Question

The table below shows unemployment rates in the capital (London) and the rest of the country. One-third of the national population lives in the capital. Construct an index of national unemployment, treating 2000 as 100 . What weights did you use for the two unemployment rates? Why? $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \begin{array}{l} \text { Unemployment } \\ \text { (\%) } \end{array} & 1997 & 1998 & 1999 & 2000 & 2001 & 2002 & 2003 \\ \hline \text { London } & 7 & 6 & 5 & 4 & 6 & 5 & 4 \\ \hline \text { Rest of country } & 10 & 9 & 8 & 8 & 9 & 8 & 8 \\ \hline \end{array} $$

Short answer

Weights are \( \frac{1}{3} \) for London and \( \frac{2}{3} \) for the rest. These reflect population shares. National index values are derived from weighted unemployment relative to the base year 2000.

Step-by-step solution

Understand the Problem

We want to construct an index of national unemployment rates with 2000 as the base year (index = 100). The task requires weighing the unemployment rates of London and the rest of the country using population distribution.

Determine Weights

Since one-third of the national population lives in London, the weight for London is \( \frac{1}{3} \), and the weight for the rest of the country is \( \frac{2}{3} \). This reflects their proportion of the total population.

Calculate Weighted Unemployment Rates for Each Year

Calculate the national weighted unemployment rate for each year using the formula: \[ \text{National Rate} = (\text{London Rate} \times \frac{1}{3}) + (\text{Rest Rate} \times \frac{2}{3}) \]. Apply this formula for all years from 1997 to 2003.

Example Calculation for 1997

For 1997: \[ (7 \times \frac{1}{3}) + (10 \times \frac{2}{3}) = 2.33 + 6.67 = 9 \]. Repeat similar calculations for each year.

Reference Year Calculation

Identify the national unemployment rate for the year 2000, used as the base. Using weighted calculations: \[ (4 \times \frac{1}{3}) + (8 \times \frac{2}{3}) = 1.33 + 5.33 = 6.66 \].

Construct the Index

Use the formula: \[ \text{Index Value for Year} = \left(\frac{\text{National Rate for Year}}{\text{National Rate for 2000}}\right) \times 100 \]. Calculate this for each year to transform them into index numbers based on 2000.

Present Index Values

Index values are calculated by scaling each year's weighted unemployment rate relative to the year 2000 rate (6.66), then multiplying by 100. For example, for 1997 it would be \[ \left(\frac{9}{6.66}\right) \times 100 \approx 135 \].

Key concepts

Weighted Average

A weighted average is a method used to calculate a single representative value from different data sets that have varying levels of importance. In simple terms, it’s like finding the balance or "average" of numbers, but giving more weight to certain numbers that you deem more significant. This is particularly helpful when handling data from different categories that don’t contribute equally to an outcome.

In the context of an unemployment index, weighting allows us to incorporate the different population distributions of regions like London and the rest of the country. By applying a weight, we can create a national rate that truly reflects the influence of each region.

For instance, since one-third of the population lives in London, its unemployment rate gets a weight of \( \frac{1}{3} \), while the larger rest of the country gets a weight of \( \frac{2}{3} \). This ensures that the calculated national unemployment rate accounts for the more substantial influence of the larger population living outside London. The formula for the weighted average becomes:

• Weighted National Rate = (London Rate \( \times \frac{1}{3} \)) + (Rest Rate \( \times \frac{2}{3} \)).

Unemployment Rate

The unemployment rate is a critical economic indicator that represents the percentage of the labor force that is jobless and actively seeking employment. It provides a snapshot of the job market and overall economic health.

Higher unemployment rates indicate a faltering economy where jobs aren’t growing fast enough to accommodate those seeking employment. Conversely, lower rates suggest a thriving job market. In this exercise, the unemployment rates are yearly figures for London and the rest of the country, showing how these rates change over time.

Understanding these figures helps policymakers and researchers evaluate which regions might require more resources or intervention. For example, consistently high unemployment in the rest of the country compared to London could signal different economic challenges or opportunities in these areas.

• London Rate: Reflects the job market condition in the capital.
• Rest Rate: Shows the employment status outside of London, covering a broader and diverse area.

Population Distribution

Population distribution refers to how people are spread across different geographical areas. It can significantly impact how various statistics are calculated and interpreted, especially in economic studies.

In the unemployment index context, knowing that one-third of the population is in London affects how unemployment figures are viewed. This distribution influences the weighting in the national unemployment index, ensuring that disparities in population don’t skew the results.

Here, the distribution impacts the weights used to calculate the national unemployment rate. Since the populations are not evenly distributed, the contribution of each region to the overall unemployment index is adjusted accordingly. This approach helps ensure a more accurate reflection of reality:

• London, despite having a smaller population, is significant because of its unique economic characteristics.
• The larger, more varied population of the rest of the country exerts a stronger influence on the national rate.

Understanding such distributions allows better decision-making and policy planning to address regional disparities.