3.3  The model

The initial year for the model is 2001. We follow the approach in Cutler et al. (1990) in assuming that all variables, including demographic variables, are initially on steady state paths. As Cutler et al. (1990) acknowledge, this does not represent reality because demographic variables are not stable at the present time; but as they say “it is not obvious how best to model [demographic change] as a single shock” (p.23). The approach they and others (e.g. Elmendorf and Sheiner, 2000) adopt is to assume that the population has been stable—that all age groups have been growing at the same rate—until 2001, at which point an unanticipated demographic shock occurs so that employment and population follow the projections described above. Table 5 sets out the definitions of the variables used in the model.

It is assumed that a central planner maximises an inter-temporal, time-additive social welfare function of general form:

(2)    

Output is produced according to a Cobb-Douglas production function with constant returns to scale:[6]

(3)    

which can be expressed in terms of the labour force in efficiency workers as:

(3a)    

Variables are defined in Table 5. Labour, L,is assumed to be supplied exogenously. This ensures that the labour market clears: shifts in the marginal product of labour lead to equal shifts in the real wage. Capital and debt accumulate according to the following accumulation equations:[7]

Table 5 - Symbols for variables used in the model

N	Population in natural units
L	Labour force in equivalent worker units (refer to Section 2.2.1)
P	Population in equivalent person units (refer to Section 2.2.1)
a	Support ratio = L/P*
n	Growth rate of N
l	Growth rate of L
p	Growth rate of P
A	Total factor productivity
g	Rate of Harrod-neutral technical progress or labour productivity growth
LE	Labour force in efficiency units
PE	Population in efficiency units
Y	Output
K	Aggregate capital stock
D	Aggregate foreign liabilities, denominated here as debt
C	Aggregate consumption
y	Output per worker measured in efficiency units (Y/ LE)
k	Capital stock in efficiency units per worker ()
d	Debt in efficiency units per worker ()
c	Consumption in efficiency units per worker ()
V	Measure of welfare maximized by social planner
r	Rate of interest
θ	The social planner’s rate of time preference
d	Rate of depreciation
γ	Capital elasticity of output
β	Degree of aversion to variability in average living standards over time[8]
q	The shadow price of capital
i	Investment in efficiency units
J(i)	The units of output required to increase the capital stock by i units, measured in efficiency units
μ	Parameter in the adjustment cost function
λ	Interest rate premium

(4)    

and

(5)    

Equation (4) is the national income accounting identity in efficiency units per worker. The left hand side is the change in net foreign assets, or current account deficit. The first term on the right hand side, r(d), is the interest charge on the outstanding stock of foreign liabilities; the term (l+g)d is subtracted because the growth of effective workers reduces debt per effective worker; the term (c/α) is consumption converted into per worker units by dividing by the support ratio; and J(i) represents investment and is the units of output required to increase the capital stock by i units. Hence J(i) – i represents the adjustment costs in terms of output required to transform goods into output. Equation (5) describes the additions to the capital stock from new investment after deducting depreciation and the growth rate of effective workers. In the simulations we adopt an adjustment cost function of the form:

(6)    

and a simple linear function for r(d):[9]

(7)    

Investment, consumption and debt are determined simultaneously. Debt is determined by (4), and consumption by

(8)    

Equation (8) is the standard Euler equation which is derived by equating the marginal rate of substitution in consumption with the return to saving. Investment is determined by equating the social value of an additional unit of investment with the social cost of capital, which gives[10]

(9)    

where

(10)    

and q is the shadow price of capital.