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, lÌ„) + βu(ct+1, lÌ„) + …
Firm’s Production and Investment Technology: f(kt, nÌ„); 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, lÌ„) + λt+1 (f(kt+1, nÌ„) − 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
- Chapter 18: Economic Efficiency
- Efficiency in the Infinite-Period General Equilibrium Framework
