Question

Go through the information given below, and answer the questions that follow.
The three graphs below capture the relationship between economic (and social) activities and subjective well-being. The first graph (Graph-1) captures the relationship between GDP (per capita) and Satisfaction with life, across different countries and four islands: Gizo, Roviana, Niijhum Dwip, and Chittagong. Graph 2 captures three measures of subjective well-being (Satisfaction with life, Affect Balance and Momentary Affect) across the four islands, which have different monetization levels (Index). Graph 3 captures levels of thirteen different socio-economic activities across four islands.


Which of the following will BEST capture the relationship between GDP (x-axis) and Life Satisfaction (y-axis) of countries?

• A. y=x
• B. y=x²
• C. y = log (x)
• D. y = \(\frac{1}{x^2}\)
• E. y = ex

Answer 1

The Correct Option is C

The question requires us to determine the best mathematical relationship that captures the connection between GDP per capita (x-axis) and Life Satisfaction (y-axis) based on Graph-1.

Upon analyzing Graph-1, we observe:

1. As GDP per capita increases, Life Satisfaction also increases.
2. The curve levels off at higher GDP values, suggesting that the rate of increase in Life Satisfaction diminishes as GDP continues to rise.

This kind of relationship is typically logarithmic. A logarithmic function, \(y = \log (x)\), shows a rapid increase initially that slows down as x increases, matching our observation from the graph.

Let’s review the other options:

• \(y = x\): This represents a linear relationship, which is not evident in the graph because the curve is not a straight line.
• \(y = x^2\): This is a quadratic relationship, indicating increasing acceleration, which does not match the plateauing effect seen in the graph.
• \(y = \frac{1}{x^2}\): This represents an inverse quadratic relationship, where y decreases with increasing x, which is contrary to the observed trend.
• \(y = e^x\): This exponential function does not apply as it suggests a rapidly increasing rate without leveling off, unlike what we see in the graph.

Therefore, the best fit for the relationship between GDP per capita and Life Satisfaction in this context is \(y = \log(x)\).

Answer 2

The problem presented involves understanding the relationship between GDP (per capita) and Satisfaction with life, as depicted in the first graph mentioned. To identify the best mathematical model that captures this relationship among the given options, let's examine each choice regarding typical economic and well-being correlations: 

• y=x: This represents a linear relationship where life satisfaction increases directly and proportionally with GDP. However, real-world data often show diminishing returns on life satisfaction as GDP increases.
• y=x²: Represents a quadratic relationship. This model implies satisfaction increases more dramatically as GDP increases, which is not typical in socio-economic studies.
• y=log(x): A logarithmic relationship suggests that as GDP increases, life satisfaction also increases, but at a decreasing rate, which aligns well with common economic observations where initial GDP increases bring significant satisfaction, but the rate of satisfaction gain decreases over time.
• y=\(\frac{1}{x^2}\): An inverse square relationship implies that as GDP increases, satisfaction decreases, which contradicts typical observations.
• y=e^x: An exponential model suggests rapid growth in satisfaction with increasing GDP, which is unrealistic over a large GDP range.

Given the analysis above, the most appropriate relationship that matches economic theories and observations is the logarithmic relationship, y=log(x), as it best reflects the diminishing returns observed in life satisfaction with increasing GDP across different countries and regions.