Hansen and Wright, 1992

#economics #macro

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.
2024 Spring, instructed by prof. Eo, Yunjong.

Standard Model

Assumptions

Equilibrium

: consumption of household
: investment through savings
: output from production of the firm.
no governmental spending and closed economy is assumed.

Households

: discount factor
: instantaneous utility function, where
: consumption of households
: leisure of households, where households have one unit of time for each to divide between leisure and hours of work:

Firms

: capital share
: log of technological progress, following process: where and .
: capital, following the accumulation process: is depreciation rate and is investment
: hours of work

Budget Constraint

Note that the real interest rate and the wage can be driven as
and Then, from the competitive equilibrium condition and the capital accumulation,

Problems and Solutions

UMP

Lagrangian: F.O.C.
Thus we have and

PMP

F.O.C. Thus we have and

Frisch Elasticity

Definition 1 (Frisch Elasticity).

The Frisch elasticity of labor supply measures the percentage change in hours worked due to the percentage change in wages, holding the marginal utility of wealth (i.e. the Lagrangian multiplier) as constant. where denotes real wage.

From the labor supply equation: taking total derivation on both and , we have

Thus for the criteria of , the steady state of hours of worked will be

Log-linearization

and the exogenous variable is

Euler equation:
Consumption-Labor Substitution:
Capital accumulation:
Resource constraint:
Production function:
Real interest rate:
Real wage:

Indivisible Labor

Additional Assumptions

Now, each individuals work either zero or hours, and the household has infinitely many homogeneous agents. Then the expected payoff is

: probability that the given agents work at time .
: consumption of unemployed agent
: consumption of employed agent
: hours of working if employed
: hours worked in household

Household's Problem

The objective function is
where the constraints are Thus the budget constraint is

Solution

F.O.C. Note that we have thus .

and therefore, we have and

Utility Function

Since , we therefore have Then,

F.O.C. Then, we have

Euler equation:
Labor supply:

Solutions

Frisch Elasticity

From Review of RBC > ^66191d Review of RBC > Theorem 2 (calculate frisch elasticity), we have Then,

Government Spending

Assuming is the government spending, and is governed by where and .

and the resource constraint is Note that the solutions are
and the exogenous variables are