47
defined as zero change in the present discounted value of current and future utility, are well
understood (Asheim and Weitzman, 1991; Sefton and Weale, 2006). In contrast to national
accounting practice, income must be deflated with the Divisia consumption price index rather
than the price index of output. Aggregation across multiple infinitely-lived households with
heterogeneous consumption preferences is feasible under constant returns to scale. The return
on the increasingly scarce natural resource increases at the expense of the increasingly
abundant other factors of production. Capital gains then represent capitalization of those
future changes in factor prices and are effectively a transfer from one factor to another rather
than a change in resources available to the whole economy. As a result, in a closed economy
where all factors are entirely owned by households, the net gains are zero and capital gains
should not be included in real income.
Appendix 4: Absorption constraints and Dutch disease dynamics
Assume a small open dependent economy with perfect access to the international capital
market. The traded good is the numeraire. Production in the traded sector only used labour, so
normalizing productivity at one we have YT = LT and W = 1. The non-traded sector has a
Cobb-Douglas production function, YN = Kα LN1-α, 0 < α < 1. Profit maximization yields the
demand for labour in the non-traded sector, LN = K[(1-α)P]1α, where P is the relative price of
non-traded goods. Labour market equilibrium then gives LT = 1- K[(1-α)P]1/a. Output of
non-traded goods is given by KN = K[(1-α)P](1-α)α Denoting the unit-cost function for
producing capital goods by c(P) = Pγ with 0 < γ < 1 the share of non-traded goods in the
production of home-grown capital, profit maximization requires that the marginal product of
capital, r(P ) = α[(1-α) P ](1-α)α, must equal the rental change, r*, plus the depreciation charge,
δ, minus the expected capital gains, c(P) / c(P). Preferences are homothetic and e(P) = Pβ,
0< β < 1, denotes the unit-expenditure function, hence consumption is non-traded goods is
given by CN = e'(P)U, where U denotes real consumption (or utility). Equilibrium on the
market for non-traded goods is given by CN + c'(P )I = KN, where I = K + δK denotes gross
investment. The representative consumer maximizes utility, ∫∞ ln(U) exp(-ρt)dt, subject to
0
the constraint, ∫∞[e(P)U+c(P)I]exp(-r*t)dt≤F0+V0+∫∞(KT+PKN)exp(-r*t)dt, where F
indicates foreign assets (bonds) and V the present value of natural resource revenues (i.e.,
natural resource wealth). The budget constraint states that the present value of the stream of
current and future consumption and investment spending on traded and non-traded goods
cannot exceed initial foreign assets plus initial resource wealth plus the present value of
current and future traded and non-traded production. If we suppose that r* = ρ, the optimality
condition for the consumer is 1/U = λ e(P), where the marginal utility of wealth λ has to be
constant over time. At the time the resource windfall becomes known (upward jump in V0), λ
jumps down and stays at this lower value forever after. A resource windfall thus corresponds
to an unanticipated, permanent fall in the marginal utility of wealth λ.
The adjustment path follows from the system of differential equations describing,
respectively, equilibrium in the market for non-traded goods and equity arbitrage: