Download 11 November 2008

11 November 2008
ECON4925 Resource economics, Autumn 2008
Seminar 6, 19 November 2008
/If it is difficult to crack these exercises it is perfectly ok to ask for a hint. But I
will be travelling in Germany Friday-Sunday and may not be able to respond
promptly/
Exercise 16
Consider a fishery with effort measured in the number of fishing vessels. Let E be the
number of fishing vessels and S the stock of fish. The harvest per fishing vessel is
then given by
h  kE  S 
with k > 0 and 0 <  ,  < 1
-Derive the aggregate cost function for the fishery under the assumption that the cost
per fishing boat is w.
-Show how total catch H depends on S in an open access fishery with zero profit.
-Discuss under what conditions the resource may become extinct.
-When the equilibrium has a positive stationary resource stock, show how this stock
depends on the exogenous parameters of your model.
-Derive the conditions for the social optimum and in particular for the steady-state
with a positive resource stock when such a steady-state exists.
Exercise 17
Consider the standard (commercial) forest model. Assume that there is a tax/subsidy
scheme such that the growth of the forest is continually subsidized at a rate σ while a
decline in the forest is taxed at the same rate σ. Let the present value for the forest
owner (over a full cycle) of such a subsidy/tax scheme be denoted N(T).
-Show that N(T) is positive.
-What does this imply for the value of land when used as a forest?
It can be shown that
N (T )
re rT
.

N (T ) 1  e rT
-What does the introduction of the subsidy/tax-scheme imply for the optimal time T of
cutting a tree (compared to optimal time before the scheme was introduced)?
Exercise 15
Consider an economy with an aggregate production function given as
Yt  Kt ( X t Rt )1
0  1
where Kt is capital, Rt is the use of an exhaustible resource (with zero extraction costs)
and Xt is a variable representing the technology level of the economy.
-Show that the interest rate in the economy, i.e. the marginal productivity of capital, is
constant provided X t Rt / K t is constant.
-Show that when the interest rate is constant the “resource price”, i.e. the marginal
productivity of the resource, is proportional to X.
-Explain what is meant by intertemporal efficiency (wrt. the use of the exhaustible
resource).
-Assume that Xt grows at a constant rate x  0 . Show that it in this case is possible to
have an intertemporally efficient growth path along which the interest rate is constant
and output grows at a constant rate g.
-What is the interest rate along the path, as just described, and what is the relationship
between the growth rate g and the saving rate s  K / Y along this path?
-Assume that population is constant and that the social welfare function is

 u (C )e
t
 t
dt
where
Ct  Yt  Kt
0
-Derive the conditions for the optimal outcome, and show under what conditions a
growth path of the type described above is optimal (given that
u C
is constant).
u