Macro assignment

Macroeconomic Theory

1 Chapter 18: Economic E�ciency

• Economic E�ciency, roughly speaking, is an equilibrium where nothing can be improved without something else being made worse o�.

→ The concept of e�ciency has nothing to with fairness. • The Social Planner: a �ctitious, `all-knowing', benevolent entity that is perfectly able to control and allocate all resources of the economy.

→ Theoretical construct used to obtain e�cient outcomes, which can be used as a benchmark against sub-optimal outcomes.

1.1 E�ciency in the In�nite-Period General Equilibrium Framework

→ For simplicity, we will assume that labor supply is exogenously �xed at n̄ so that we do not have to optimize over it (which also means leisure is �xed at l̄)

→ If we had endogenous labor, we would have to optimize over leisure/labor as usual

1.1.1 The Social Planner

→ The Social Planner uses the households' preferences and the �rms' production technology to allocate the economy's resources to obtain an e�cient equilibrium outcome.

For t = 1, 2, ...

Household's Preferences: V = u(ct, l̄) + βu(ct+1, l̄) + ...

Firm's Production and Investment Technology: f(kt, n̄); invt = kt+1 − (1 −δ)kt

Resource Constraint: f(kt, n̄) = ct + invt

⇒ The Social Planners Optimization Problem:

max {ct+s,kt+1+s}∞s=0

V =

∞∑ s=0

βsu(ct+s, l̄)

subject to:

f(kt, n̄) − ct − (kt+1 − (1 −δ)kt) = 0 for all t

→ Note that we are only optimizing over the intertemporal dimension. Thus we only need to take �rst-order condition with respect to ct,ct+1, and kt+1

1

→ One can obtain the e�ciency condition by setting up a sequential Lagrangian and taking the relevant �rst-order conditions:

L = ∞∑ s=0

{βsu(ct+s, l̄) + λt+s (f(kt+s, n̄) − ct+s − (kt+1+s − (1 − δ)kt+s))}

Writing this out for s = 0 and s = 1:

L = u(ct, l̄) + λt (f(kt, n̄) − ct − (kt+1 − (1 − δ)kt)

+ βu(ct+1, l̄) + λt+1 (f(kt+1, n̄) − ct+1 − (kt+2 − (1 −δ)kt+1)) + ...

FOCs:

∂L ∂ct

= 0 −→ ∂u

∂ct −λt = 0 −→

∂u

∂ct = λt (1)

∂L ∂ct+1

= 0 −→ β ∂u

∂ct+1 −λt+1 = 0 −→ β

∂u

∂ct+1 = λt+1 (2)

∂L ∂kt+1

= 0 −→−λt + λt+1 (

∂f

∂kt+1 + (1 − δ)

) = 0 −→ λt+1

( ∂f

∂kt+1 + (1 −δ)

) = λt (3)

Using Equation (1) and (2) into Equation (3):

λt+1

( ∂f

∂kt+1 + (1 −δ)

) = λt

β ∂u

∂ct+1

( ∂f

∂kt+1 + (1 −δ)

) = ∂u

∂ct

⇒ ∂u/∂ct

β∂u/∂ct+1 =

∂f

∂kt+1 + (1 −δ) (4)

⇒ Equation (4) is the E�ciency condition from the Social Planner's problem. → Economic intuition: It is e�cient when the households' marginal rate of substitution of ct

for ct+1 is equal to the �rm's marginal product of capital in period t+1 plus the undepreciated

portion of that marginal unit of capital.

⇒ Note that this is equivalent to the �nancial market equilibrium condition from the `de- centralized' in�nite-period framework where households and �rms had their own optimization

problems, and equilibrium obtained from market forces.

• First Welfare Theorem of Economics: The equilibrium behavior of households and �rms in a decentralized framework is economically e�cient.

→ This fails if:

1. distortionary taxes are present (e.g. wage income, interest income taxes)

2. `externalities' are present

3. �rms or households have `market power' to in�uence prices

2