kt+1 + (1−δ)

) = 0 ⇒

∂f

∂kt+1 = rt+1 + δ (13)

where Equations (12) and (13) are the firms intratemporal and intertemporal optimality condi- tions, which hold for all t.

→ The solution to the model, {nt+s∗,kt+1+s∗}∞s=0, is fully characterized by the optimality conditions for each period, and initial and terminal conditions of the capital stock given demand for the firms’ output.

⇒ If the economic intuition from two-period models is the same as that from infinite period models, why bother with more complex infinite period models? When using macroeconomic mod- els to make quantitative statements about the dynamics of aggregates — such as consumption, savings, capital, labor, and output — the infinite period framework is more useful because it allows for multiple periods that reasonably correspond to a monthly, quarterly, or annual frequency.

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