(a) On the basis of the figures given below, calculate the age-specific death rates (ASDRs) for all the age groups. Also…

(a) ASDR = (deaths in age group / population in age group) × 1000. Total population = 10000 + 18000 + 26000 + 20000 + 5000 = 79000. Total deaths = 220 + 40 + 62 + 350 + 2000 = 2672.

Original ASDRs (per 1000):

• 0–10: 220/10000 × 1000 = 22.000
• 10–30: 40/18000 × 1000 = 2.222
• 30–50: 62/26000 × 1000 = 2.385
• 50–70: 350/20000 × 1000 = 17.500
• 70+: 2000/5000 × 1000 = 400.000

CDR = Σ(ASDRᵢ × Pᵢ)/ΣPᵢ = (22.000×10000 + 2.222×18000 + 2.385×26000 + 17.500×20000 + 400.000×5000)/79000 = 2672000/79000 = 33.823 per 1000.

Corrected age data: recorded ages 37 and 45 lie in 30–50. True ages are 47 and 54. Since 47 remains in 30–50, but 54 moves to 50–70, one death moves from 30–50 to 50–70. New deaths: 30–50 = 61, 50–70 = 351. Population unchanged.

Corrected ASDRs (per 1000):

• 0–10: 22.000
• 10–30: 2.222
• 30–50: 61/26000 × 1000 = 2.346
• 50–70: 351/20000 × 1000 = 17.550
• 70+: 400.000

Corrected CDR = 2672/79000 × 1000 = 33.823 per 1000 (unchanged, as total deaths and population are unchanged).

(b) Identification means whether the structural parameters can be uniquely recovered from the reduced-form parameters. For example, in the demand-supply model Q_d = α + βP + u, Q_s = γ + δP + v, with equilibrium Q_d = Q_s = Q, only P and Q are observed. Different pairs (α, β, γ, δ) can yield the same reduced-form equations for P and Q, so the structural coefficients are not identified without an exogenous shifter.

Let G be the number of endogenous variables and K the number of predetermined variables (including intercept). For the i-th equation, let gᵢ be the number of endogenous variables appearing in it and kᵢ the number of predetermined variables appearing in it. Order condition: K − kᵢ ≥ gᵢ − 1. Rank condition: delete row i from the structural coefficient matrix; take the columns corresponding to variables excluded from equation i; this submatrix must have rank G − 1.

Here G = 4: y₁, y₂, y₃, y₄. Predetermined variables: intercept, x₁, x₂, x₃, so K = 4.

Equation 1: g₁ = 3, k₁ = 3. Order: 4 − 3 = 1 < 3 − 1 = 2, so it fails. Excluded variables y₄, x₃ give rank 1 < 3. Equation 1 is underidentified.

Equation 2: g₂ = 2, k₂ = 3. Order: 4 − 3 = 1 = 2 − 1. Excluded variables y₁, y₄, x₃ give a submatrix of rank 2 < 3. Equation 2 is underidentified.

Equation 3: g₃ = 2, k₃ = 3. Order: 4 − 3 = 1 = 2 − 1. Excluded variables y₂, y₄, x₃ give rank 2 < 3. Equation 3 is underidentified.

Equation 4: g₄ = 3, k₄ = 1 (only x₃; no intercept). Order: 4 − 1 = 3 ≥ 3 − 1 = 2. Excluded variables y₃, intercept, x₁, x₂ give rank 3 = G − 1. Equation 4 is overidentified and identified.

Thus the first three equations are underidentified, while the fourth equation is identified.

(c) qₓ = 0.001 + 0.0002x. Starting with l₅₀ = 10000. Use dₓ = lₓ qₓ, lₓ₊₁ = lₓ − dₓ, Lₓ = (lₓ + lₓ₊₁)/2.

• x = 50: l₅₀ = 10000.000, q₅₀ = 0.0110, d₅₀ = 110.000, L₅₀ = 9945.000
• x = 51: l₅₁ = 9890.000, q₅₁ = 0.0112, d₅₁ = 110.768, L₅₁ = 9834.616
• x = 52: l₅₂ = 9779.232, q₅₂ = 0.0114, d₅₂ = 111.483, L₅₂ = 9723.490
• x = 53: l₅₃ = 9667.749, q₅₃ = 0.0116, d₅₃ = 112.146, L₅₃ = 9611.676
• x = 54: l₅₄ = 9555.603, q₅₄ = 0.0118, d₅₄ = 112.756, L₅₄ = 9499.225
• x = 55: l₅₅ = 9442.847, q₅₅ = 0.0120, d₅₅ = 113.314, L₅₅ = 9386.190
• x = 56: l₅₆ = 9329.533, q₅₆ = 0.0122, d₅₆ = 113.820, L₅₆ = 9272.622
• x = 57: l₅₇ = 9215.712, q₅₇ = 0.0124, d₅₇ = 114.275, L₅₇ = 9158.575
• x = 58: l₅₈ = 9101.437, q₅₈ = 0.0126, d₅₈ = 114.678, L₅₈ = 9044.098
• x = 59: l₅₉ = 8986.759, q₅₉ = 0.0128, d₅₉ = 115.031, L₅₉ = 8929.244
• x = 60: l₆₀ = 8871.729, q₆₀ = 0.0130, d₆₀ = 115.332, L₆₀ = 8814.063

For x = 60, l₆₁ = 8871.729 − 115.332 = 8756.396, so L₆₀ = (8871.729 + 8756.396)/2 = 8814.063.