Environment. The economy consists of I industries. A unique final good is produced competitively by combining the output of a continuum of industries using the following constant elasticity of substitution aggregate:
where G(i) denotes the net output in industry i, and σ is the elasticity of substitution between industries.
Throughout the analysis, we use the price of the final good G as the numeraire.
In industry i, there is a monopolist innovating new technologies and manufacturing intermediates that embody the patented technology to a representative production firm. Output in each industry is produced by combining two types of tasks and one intermediate good embodying this technology:
Gg(i) = η−η(1 − η)−1 D(i)α(i)A(i)1−α(i)η
q(θ(i))1−η (D.2)
where D(i) and A(i) denote age-deteriorating and the other tasks (hereafter, D-tasks and A-tasks, re-spectively) in industry i; q(θ(i)) is the quantity of intermediates embodying technology θ(i) used in the industry; α(i) ∈ (0, 1) designates the relative importance of D-tasks in industry i; 1 − η ∈ (0, 1) is the share of intermediates. The term η−η(1 − η)−1 is included as a convenient normalization.
We assume that the unit cost of producing the intermediate good is 1 − η. Hence, the industry’s net output can be written as:
G(i) = Gg(i) − (1 − η) · q(θ(i)). (D.3)
There are two types of labors: Y young and O old workers. We assume that young workers fully specialize in age-deteriorating tasks and old workers fully specialize in the other tasks. We denote the wage of young, old workers, as well as the rental price of machines by W , V , and R, respectively.
The aggregate of D-tasks is produced by combining a unit measure of industry-specific tasks:
D(i) =
where ζ is the elasticity of substitution between these tasks. We assume that each task d(i, z) can be produced either by young workers or machines:
d(i, z) =
(γy(z)y(i, z) + m(i, z) if z ∈ [0, θM(i)]
γy(z)y(i, z) if z ∈ (θM(i), 1], (D.5) where y(i, z) denotes the number of young workers employed in task z in industry i; γy(z) corresponds to the marginal productivity of young workers relative to machines in producing task z. We assume that γy(z) is continuous and increasing in z — human workers have comparative advantage in higher indexed D-tasks.
To minimize production costs, D-tasks will be allocated to factors depending on their comparative advantage. In particular, θM(i) denotes the automation threshold in industry i. The assignments of D-tasks to factors can be illustrated diagrammatically. In Figure D1, R and W/γy(z) are the effective costs of machines and young labor, respectively. In the equilibrium, tasks indexed z < θM(i) will be assigned to machines only, and those indexed z ≥ θM(i) will be performed exclusively by younger workers.
Figure D1 Equilibrium Allocation of Factors to Age-Deteriorating Tasks
This figure plots the effective cost of producing age-deteriorating tasks. In the equilibrium, tasks will be allocated to factors that have the lowest effective cost of producing them — tasks with index z < θM will be automated, and those with z ≥ θM will be produced by young workers only.
Factor markets clearing requires the following:
Y = Yd=
Optimization Problems. Let us turn to the optimization problems of the technology monopolist and the production firm in industry i. For the technology monopolist, we assume that the cost of innovation is
1−η
2−ηP (i)G(i)C(θ(i)), where C(θ(i)) is assumed to be a increasing and convex function. This is a one-time setup cost for the technology monopolist on, for example, researches and experiments. Once developed and patented, the monopolist makes machines embodying technology θ(i) at a per-unit cost 1 − η, and charges a monopoly unit price χ(θ(i)). Hence, the monopolist’s optimization problem can be written as:
max
θ(i),χ(θ(i))
ΠM(i) = χ(θ(i)) − (1 − η)q(θ(i)) −1 − η
2 − ηP (i)G(i)C(θ(i)). (D.6) For the production firm, let P (i) denote the price of output, PD(i) be the price of D-tasks. Assuming that the marginal product of older workers in producing A-tasks is one, the price of A-tasks will be exactly V . Then firm’s optimization problem can be written as:
max
D(i),A(i),q(θ(i))
Π(i) = P (i) · Gg(i) − PD(i) · D(i) − V · A(i) − χ(θ(i)) · q(θ(i)). (D.7)
First-order condition with respect to q(θ(i)) implies that the demand for intermediates as a function of technology level θ(i) is given by:
q∗(θ(i)) = 1
ηD(i)α(i)A(i)1−α(i) χ(θ(i)) P (i)
−η1
. (D.8)
Given the demand for intermediate goods in (D.8), the first-order condition of the monopolist’s problem
with respect to price of the intermediate goods gives:
χ∗(θ(i)) = P (i). (D.9)
Substituting this price into equations (D.8), (D.2), and (D.3), we derive the demand for intermediates, gross output, and net output of industry i as:
q∗(θ(i)) =1 Firm’s profit maximization problem can be simplified to:
max
D(i),A(i)Π(i) = 1
1 − ηP (i)D(i)α(i)A(i)1−α(i)− PD(i)D(i) − V A(i),
Taking first-order conditions with respect to D(i) and A(i), we can derive the price of the output in industry (i):
P (i) = (1 − η)α(i)−α(i) 1 − α(i)α(i)−1
· PD(i)α(i)· V1−α(i). (D.11) The ideal price index in the economy is:
1 =
D1 Technology Decisions of the Production Firm.
For the representative production firm in each industry, the decision of adopting existing machines (as created by the technology monopolist) will depend on the cost savings from using machines, which in turn are determined by relative factor prices. In particular, let δ(i) denote the cost savings from using machines instead of younger workers in industry i:
δ(i) = 1
When the effective cost of using young workers is higher than the cost of using machines — W/γy(z) > R
— the cost savings from using machines will be positive and the task will be assigned to machines only;
conversely, when R ≥ W/γy(z), it is more costly to use machines and thus the firm will use young workers only. Therefore, we can summarize these decisions by defining a technology threshold, θM(i), which satisfies
θM(i) =
(θ(i), if δ(i) > 0 0, if δ(i) ≤ 0.
We can now compute the cost of producing D-tasks and the wage-bill share of young workers in the pro-duction of D-tasks as:
The industry net output G(i) can be written as:
G, the demand for D-task D(i) is:
D(i) = 1 − η
The amount of younger workers required to produce D-tasks is given by:
Yd(i) =SY(i)PD(i)D(i)
W = 1
(2 − η)W(P (i))1−σα(i)SY(i). (D.18) Therefore, factor demands in the economy can be written as:
Od= Y
Aging and Equilibrium Wages. Let φ = Y +OO be the share of older workers in the working population.
The relative demand for old and young workers is:
O Let C(W, V, R) denote the unit cost of the final good. The equilibrium wages W and V are then given by the point on iso-cost curve C(W, V, R) = 1 that satisfies the relative demand of workers in equation (D.19).
The characterization of the equilibrium is shown in Figure D2. The horizontal and vertical axises represent young and old wages, respectively. The downward-sloping curve is the iso-cost curve. The equilibrium is achieved where ∂C/∂W∂C/∂V = 1−φφ . Population aging — an increase in φ — raises equilibrium young wages and lowers equilibrium old wages.
D2 Innovation Decisions of the Technology Monopolist
Now we turn to the innovation decision of the technology monopolist in each industry. Recall that the cost function of developing new technologies, C(θ(i)), is convex and increasing. We denote h(θ) = 1−C(θ)C0(θ) and further assume that h(θ) is positive and increasing. Using G(i) = (P (i))−σG and the fact that the monopolist takes wages (W and V ), machine rental price (R), and output (G) as given, her problem can be further simplified to:
max
θ(i)ΠM(i) = (1 − σ)α(i) ln PD(i) + ln(1 − C(θ(i))) (D.20) We can show that this profit maximization problem has a unique solution (in Appendix A2). Furthermore, the innovation decision θ∗(i) is increasing in young wages — the equilibrium innovation decision θ∗(i) is increasing in equilibrium wage for the younger workers W (shown in Appendix A2).
Aging and Innovation Decision. Finally, we can derive the effect of population aging on monopolist’s
Figure D2 Aging and Equilibrium Wages.
This figure illustrates how equilibrium wages, W and V , respond to the severity of aging, φ. The horizontal and vertical axises represent wages for older and younger workers, respectively. The downward-sloping curve is the iso-cost curve. The equilibrium is given by the the point at which the iso-cost curve is tangent to the relative demand of workers. An increase in φ — aging — will lead to higher young wage and lower old wage in the equilibrium.
innovation decision. In particular,
∂θ∗(i)
∂φ = ∂θ(i)
∂ ln(W∗)
∂ ln(W∗)
∂φ (D.21)
If this partial derivative is positive, equilibrium innovation is increasing with the severity of aging. Recall that equilibrium wage for younger workers is increasing in aging (Figure D2), it is sufficient to show that the semi-elasticity of innovation level θ∗(i) with respect to equilibrium old wages W∗ is positive (shown in Appendix A2):
∂θ∗(i)
∂ ln(W∗)= α(i) · (σ − 1)SY∗(i)(1 − SY∗(i))
h0(θ∗(i)) + h(θ∗(i))(1 − SY∗(i) − θ∗(i))> 0. (D.22) Therefore, the partial derivative in Equation (D.21) is positive, unambiguously. An increasing in φ — aging
— would induce technology change. In addition, the effect is more pronounced for industries with higher α(i) — that is, industries that rely more on age-deteriorating tasks are more responsive to population aging.
D3 Omitted Proofs of the Model
D3.1 Existence and Uniqueness of the Equilibrium
We start by proving that any critical point of Problem (D.20) is a local maximum and then the uniqueness of the maximum. Recall that the Monopolist’s profit maximization problem can be simplified as:
max
θ(i)ΠM(i) = (1 − σ)α(i) ln PD(i) + ln(1 − C(θ(i))) Take first and second-order condition of equation with respect to θ(i):
∂ΠM(i)
At every critical point, equation (D.23) holds with equality. Since (η − 1)π(i)SY(i) < 1 and h(θ) <
(1 − θ)h0(θ), equation (D.24) takes a negative value. Hence, any critical point is a local maximum.
Now let us suppose that ΠM(i) has two local maxima, θ0 and θ1 > θ0. By the intermediate value theorem, ΠM(i) has a local minimum in (θ0, θ1) which contradicts the fact that any critical point of ΠM(i) is a local maximum. Therefore, ΠM(i) is single peaked, and the critical point is the unique global maximum.
D3.2 Comparative Statics with respect to W
In this section, we show that the monopolist’s equilibrium innovation decision, θ∗(i), is nondecreasing in young wages, W . The cross-partial derivative of Problem (D.20) with respect to θ(i) and W is:
∂2ΠM(i)
Therefore, ∂∂θ(i)∂W2ΠM(i) ≥ 0 regardless of the value of η — the monopolist’s equilibrium innovation decision, θ∗(i), is nondecreasing in equilibrium wage for younger workers, W .
D3.3 Comparative Statics with respect to φ
Lastly, we show that the monopolist’s equilibrium innovation decision (θ∗(i)) is increasing in the severity of aging (φ); such effect is more pronounced if the industry she serves in relies more on age-deteriorating tasks (α(i)). Specifically,
∂θ∗(i)
∂φ = ∂θ∗(i)
∂ ln(W )×∂ ln(W )
∂φ > 0 (D.25)
Recall that young wages W is increasing in aging φ, it is sufficient to show that the semi-elasticity of innovation with respect to young wages is positive, i.e., ∂θ∗(i)/∂ ln(W ) > 0.
In the equilibrium, Equation (D.23) holds with equality:
h(θ∗(i)) = (σ − 1)α(i) SY(i)
1 − θ∗(i)δ(i). (D.26)
Take partial derivative of h(θ∗(i)) with respect to θ∗(i) and ln(W ):
∂h(θ∗(i))
θ∗(i) = SY(i)
1 − θ∗(i)(η − 1)h(θ∗(i))δ(i) − h0(θ∗(i)), (D.27)
∂h(θ∗(i))
ln(W ) = SY(i)
1 − θ∗(i)(θ − 1)α(i)(1 − SY(i)). (D.28) The implicit function theorem implies that the derivative of θ∗(i) with respect to ln(W ) is:
∂θ∗(i)
∂ ln(W )= α(i) · (σ − 1)SY∗(i)(1 − SY∗(i))
h0(θ∗(i)) + h(θ∗(i))(1 − S∗Y(i) − θ∗(i)) > 0. (D.29) Therefore, the equilibrium innovation choice, θ∗(i), is increasing in the severity of aging, φ. And the choice is more responsive to aging if the technology monopolist operates in an industry that relies more on age-deteriorating tasks, i.e., higher α(i).