U
t 0
e
t
Lt u C t dt H
The lifetime utility function

U

t 0
e
t
Lt u C t dt H
暨 南 大 学 经 济 学 院 周 泳 宏


C(t): consumption of each member of the household at time t L(t): total population of the economy L(t)/H:the number of members of the household u(·): each member’s utility at a given date uL/H: household’s total utility
暨 南 大 学 经 济 学 院 周 泳 宏

There is no uncertainty in this model, risk aversion is irrelevant θ determines the househould’s willingness to shift consumption between different periods
Chapter 2 Infinite-Horizon and Overlapping-Generations Model
周泳宏 tzhouyhjnu@
Introduction

This chapter investigates two models

Ramsey-Cass-Koopmans Model Diamond Model


暨 南 大 学 经 济 学 院 周 泳 宏
Each household supplies one unit of labor at each point in time. Capital does not depreciate
K 0
The initial stock of capital
K 0 H
☆ prove: Discount factor

U

暨 南 大 学 经 济 学 院 周 泳 宏
t 0
e
t
Lt u C t dt H

ρ:discount rate The e t acts to discount the future The greater is the discount rate, the less the household values future consumption relative to current consumption.
If this condition does not hold, the household can attain infinite lifetime utility, and its maximization problem does not have a well-defined solution
模型假设

完全竞争市场结构 长生不老的不断扩展的家庭 家庭和个人完全同质 忽略资本的折旧
2.1 Assumptions
1）Firms


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The economy is viewed as consisting of a large number of identical firms. With perfectly competitive markets, factor returns are equal to the marginal products Firms take A as given
each household initially owns and rents to firms

暨 南 大 学 经 济 学 院 周 泳 宏

The household chooses the consumption/saving plan that maximises their discounted ‘lifetime’ utility. The lifetime utility function for the household is
instantaneous utility function
C t u C t ， n (1 ) g 0 1
1

暨 南 大 学 经 济 学 院 周 泳 宏

This function form is needed for the economy to convergence to BGP. The utility function is known as constant relative risk aversion (CRRA) utility.

暨 南 大 学 经 济 学 院 周 泳 宏

The production side (firm) of the economy is identical to that used in the Solow model. The household side consists of a ‘representative agent’ who owns the capital, supplies the labor used by firms, and consumes or saves the resulting output.

Given the parameters,the representative agent chooses the consumption/saving plan that maximises their (expected) discounted lifetime utility.
暨 南 大 学 经 济 学 院 周 泳 宏

暨 南 大 学 经 济 学 院 周 泳 宏

Both the models continue to treat the growth rates of labor and knowledge as exogenous But derive the evolution of the capital stock from the interaction of maximizing households and firms in competitive markets
2.2 The Behavior of Households and Firms
1）Firms

Markets are competitive, with no depreciation, the real interest rate (real rate of return) on capital is

The smaller isθ, the more slowly marginal utility falls as consumption rises, and so the more willing the household is to allow its consumption to vary over time to take advantage of differences between its discount rate and the rate of return it gets on its saving

暨 南 大 学 经 济 学 院 周 泳 宏

Ramsey: there is a fixed number of infinitely lived households Diamond:there is turnover in the population

New individuals are continually being born, and old individuals are continually dying

A grows exogenously at rate g
Y t F K t , At Lt
2）Household


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There is a large and fixed number of infinitely lived identical households. The size of each household grows at rate n The number of households, H, remains fixed.
Part A
Ramsey-Cass-Koopmans model
Introduction

The Ramsey model is a general equilibrium model in which households and firms interact in competitive markets.

The Ramsey-Cass-Koopmans model The Overlapping Generations or Diamond model

The central difference between the Diamond model and the Ramsey-CassKoopmans model is that

暨 南 大 学 经 济 学 院 周 泳 宏

Firms pay rent and wages to the households who decide how much to spend on consumption (revenue for the firms) and how much to save (invest in new capital). The heart of the model is the consumption/saving decision and this is based on micro-optimising behaviour.