Problem 34 - Applied Calculus

Question

An Australian \(^{23}\) study found that, if other factors are constant (education, experience, etc.), taller people receive higher wages for the same work. The study reported a "height premium" for men of \(3 \%\) of the hourly wage for a \(10 \mathrm{~cm}\) increase in height; for women the height premium reported was \(2 \%\). We assume that hourly wages are a linear function of height, with slope given by the height premium at the average hourly wage for that gender. (a) The average hourly wage \(^{24}\) for a \(178 \mathrm{~cm}\) Australian man is AU$$\$29.40$$. Express the average hourly wage of an Australian man as a function of his height, \(x\) \(\mathrm{cm} .\) (b) The average hourly wage for a \(164 \mathrm{~cm}\) Australian woman is AU$$\$24.78$$. Express the average hourly wage of an Australian woman as a function of her height, \(y \mathrm{~cm}\). (c) What is the difference in average hourly wages between men and women of height \(178 \mathrm{~cm}\) ? (d) Is there a height for which men and women are predicted to have the same wage? If so, what is it?

Step-by-Step Solution

Answer

(a) \(w_m(x) = 29.40 + 0.0882(x - 178)\) (b) \(w_w(y) = 24.78 + 0.04956(y - 164)\) (c) Approx. AU$3.93 (d) No realistic height for equal wages.

Step 1: Define Men's Wage Function

For men, we know the average hourly wage for a 178 cm man is AU$29.40. The height premium is 3% for every 10 cm increase in height. This means the slope of the function for men's wages is \(0.03 \times \frac{29.40}{10}\). Let's calculate the slope and define the wage function. Let \(x\) be the height:\[ w_m(x) = 29.40 + 0.003 \times (x - 178) \times 29.40 \] Simplifying:\[ w_m(x) = 29.40 + 0.0882(x - 178) \]

Step 2: Define Women's Wage Function

For women, the average hourly wage for a 164 cm woman is AU$24.78. The height premium is 2% per 10 cm. Thus, the slope for women's wages is \(0.02 \times \frac{24.78}{10}\). Calculate the slope and define the wage function. Let \(y\) be the height:\[ w_w(y) = 24.78 + 0.002 \times (y - 164) \times 24.78 \] Simplifying:\[ w_w(y) = 24.78 + 0.04956(y - 164) \]

Step 3: Calculate Wage Difference for 178 cm

To find the difference in wages between a 178 cm man and woman, evaluate both functions at 178 cm:For men:\[ w_m(178) = 29.40 + 0.0882(178 - 178) = 29.40 \]For women:\[ w_w(178) = 24.78 + 0.04956(178 - 164) \approx 25.47184 \]The wage difference:\[ 29.40 - 25.47184 = 3.92816 \]

Step 4: Equalize Men's and Women's Wages

We set \(w_m(x) = w_w(y)\) to find the height where men's and women's wages are equal:\[ 29.40 + 0.0882(x - 178) = 24.78 + 0.04956(y - 164) \]Since we want the same height for both, set \(x = y\):\[ 29.40 + 0.0882(x - 178) = 24.78 + 0.04956(x - 164) \]Simplify and solve for \(x\):\[ 0.0882x - 15.6976 = 0.04956x + 8.13024 \]\[ 0.03864x = 23.82784 \]\[ x \approx 616.449 \]This height is impractical, indicating no realistic height where wages are equal.

Key Concepts

Wage Function Modeling

Wage function modeling is an insightful approach that allows us to express wages as a function of different variables or factors, such as height in this context. Here, wage is considered a linear function, which means it is represented as a straight line on a graph. The idea is that certain factors, like height, consistently impact wages in a predictable fashion.

For instance, in this exercise, we are given specific wage premiums based on height — 3% for every additional 10 cm for men, and 2% for the same increment for women. This translates into a slope when we set up the linear functions for wages. The slope indicates how rapidly wages increase with height. To create these functions, we start with the known average hourly wage and adjust for height using the defined premium:

• Men: Starting wage is AU$29.40 for 178 cm. The wage increases by 0.3% per cm with a calculated slope of 0.0882.
• Women: Starting wage is AU$24.78 for 164 cm. The wage increases by 0.2% per cm with a slope of 0.04956.

The equations formed resemble typical linear equations in mathematics, where the starting wage corresponds to the y-intercept, and the premium per cm is analogous to the slope in these lines.

Height Premium

The term 'height premium' refers to the extra compensation or advantage in wages that taller individuals receive compared to their shorter counterparts. This concept helps us explore the economic implications of physical characteristics like height on earnings.

According to the exercise, the height premium applies differently to men and women. Men enjoy a greater premium than women at 3% per 10 cm compared to 2% for women. This means men see a higher increase in their wages with height than women do. The premium essentially acts as an adjustment factor in the wage function, reflecting how incremental changes in height create variations in pay.

It is crucial to understand that such premiums are not arbitrary but based on observed data and research, like the Australian study mentioned. Height influences perceptions of individuals in the workplace and can impact job opportunities and salary negotiations, which explains the existing height premium in various industries.

Gender Wage Gap

The gender wage gap is a widely discussed social and economic issue referring to the difference in earnings between men and women. This exercise illustrates an intersection of the gender wage gap with the concept of height premium.

Despite adjustments for height, wage discrepancies between genders persist. Men, irrespective of height, earn more on average than women, highlighting a component of the gender wage gap. For example, at the comparison point of 178 cm, men earn AU$29.40, while women earn approximately AU$25.47, showcasing a gap of AU$3.93.

This difference could be attributed to various factors, including discrimination, societal norms, and historical pay structures that may undervalue women's work. Moreover, in this exercise, the hypothetical scenario posits a shared height where wages would be equal, but it reveals an impractical solution (at 616.44 cm), further emphasizing the ingrained wage disparity.

Tackling the gender wage gap requires multi-faceted solutions, including policy reform and changing workplace cultures to ensure fairness regardless of gender.