Inflation, Unemployment, CPI, and Real GDP: Study Notes

Short-run relationships between inflation and unemployment

Rate of inflation can decrease; and when unemployment is too low, prices rise due to higher demand.
Interplay: both affect each other; inflation and unemployment are linked.
In the short term, the link between inflation and unemployment is negative: as one goes up, the other tends to go down.
Terminology:
- Negative relationship: variables move in opposite directions (one up, the other down).
- Positive relationship: variables move in the same direction (to be used later in the course).

Unemployment rate: definition and discouraged workers

Unemployment rate definition (in simplest terms): the percentage of people who are actively looking for a job and are available for work.
In class terms: people who have been actively looking for a job within the last four weeks.
Discouraged workers: people who have stopped looking for a job in the last four weeks and are not contacted; they are not counted as unemployed.

Key concepts: inflation, CPI, and related indices

Inflation = sustained increase in the general price level.
Market basket approach: we cannot track all goods and services, so we track a representative basket of goods and services.
Common price index: CPI (Consumer Price Index); PPI (Producer Price Index) is also used but CPI is more common.
CPI definition (in words): the price of a fixed basket of goods and services purchased by typical households, measured over time.
CPI vs. wage measures:
- Real wages capture purchasing power and living standards; nominal wages are measured in current dollars and may not reflect changes in prices.
Why economists focus on real wages: to assess changes in living standards rather than what nominal dollars buy at current prices.

Measuring inflation with the CPI: formulas and base year concepts

CPI for year t:
CPI_t = rac{Cost\ of\ basket\ in\ year\ t}{Cost\ of\ basket\ in\ base\ year} \times 100
Base year property:
- The base year CPI is set to 100 by construction, i.e., CPI_{base} = 100.
Inflation rate (annual) relative to the previous year:
\pit = \frac{CPIt - CPI{t-1}}{CPI{t-1}} \times 100
Note about wording in class material: the general idea is to compare current basket cost to base-year basket cost and scale to 100; then compute year-over-year changes for inflation.
Practical note: you can compute CPI for multiple years by calculating the basket cost in each year and applying the CPI formula; base year remains fixed.

Example: CPI calculation with a fixed basket (base year = 2020)

Basket contents (example): 1 loaf of bread, 1 movie ticket, 1 gallon of gasoline, 1 scented candle.
Basket costs by year (example):
- Base year cost = 26 → CPI_{2020} = \frac{26}{26} \times 100 = 100.
- Year 2 basket cost = 29 → CPI_{2021} = \frac{29}{26} \times 100 = 111.54.
- Year 3 basket cost = 30 → CPI_{2022} = \frac{30}{26} \times 100 = 115.38.
- Year 4 basket cost = 30.50 → CPI_{2023} = \frac{30.50}{26} \times 100 = 117.31.
Annual inflation (using year-to-year changes):
- Year 2021 inflation: \pi_{2021} = \frac{111.54 - 100}{100} \times 100 = 11.54\%.
- Year 2022 inflation: \pi_{2022} = \frac{115.38 - 111.54}{111.54} \times 100 \approx 3.46\%.
- Year 2023 inflation: \pi_{2023} = \frac{117.31 - 115.38}{115.38} \times 100 \approx 1.68\%.
Summary: CPI provides a way to translate a fixed basket of goods into a price index over time and then derive inflation rates.

Real vs nominal wages

Nominal wages: wages measured in current dollars (nominal terms).
Real wages: wages adjusted for inflation to reflect purchasing power.
Why real wages matter: real wages reveal changes in living standards and purchasing power, not just dollar amounts.
Formula:
RealWaget = \frac{NominalWaget}{\frac{CPIt}{100}} = NominalWaget \times \frac{100}{CPI_t}

Unemployment: types and causes

Three types of unemployment:
- Frictional unemployment: short-term unemployment as workers search for new jobs or transition between jobs.
- Structural unemployment: long-term unemployment due to mismatches between skills and jobs or due to technological/sectoral changes.
- Cyclical unemployment: unemployment linked to the business cycle; rises during recessions and falls during expansions.
Note on crises: cyclical unemployment tends to rise during recessions (economic downturns) and fall during recoveries; structural and frictional unemployment persist independently of the business cycle.
Pandemic example (from class discussion): during the pandemic, unemployment was largely frictional in nature because many positions were temporarily paused rather than permanently lost.

Part 2: Calculations using CPI (step-by-step approach)

Steps:
1) Determine basket cost in base year.
2) For each year t, determine basket cost with current prices.
3) Compute CPIt = \frac{Cost\ of\ basket\ in\ year\ t}{Cost\ of\ basket\ in\ base\ year} \times 100. 4) Compute annual inflation: \pit = \frac{CPIt - CPI{t-1}}{CPI{t-1}} \times 100 for t ≥ 2; \pi1 is NA (no previous year).
Example values (from the discussion):
- Base year: cost = 26 → CPI_{base} = 100. (So Year 1 CPI = 100 by construction.)
- Year 2: cost = 29 → CPI_{Year2} = 111.54. (29/26 × 100)
- Year 3: cost = 30 → CPI_{Year3} = 115.38. (30/26 × 100)
- Year 4: cost = 30.50 → CPI_{Year4} = 117.31. (30.50/26 × 100)
Resulting inflation rates (from year-to-year):
- Year 2: \pi_{Year2} = 11.54\%.
- Year 3: \pi_{Year3} \approx 3.46\%.
- Year 4: \pi_{Year4} \approx 1.68\%.

Real GDP: inflation-adjusted measure

Concept: Real GDP removes the effects of inflation to compare true output across years.
Method (using CPI):
RealGDPt = \frac{NominalGDPt}{\frac{CPIt}{100}} = NominalGDPt \times \frac{100}{CPI_t}
Example (using numbers from the discussion):
- Base year: NominalGDP{Year1} = 5; CPI{Year1} = 100 → RealGDP_{Year1} = 5.00.
- Year 2: NominalGDP{Year2} = 6.1; CPI{Year2} = 111.54 → RealGDP_{Year2} ≈ 6.1 × 100 / 111.54 ≈ 5.47.
- Year 3: NominalGDP{Year3} = 6.5; CPI{Year3} = 115.38 → RealGDP_{Year3} ≈ 6.5 × 100 / 115.38 ≈ 5.63.
- Year 4: NominalGDP{Year4} = 6.5; CPI{Year4} = 117.31 → RealGDP_{Year4} ≈ 6.5 × 100 / 117.31 ≈ 5.54.
Important use: Real GDP is central to tracking business cycles and recessions, since it reflects true production after removing price changes.
Connection to wages: the same inflation-adjustment method applies to wages and other economic variables when you want to compare across time in real terms.

Practical implications and connections

Real-world relevance: CPI and real vs nominal concepts are used to assess living standards, growth, and policy impacts.
Historical context: references to business cycles such as recessions, the Great Recession, and discussions of how unemployment reacts to economic events (e.g., crises, pandemics).
Methodological note: the approach shown (CPI-based deflators) can be used to adjust any nominal variable to real terms for comparison over time.

Quick reference: key formulas to remember

CPI (year t):
CPI_t = \frac{Cost\ of\ basket\ in\ year\ t}{Cost\ of\ basket\ in\ base\ year} \times 100
Inflation rate (year t relative to t-1):
\pit = \frac{CPIt - CPI{t-1}}{CPI{t-1}} \times 100
Real GDP from nominal GDP (using CPI):
RealGDPt = NominalGDPt \times \frac{100}{CPI_t}
Real wages from nominal wages (using CPI):
RealWaget = NominalWaget \times \frac{100}{CPI_t}
Unemployment rate (standard definition):
UR = \frac{U}{LF} \times 100
Unemployment types (descriptions in words): frictional, structural, cyclical

Notes on terminology used in class

Negative vs positive relationships:
- Negative: when one variable rises and the other falls (e.g., short-run unemployment vs inflation).
- Positive: when both variables move in the same direction (used later in the course).
Discouraged workers are not counted among the unemployed because they are not actively seeking work.
The base-year CPI is set to 100, and all subsequent CPIs are expressed relative to that base.

Summary takeaway

CPI provides a controlled way to measure how the cost of living changes over time.
Inflation rates are derived from year-to-year changes in the CPI.
Real GDP and real wages are obtained by adjusting nominal values with the CPI, allowing comparisons that reflect true purchasing power and production levels across years.
Understanding unemployment requires distinguishing frictional, structural, and cyclical causes, with cyclical unemployment being tied to the business cycle.