Macroeconomic Theory

1 Chapter 18: Economic Eï¿½ciency

â€¢ Economic Eï¿½ciency, roughly speaking, is an equilibrium where nothing can be improved

without something else being made worse oï¿½.

â†’ The concept of eï¿½ciency has nothing to with fairness.

â€¢ The Social Planner: a ï¿½ctitious, `all-knowing’, benevolent entity that is perfectly able to

control and allocate all resources of the economy.

â†’ Theoretical construct used to obtain eï¿½cient outcomes, which can be used as a benchmark

against sub-optimal outcomes.

1.1 Eï¿½ciency in the Inï¿½nite-Period General Equilibrium Framework

â†’ For simplicity, we will assume that labor supply is exogenously ï¿½xed at nÌ„ so that we do not

have to optimize over it (which also means leisure is ï¿½xed at lÌ„)

â†’ If we had endogenous labor, we would have to optimize over leisure/labor as usual

1.1.1 The Social Planner

â†’ The Social Planner uses the households’ preferences and the ï¿½rms’ production technology to

allocate the economy’s resources to obtain an eï¿½cient equilibrium outcome.

For t = 1, 2, …

Household’s Preferences: V = u(ct, lÌ„) + Î²u(ct+1, lÌ„) + …

Firm’s Production and Investment Technology: f(kt, nÌ„); invt = kt+1 âˆ’ (1 âˆ’Î´)kt

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

â‡’ The Social Planners Optimization Problem:

max

{ct+s,kt+1+s}âˆžs=0

V =

âˆžâˆ‘

s=0

Î²su(ct+s, lÌ„)

subject to:

f(kt, nÌ„) âˆ’ ct âˆ’ (kt+1 âˆ’ (1 âˆ’Î´)kt) = 0 for all t

â†’ Note that we are only optimizing over the intertemporal dimension. Thus we only need to

take ï¿½rst-order condition with respect to ct,ct+1, and kt+1

1

â†’ One can obtain the eï¿½ciency condition by setting up a sequential Lagrangian and taking the

relevant ï¿½rst-order conditions:

L =

âˆžâˆ‘

s=0

{Î²su(ct+s, lÌ„) + Î»t+s (f(kt+s, nÌ„) âˆ’ ct+s âˆ’ (kt+1+s âˆ’ (1 âˆ’ Î´)kt+s))}

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

L = u(ct, lÌ„) + Î»t (f(kt, nÌ„) âˆ’ ct âˆ’ (kt+1 âˆ’ (1 âˆ’ Î´)kt)

+ Î²u(ct+1, lÌ„) + Î»t+1 (f(kt+1, nÌ„) âˆ’ ct+1 âˆ’ (kt+2 âˆ’ (1 âˆ’Î´)kt+1)) + …

FOCs:

âˆ‚L

âˆ‚ct

= 0 âˆ’â†’

âˆ‚u

âˆ‚ct

âˆ’Î»t = 0 âˆ’â†’

âˆ‚u

âˆ‚ct

= Î»t (1)

âˆ‚L

âˆ‚ct+1

= 0 âˆ’â†’ Î²

âˆ‚u

âˆ‚ct+1

âˆ’Î»t+1 = 0 âˆ’â†’ Î²

âˆ‚u

âˆ‚ct+1

= Î»t+1 (2)

âˆ‚L

âˆ‚kt+1

= 0 âˆ’â†’âˆ’Î»t + Î»t+1

(

âˆ‚f

âˆ‚kt+1

+ (1 âˆ’ Î´)

)

= 0 âˆ’â†’ Î»t+1

(

âˆ‚f

âˆ‚kt+1

+ (1 âˆ’Î´)

)

= Î»t (3)

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

Î»t+1

(

âˆ‚f

âˆ‚kt+1

+ (1 âˆ’Î´)

)

= Î»t

Î²

âˆ‚u

âˆ‚ct+1

(

âˆ‚f

âˆ‚kt+1

+ (1 âˆ’Î´)

)

=

âˆ‚u

âˆ‚ct

â‡’

âˆ‚u/âˆ‚ct

Î²âˆ‚u/âˆ‚ct+1

=

âˆ‚f

âˆ‚kt+1

+ (1 âˆ’Î´) (4)

â‡’ Equation (4) is the Eï¿½ciency condition from the Social Planner’s problem.

â†’ Economic intuition: It is eï¿½cient when the households’ marginal rate of substitution of ct

for ct+1 is equal to the ï¿½rm’s marginal product of capital in period t+1 plus the undepreciated

portion of that marginal unit of capital.

â‡’ Note that this is equivalent to the ï¿½nancial market equilibrium condition from the `de-

centralized’ inï¿½nite-period framework where households and ï¿½rms had their own optimization

problems, and equilibrium obtained from market forces.

â€¢ First Welfare Theorem of Economics: The equilibrium behavior of households and ï¿½rms

in a decentralized framework is economically eï¿½cient.

â†’ This fails if:

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

2. `externalities’ are present

3. ï¿½rms or households have `market power’ to inï¿½uence prices

2

- Chapter 18: Economic Efficiency
- Efficiency in the Infinite-Period General Equilibrium Framework