EMEA6: Growth Models

1
EMEA6 Growth Models

• The Ramsey-Cass-Koopmans model of economic growth
• The DICE model of optimal greenhouse gas emission
  reduction

2
The Ramsey-Cass-Koopmans Model

• This model goes back to the 1928 paper of Frank
  Ramsey, which was long neglected
• Cass (1965) and Koopmans (1965) developed a model
  that looks the same but it is much better founded
• The crucial difference between Ramsey and Solow
  is that savings are derived from welfare
  maximisation
• An essential assumption in Ramsey models is that
  consumers live forever (or rather paternalistic
  dynasties)

3
Firms

• There are a large number of identical firms. Each
  has a production function
• The firms hire workers and rent capital in
  competitive markets, and pay them their marginal
  products. Firms sell their output in competitive
  markets. Firms take A as given. Because the
  production function has constant returns to
  scale, and the economy is competitive, firms earn
  zero profits.

4
Firms -2

• In a competitive market, capital earns its
  marginal product. Ignoring depreciation, the real
  rate of return on capital equals its earnings per
  unit of time. Thus, the real rate of interest is
• The real wage of effective labour equals its
  marginal product

5
Households

• There are a large number of identical households.
  The size of each household grows at rate n. Each
  household member supplies one unit of labour at
  every point in time. The household rents its
  capital to the firms. Households use their wage,
  interests and profits for consumption and saving,
  so as to maximise life time utility

6
Households -2

• Because the interest rate varies over time,
  define compounded interest as
• The budget constraint is that the net present
  value of consumption may not exceed the net
  present value of initial capital and wages earned

7
Households -3

• The budget constraint states that total
  consumption cannot exceed total earnings,
  considering that earnings can be shifted back and
  forth in time, accounting for the return on
  investment

8
Households -4

• As w-c is investment, this essentially says that
  the present value of the households assets
  cannot be negative in the limit

9
Households -5

• We maximise this, under the budget constraint
  derived earlier
• Note that ßgt0

10
Households -6

• The first order condition is

11
Households -7

• Eulers equation
• essentially says that if the rate of return on
  capital exceeds the time preference, consumption
  per worker grows over time
• What are the dynamics of the system?
• It is saddle-point stable, until one imposes the
  budget constraint
• Convergence is faster, as there is an extra
  control variable

12
The DICE Model

• The Dynamic Integrated Climate Economy model of
  Bill Nordhaus, dating back to 1990, is the
  prototype of most coupled climate-economy models
• DICE combines the simplest of economic growth
  models with the simplest of carbon cycle and
  climate models
• Indeed, the only interesting part of DICE is the
  interaction between the natural and the
  socio-economic system
• Let us go through the equations ...

13
DICE Objective

• Maximise total welfare
• Total welfare equals average welfare times the
  number of people
• Since the number of people is exogenous, total
  welfare equals average welfare
• Or, total welfare is the sum of individual
  welfare distribution does not matter
• Future welfare is worth less than current welfare

14
DICE Economy

• Production is a combination of labour and capital
• Technology progresses exogenously
• Income is spread over consumption and investment
• Wages and working hours vary freely with labour
  demand and supply
• Labourers own the stocks of the companies they
  work for
• There are a large number of firms, none of which
  has market power
• All products are substitutable, and can be
  divided into small units

15
DICE Carbon and Climate
16
DICE Impact

• Climate change damages are a power function of
  temperature
• Emission reduction costs are a power function of
  emission reduction
• Both climate change damages and emission
  reduction costs lower income, hence consumption
  and investment, hence welfare and growth

17
Costs of Emission Reduction
18
Marginal cost estimates

• 0 1 3
• Nordhaus 1994 BG 5
• EV 12
• Fankhauser 1994 EV 20 (6-45)
• Tol 1999 BG 73 23 9
• EW 171 60 23
• EVW 244 82 35
• (143) (51) (22)
• Downing 2000 BG 75 46 16
• Tol 2000 BG 20 4 -7

19
DICE Overview

• 10 equations that can be solved in a matter of
  minutes on a PC
• Balances the costs of emission reduction with the
  costs of climate change
• Thus yields optimal abatement, that is, emission
  reduction that maximises social welfare
• If the world were ruled by a benevolent dictator,
  what would she do?

20
DICE Outcomes

• If the world were ruled by a benevolent dictator,
  what would she do?
• A little emission reduction in the beginning,
  more later, but not enough to stabilise
  concentrations
• The first part is robust, the second part is very
  sensitive to assumptions
• The last part is robust, unless there is a cheap
  alternative to fossil fuels
• If we consider many countries, optimal emission
  reduction goes down!