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Consider the optimal conversion of depleting exhaustible resources into foreign assets
for a small open economy which uses capital and resources in production, obtains an
exogenous return on investment abroad, and faces elastic demand for its resources on the
global market (Dasgupta et al., 1978). Maximizing social welfare yields the Hotelling rule and
the efficiency conditions that the marginal product of capital must equal the world interest
rate and that of resources the world price of resources. The optimal rate of resource depletion
thus equals the elasticity of world demand for its resources times the interest rate. The initial
price and the resulting depletion path of natural resources are set so that reserves are
eventually completely exhausted. A resource discovery thus leads to an immediate fall in the
resource price and increase in the rate of resource depletion. Suppose world demand for
resources is given by E=E(Q), where Q is the price of natural resources and ε ≡ - QE'/E > 1
the constant elasticity of demand. The social planner maximizes utilitarian social welfare,
U(C(t))exp(-ρt)dt, subject to the equations describing natural resource depletion, the
dynamics of the current account and the Cobb-Douglas production function, i.e.,
S=-E-R, A =r(A-K)+Y+QE(Q)-Cand Y=F(K,R)=KαRβ,
where C, S, R, A, K, Y, r and ρ denote consumption, the resource stock, resource use in
production, national assets, the capital stock, domestic production, the exogenous world
interest rate and the subjective rate of time preference, respectively. The production function
has decreasing returns to scale with respect to K and R (0 < α+β < 1). It follows that:
F = r, Fr = Q(1 - 1Zε), Q/ Q = r, E/ E = -εr, C/ C = σ(r - ρ),
where σ is the elasticity of intertemporal substitution. The first and second equation equate
the marginal products of capital and resource to the interest rate and the marginal revenue of
natural resources, the third equation is the Hotelling rule which (given that demand for
resources is iso-elastic) says that capital gains on natural resources must equal the rate of
interest r, and the fourth equation is the Keynes-Ramsey rule. Effectively, the first three
equations result from maximizing the present value of natural resource and other income
while the fourth equation results from choosing the timing of consumption to maximize utility.
It follows that capital and natural resource use must decline over time: