Question

Valuing human capital with wage growth: To make the calculation of the present discounted value of a worker's human capital more realistic, suppose labor income starts at $50,000 initially, but then grows at a constant rate of 2% per year after that. Let wₜ be labor income in year t, so that wₜ = w₀(1+g)ₜ where w₀ = $50,000 and g = 0.02. The steps below will walk you through the problem.

(a) If the interest rate is R, what is the formula for the present discountec value today (in year 0 ) of labor income from a particular future year t ?
(b) Now add up these terms from t=0 to t=45 to get a formula for the present discounted value of labor income.
(c) Write your answer to part (b) so that it takes the form of the geometric series
pdv = w₀(1+a+a²+a³+⋯+a⁴⁵).
What is the value of a that you find? (d) Apply the geometric series formula to compute the present discounted value for the case of R=0.04,R=0.03, and R=0.02. What weird thing happens (and why) when R=0.02 ?
(e) Comment on your results.

Answer 1

The formulas for the present discounted value (PDV) of labor income in the future are derived. By adding up PDV terms from t=0 to t=45, a formula for the total PDV of labor income is obtained. The formula is written in the form of a geometric series, and the value of 'a' is calculated. Applying the geometric series formula for different interest rates reveals that when the interest rate equals the growth rate, the PDV becomes infinite.

Step-by-step explanation:

(a) The formula for the present discounted value (PDV) today (in year 0) of labor income from a particular future year t is given by:

PDV = wₜ / (1 + R)ⁿ

where wₜ is the labor income in year t, R is the interest rate, and ⁿ is the number of periods (years) from year 0 to year t.

(b) Adding up the PDV terms from t=0 to t=45, we get the formula for the present discounted value of labor income as:

PDV = w₀(1/(1+R)⁰ + 1/(1+R)¹ + 1/(1+R)² + ... + 1/(1+R)⁴⁵)

(c) Rewriting the answer from part (b) in the form of a geometric series, we have:

PDV = w₀(1 + a + a² + a³ + ... + a⁴⁵)

where a = 1/(1+R).

(d) Applying the geometric series formula to compute the PDV for R=0.04, R=0.03, and R=0.02, we find:

• PDV for R=0.04 is w₀ / (1 + 0.04)
• PDV for R=0.03 is w₀ / (1 + 0.03)
• PDV for R=0.02 is w₀ / (1 + 0.02)

When R=0.02, the PDV becomes infinite because the denominator (1+R) equals 1, making the PDV equal to w₀. This happens because the growth rate of labor income (2%) is equal to the interest rate (2%), resulting in the infinite accumulation of income.

(e) The results indicate that as the interest rate approaches the growth rate of labor income, the present discounted value approaches infinity, implying that the value of labor income becomes unbounded.

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Answer 2

To calculate the present discounted value of labor income, the formula PDVt = wt / (1+R)t is used, summing this from t=0 to t=45 to get the total PDV. An interesting result occurs when the interest rate R is equal to the income growth rate g, leading to an indefinite increase in PDV, which demonstrates the significant impact of these rates on valuation.

Step-by-step explanation:

To calculate the present discounted value (PDV) of a worker's labor income that starts at $50,000 and grows at 2% per year, we begin with the formula for labor income in year t, which is wt = w0(1+g)t, where w0 = $50,000 and g = 0.02. The formula for the PDV of labor income for a particular future year t is given by PDVt = wt / (1+R)t where R is the interest rate.

To find the total PDV from t=0 to t=45, we add up all these terms, resulting in a sum of a geometric series. The formula of this series is pdv = w0(1+a+a2+a3+ … +a45), where a = (1+g) / (1+R). The value of a represents how much one year's labor income is worth in today's dollars, factoring in the growth of income and the interest rate.

For computation, we apply the geometric series formula with different values of R such as 0.04, 0.03, and 0.02. An interesting scenario arises when R = 0.02, which is equal to the growth rate of the labor income; here, the PDV increases indefinitely since the growth rate of the income is canceling out the discounting effect of the interest rate. This illustrates the importance of the relationship between wage growth and interest rates in determining the PDV of human capital.