Macro Lect 4 2015 Ramsey - WordPress.com · Prof. George Alogoskouﬁs, Dynamic Macroeconomic...

Prof. George Alogoskoufis, Dynamic Macroeconomic Theory

The Representa,ve Household Model

Ramsey-‐Cass-‐Koopmans

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The Ramsey Model

The representa,ve household model predates the Solow model. The first such model is due to Ramsey (1928), who set out to analyze the op,mal savings behavior of a household with a long ,me horizon. The Ramsey model remained in rela,ve obscurity for many years. It re-‐surfaced in the 1960s, with the extensions of Cass (1965) and Koopmans (1965), and has since evolved as the standard inter-‐temporal model in macroeconomics.

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The Ramsey Model• Assump,ons about technology and market structure in the Ramsey model are similar to the assump,ons of the Solow model.

• Things differ in the determina,on of savings. Instead of the fixed and exogenous saving rate of the Solow model, in the Ramsey model savings are determined as a result of the op,mal inter-‐temporal behavior of a representa,ve household. Consequently, savings behavior is determined endogenously.

• The representa,ve household model is thus theore,cally more sa,sfying than the Solow model, as it is based in inter-‐temporal op,miza,on, and equilibrium paths depend solely on parameters related to the preferences of households, the technology of produc,on, popula,on growth and market structure.

• Moreover, as the typical form of the model assumes complete and compe,,ve markets, and that all households are alike, the Ramsey model determines the socially op,mal savings behavior in the sense of the maximiza,on of social welfare.

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Proper,es of the Ramsey Model

• The savings rate in the Ramsey model is not constant, as in the Solow model, but a func,on of the the state of the economy.

• In the representa,ve household model there is no possibility of dynamic inefficiency, in the sense of an excessively high savings rate that leads the economy to a level of capital beyond the “golden rule”. The representa,ve household chooses its individually op,mal level of savings, which, because of the assump,on of full compe,,ve markets, is also socially op,mal.

• As it turns out, the steady state capital stock in this model is below the golden rule capital stock, because of the assump,on of a posi,ve pure rate of ,me preference. This op,mal steady state capital stock defines the so called modified golden rule.

• However, this model is also an exogenous growth model, similar in this respect to the Solow model.

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Produc,on Func,on and Compe,,ve Equilibrium

y(t) = f (k(t))

r(t) = ′f (k(t))−δ

w(t) = f (k(t))− k(t) ′f (k(t))

k•(t) = r(t)k(t)+w(t)− c(t)− (n + g)k(t)

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Preferences of the Representa,ve Household

U = e−ρtu(C(t)) L(t)Ht=0

∞

∫ dt

where,

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L(t) = L(0)ent ,θ > 0,ρ − n − (1−θ )g > 0

u(C(t)) = C(t)1−θ

1−θ


Op,miza,on of Welfare of the Representa,ve Household

maxU = B e−βt c(t)1−θ

1−θt=0

∞

∫ dt

k•(t) = r(t)k(t)+w(t)− c(t)− (n + g)k(t)

under the constraint,

where, β = ρ − n − (1−θ )g > 0

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First Order Conditions for a Maximum of the Representative Household

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c•(t)c(t)

= 1θr(t)− ρ −θg( ) = 1

θr(t)− ρ( )− g

k•(t) == r(t)k(t)+w(t)− c(t)− (n + g)k(t)

limt→∞

e−βt k(t) ′u c(t)( ) = limt→∞

e−βt k(t)c(t)−θ = 0


Interpre,ng the Equa,ons of the Ramsey Model

• The first equa,on determines the accumula,on of capital

• The second equa,on is known as the Euler Equa0on for Consump0on. The growth rate of consump,on per capita is posi,ve if the real interest rate exceeds the pure rate of ,me preference of the representa,ve household. In addi,on, the higher the elas,city of inter-‐temporal subs,tu,on in consump,on, the higher the growth rate of consump,on for a given difference in the real interest rate from the pure rate of ,me preference of the representa,ve household.

• The third equa,on is the transversality condi,on, and is derived from the household’s intertemporal budget constraint.

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Interpre,ng the Euler Equa,on for Consump,on

• The higher the real interest rate rela,ve to the pure rate of ,me preference, the greater the incen,ve for the representa,ve household to reduce current consump,on and invest in capital with a higher rate of return r(t), in order to enjoy higher future consump,on. So if the real interest rate is higher than the pure rate of ,me preference, consump,on per capita will be growing along the op,mal path.

• The higher the elas,city of inter-‐temporal subs,tu,on, the easier it is for the household, in u,lity terms, to subs,tute consump,on over ,me. So, the easier it is to subs,tute current for future consump,on. Consequently, for a given difference between the real interest rate and the pure rate of ,me preference, the growth rate of per capita consump,on is higher, the higher the elas,city of inter-‐temporal subs,tu,on.

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The Efficiency of Competitive Equilibrium in the Ramsey Model• In this model, due to the assumption of competitive markets, maximizing the inter-

temporal utility function of a representative household is under the same constraint as the one that would be used by a social planner, i.e the economy wide budget constraint. Thus, the problem of the representative household is the same as the problem of the social planner.

• Consequently, the competitive equilibrium in the model of the representative household would be fully efficient. A decentralized competitive equilibrium in which each household maximizes its own utility function over time, under its private budget constraint, would lead to the same outcome as that of the choice of a social planner who had as her objective the maximization of the inter-temporal utility function of the representative household, under the appropriate aggregate budget constraint.

• Thus, in the case of the representative household model with full and competitive markets, we have an application of the first theorem of welfare economics, which suggests that when markets are competitive and complete, and there are no externalities, the decentralized equilibrium is efficient as it maximizes social welfare.

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The Intertemporal Budget Constraint of the Representa,ve Household

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e− r

_(t )−n−g⎛

⎝⎜⎞⎠⎟tc(t)

t=0

∞

∫ dt = k(0)+ e− r

_(t )−n−g⎛

⎝⎜⎞⎠⎟tw(t)

t=0

∞

∫ dt

r_(t) = 1

tr(v)dv

v=0

t

∫where

Integra,ng the capital accumula,on equa,on of the representa,ve household, we get the inter temporal budget constraint,


The Consump,on Func,on of the Representa,ve Household Model

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where

Integra,ng the Euler equa,on for consump,on of the representa,ve household, a_er using the inter temporal budget constraint, we get the following aggregate consump,on func,on,

c(0) = γ (0) k(0)+ e− r

_(t )−n−g⎛

⎝⎜⎞⎠⎟tw(t)

t=0

∞

∫ dt⎛

⎝⎜

⎞

⎠⎟

γ (0) = er_(t )(1−θ )−ρ+θn

θ

⎛

⎝⎜⎜

⎞

⎠⎟⎟t

dtt=0

∞

∫⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−1


Interpreting the Consumption Function of the Representative Household Model

• Consumption is a proportion γ(0) of total wealth.

• The representative household consumes a share of its total wealth γ(0), that depends on the evolution of the average real interest rate, the pure rate of time preference rate ρ, the elasticity of inter-temporal substitution of consumption 1/θ, and the population growth rate n.

• The impact of the average real interest rate on the proportion of total wealth that is consumed depends on the elasticity of inter-temporal substitution of consumption 1/θ.

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Aggregate Capital Accumula,on and Consump,on Growth in the Ramsey Model

k•(t) = f (k(t))− c(t)− (n + g +δ )k(t)

c•(t)c(t)

= 1θ

′f (k(t))−δ − ρ( )− g

limt→∞

e−βt k(t) ′u c(t)( ) = limt→∞

e−βt k(t)c(t)−θ = 0

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The Dynamic Adjustment of Consump,on and the Capital Stock

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The Balanced Growth Path• The balanced growth path in the representa,ve household model is similar to the balanced growth path in the Solow model. The capital stock, output and consump,on per efficiency unit of labor are constant. Consequently, the savings ra,o (y-‐c)/y, is also constant on the balanced growth path.

• The total capital stock, total output and total consump,on are growing at a rate n+g. The per capita capital stock, per capita output and per capita consump,on are growing at a rate g.

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The Ramsey Model and the Golden Rule

• In the representa,ve household model, capital per efficiency unit of labor on the balanced growth path is always lower than the golden rule. This is because the representa,ve household has a posi,ve pure rate of ,me preference and discounts future u,lity. Thus, the representa,ve household does not seek to maximize per capita consump,on on the balanced growth path, as assumed in the golden rule, but an inter-‐temporal u,lity func,on which, given the posi,ve pure rate of ,me preference, gives a greater weight to current consump,on rela,ve to future consump,on.

• Thus, the steady state capital stock in the Ramsey model is lower than the one that corresponds to the golden rule, as the steady state real interest rate is higher than n+g. The steady state in the Ramsey model, is o_en referred to as the modified golden rule.

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Effects of a Permanent Increase in the Pure Rate of Time Preference

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The Adjustment Path Following an Increase in the Pure Rate of Time Preference

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The Ramsey Model in Discrete Time (per efficiency unit of labor)

• Cobb Douglas Production Function

• Euler Equation for Consumption

• Equilibrium Condition in the Goods Market

yt = Aktα

ct+1ct

= 1+ rt+11+ ρ

⎛⎝⎜

⎞⎠⎟

1θ 11+ g

yt = ct + (1+ n)(1+ g)kt+1 − (1−δ )kt21


Population Growth and Technical Change

• Population Growth

• Technical Change

Lt = L0 (1+ n)t

ht = h0 (1+ g)t

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The Real Interest Rate and the Real Wage

rt =αAktα−1 −δ

wt = (1−α )Aktα

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Capital Accumulation and Consumption Growth per efficiency unit of Labor

kt+1 =1

(1+ n)(1+ g)Akt

α + (1−δ )kt − ct( )

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ct+1ct

=1+αA kt+1( )α−1 −δ

1+ ρ⎛

⎝⎜

⎞

⎠⎟

1θ 11+ g


Capital and Output on the Balanced Growth Path

k*= αA(1+ ρ)(1+ g)θ − (1−δ )

⎛⎝⎜

⎞⎠⎟

11−α

y*= A αA(1+ ρ)(1+ g)θ − (1−δ )

⎛⎝⎜

⎞⎠⎟

α1−α

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Savings Rate on the Balanced Growth Path

s*= a (1+ n)(1+ g)− (1−δ )(1+ ρ)(1+ g)θ − (1−δ )

⎛⎝⎜

⎞⎠⎟

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Dynamic Simulations of the Ramsey Model

Assumptions about parameters

Α=1, α=0.333, ρ=0.02, θ=1, n=0.01, g=0.02, δ=0.03

Two Alternative Scenarios

1. An increase in the pure rate of time preference ρ by 5% (i.e. from 0.02 to 0.021)

2. An Increase in total factor productivity Α by 5% (i.e. from 1 to 1.05)

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Καθ. Γ. Αλογοσκούφης, Δυναμική Μακροοικονομική28

Καθ. Γ. Αλογοσκούφης, Δυναμική Μακροοικονομική29


Variables on the Balanced Growth Path

Initial Increase in ρ Increase in Α

k 10.275 10.056 11.055

y 2.172 2.157 2.337

c 1.554 1.551 1.672

r 0.040 0.041 0.040

w 1.449 1.439 1.559

s 0.285 0.281 0.285

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Conclusions• The Ramsey model of a representa,ve household is a very important reference model, not only for the theory of economic growth, but more generally for modern dynamic macroeconomics. As it is based on the assump,on of inter-‐temporal op,miza,on by a representa,ve household, this model describes the socially op,mal choice of savings and the socially op,mal growth path.

• This model is a dynamic general equilibrium model and represents, for dynamic macroeconomics, what the compe,,ve Arrow-‐Debreu general equilibrium model represents for microeconomics and general equilibrium theory.

• In other respects, the Ramsey model has proper,es and weaknesses similar to the Solow model.

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