RCK Model

#economics #macro

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

Ramsey-Cass-Koopmans Model

Endogenous savings model
Representing household with infinite lifetime decides the consumption and savings for every time.

Assumptions

Model Setting

Representative household: consuming while producing
Infinite lifetime decision: maximize utility over infinite lifetime
Constant population & technology growth rate: , .

Behaviors of Firms

Same as Solow Model, we have a production function

,
, : depreciation rate, : saving rate
: total consumption
: consumption per person

Market Equilibrium

From the previous discussion previous discussion

Closed Economy: thus .
real interest rate: .
composite interest rate: , then .
real wage: .
wage per unit of effective labor:

Dynamics of

From the previous discussion previous discussion,

Behaviors of Households

Each member of the household supplies one unit of labor at .

Household's utility function:

: (utility) discount rate, tomorrow's consumption is discounted.
: consumption of each member of the household at
: instantaneous utility function
: Total population
: number of households
: number of members of the household
: households' total instantaneous utility

Utility Functions

CRRA utility function:

: constant relative risk aversion coefficient
diminishing marginal utility: , .
note that if , then .

Since , therefore the lifetime utility is

Budget Constraints

Present value of lifetime consumption is no greater than the present value of lifetime wealth: Since and ,

No-Ponzi-game Condition

Present value of the household's asset holding is not negative: i.e.

Optimization Problem

Problem: where , and .

Lagrangian: F.O.C.
thus taking logs on both sides: and differentiate with respect to : thus we have the Euler's equation:

Dynamics

Steady State

Two variables problem : S.S.:

if : then , thus decreasing.
if : then , thus increasing.
if : then less savings, thus decreasing.
if : then more savings, thus increasing.

Pasted image 20240331213507.png

Saddle Path: single equilibrium trajectory leads to the fixed point.
Pasted image 20240331215550.png

Balanced Growth Path

Equilibrium in RCK is not always consumption-maximizing. Let the equilibrium defined in RCK as .

By the definition of Solow Growth Model > Golden Rule Solow Growth Model > Golden Rule in Solow model, while thus if , then note that & since the household discount the future consumption. thus to maximize the lifetime utility, the household increases the current consumption rather than the opposite.

However, the equilibrium in RCK always satisfies the Pareto Efficiency:

Remark (Pareto Efficiency).

The equilibrium is called Pareto Efficient if it is impossible to make anyone better off without making someone else worse off.

Remark (First Welfare Theorem).

If markets are competitive and complete, and there is no externality (and the number of agents are finite), then the decentralized equilibrium is Pareto Efficient.

Since RCK satisfies the conditions of ^b59238 Remark 2 (First Welfare Theorem), it is Pareto Efficient under the perspective of social planer.

Note that it is impossible to achieve the Golden-Rule in RCK, since if , then the household must decrease the current consumption to increase the investment by more savings, eventually increase the future consumption. however, since the future consumption is discounted, the lifetime utility might not be the optimal.

Impact of Discount Rate

From: if decreases,

since discount rate is decreased, future consumption is more valued than before
thus the current consumption decreases, while the future consumption increases
since the current savings increased, then the future capital level increases in the equilibrium.

Pasted image 20240331223331.png

Note that the choice variable is consumption, i.e. the consumer decides given , and thus the capital is gradually adjusted by the changes in the saving.

Effects of Government Purchases

Model Expansion

Adding Government into Model:

: total taxes
: taxes per unit of effective labor
: total government expenditure
: government expenditure per unit of effective labor
: government budget is balanced

Behavior of the Firms

Profit Maximization Conditions:

Dynamics of :

Behavior of the Households

Optimization Problem: where , and .

Lagrangian: F.O.C.

Steady State

Equilibrium Condition: Steady-State: while does not change, decreases by the amount of .

Permanent and Temporary Changes

Pasted image 20240331224908.png

If taxation is permanent: instantaneously moves to , by decreasing consumption.
If taxation is temporary: since the permanent income decreases less then the permanent taxation, the consumption decreases less then , and then moves back to .