Notes on Price and Quantity Demanded under Ceteris Paribus

Ceteris Paribus (All Else Equal)

  • Key assumption used when analyzing how one variable changes while holding all other determinants constant.
  • In this context, we analyze how quantity demanded responds to a change in price, keeping income, tastes, prices of other goods, expectations, and the number of buyers unchanged.
  • This assumption underpins the idea of the demand curve: a relationship between price and quantity demanded holding everything else constant.

Law of Demand

  • Inverse relationship between price and quantity demanded: as price falls, quantity demanded rises; as price rises, quantity demanded falls.
  • This results in a downward-sloping demand curve when plotted with price on the vertical axis and quantity demanded on the horizontal axis.
  • Intuition: lower price makes the good more affordable, leading more consumers (or existing buyers) to purchase; higher price discourages purchases.

Quantity Demanded Change When Price Changes

  • Along the demand curve: a change in price causes a movement along the curve, changing the quantity demanded.
  • If price changes but all other determinants stay the same, the curve itself does not shift; only the point on the curve moves.
  • Distinction:
    • Movement along the curve: due to price change (ceteris paribus).
    • Shift of the curve: due to changes in income, tastes, prices of related goods, expectations, or number of buyers.

Demand Curve Shifts vs Movement

  • Shifts occur when non-price determinants change:
    • Income (normal vs inferior goods)
    • Prices of related goods (substitutes and complements)
    • Tastes and preferences
    • Expectations about future prices
    • Number of buyers in the market
  • When a curve shifts, at the same price, quantity demanded changes.

Elasticity of Demand (Price Elasticity of Demand)

  • Definition (discrete changes):
    ϵ<em>d=%ΔQ</em>d%ΔP=ΔQ<em>d/Q</em>dΔP/P\epsilon<em>d = \frac{\% \Delta Q</em>d}{\% \Delta P} = \frac{\Delta Q<em>d / Q</em>d}{\Delta P / P}
  • Definition (point elasticity, for small changes):
    ϵ<em>d=dQ</em>ddP⋅PQd\epsilon<em>d = \frac{dQ</em>d}{dP} \cdot \frac{P}{Q_d}
  • Interpretation (in absolute value):
    • Elastic: |\epsilon_d| > 1 (quantity demanded responds strongly to price changes)
    • Inelastic: |\epsilon_d| < 1 (quantity demanded responds weakly)
    • Unit elastic: |\epsilon_d| = 1
  • Sign convention: usually negative for standard downward-sloping demand; we report the absolute value for classification.

Total Revenue and Elasticity

  • Total revenue (TR):
    TR=P×QTR = P \times Q
  • Relationship between elasticity and total revenue:
    • If demand is elastic (|\epsilon_d| > 1), a price decrease increases TR because the percentage rise in quantity demanded more than offsets the price drop. Conversely, a price increase decreases TR.
    • If demand is inelastic (|\epsilon_d| < 1), a price increase increases TR because the percentage gain in price dominates the smaller percentage drop in quantity.
    • If unit elastic (|\epsilon_d| = 1), TR remains unchanged when price changes (to first order).

Determinants of Elasticity

  • Availability of close substitutes: more substitutes -> more elastic.
  • Budget share: goods that consume a larger share of income tend to have more elastic demand.
  • Necessity vs luxury: necessities tend to be inelastic; luxuries tend to be elastic.
  • Time horizon: demand is more elastic in the long run as consumers adjust (find substitutes, change behavior).
  • Definition of the market: narrow definitions (e.g., Pixie Stix) yield more elastic demand than broad definitions (e.g., snacks).

Example: Pizza (Pizza as a Good)

  • Scenario: price change from $P0$ to $P1$ with quantity demanded changing from $Q0$ to $Q1$, holding all else constant.
    • Suppose $P0 = 2.00$, $Q0 = 100$ pizzas; $P1 = 2.20$, $Q1 = 80$ pizzas.
    • Changes:
      ΔP=P<em>1−P</em>0=0.20(increase)\Delta P = P<em>1 - P</em>0 = 0.20\quad (\text{increase})
      ΔQ=Q<em>1−Q</em>0=−20(decrease)\Delta Q = Q<em>1 - Q</em>0 = -20\quad (\text{decrease})
    • Percent changes:
      %ΔP=ΔPP<em>0=0.202.00=0.10\%\Delta P = \frac{\Delta P}{P<em>0} = \frac{0.20}{2.00} = 0.10%ΔQ=ΔQQ</em>0=−20100=−0.20\%\Delta Q = \frac{\Delta Q}{Q</em>0} = \frac{-20}{100} = -0.20
    • Elasticity estimate:
      ϵd=%ΔQ%ΔP=−0.200.10=−2.0\epsilon_d = \frac{\%\Delta Q}{\%\Delta P} = \frac{-0.20}{0.10} = -2.0
    • Interpretation: |\epsilon_d| = 2.0 > 1 => elastic demand at this price range.
    • Implication for TR:
    • Before: $TR0 = P0 \times Q_0 = 2.00 \times 100 = 200$.
    • After: $TR1 = P1 \times Q_1 = 2.20 \times 80 = 176$.
    • TR decreased when price increased; consistent with elastic demand.
  • Alternative scenario: if price decreased from $2.00 to $1.60 and quantity rose from 100 to 130,
    • $\%\Delta P = -20\%$, $\%\Delta Q = +30\%$, $\epsilon_d = -1.5$ (elastic), and $TR$ would increase.

Connections to Foundational Principles and Real-World Relevance

  • Microeconomic core: how consumers respond to price changes affects market prices, quantities, and welfare.
  • Pricing strategies: firms consider elasticity to decide on pricing, discounts, or promotions to maximize revenue.
  • Public policy: understanding elasticity informs tax incidence and welfare effects of price controls or tariffs.
  • Ethical/practical implications: pricing that exploits inelastic demand for essential goods can affect access and equity.

Summary of Key Points

  • Ceteris Paribus: analyze price effects holding all else constant.
  • Law of Demand: price and quantity demanded move in opposite directions; downward-sloping demand.
  • Movement vs Shift: price changes cause movement along the curve; non-price determinants shift the curve.
  • Elasticity: measures responsiveness; formulae in percent changes or derivatives.
  • TR and Elasticity: how price changes affect total revenue depending on elasticity.
  • Determinants: substitutes, income share, necessity vs luxury, time, market definitions.
  • Pizza example demonstrates calculation of elasticity and TR implications.

Notation and Formulas (LaTeX)

  • Elasticity:
    ϵ<em>d=%ΔQ</em>d%ΔP=ΔQ<em>dQ</em>d÷ΔPP\epsilon<em>d = \frac{\% \Delta Q</em>d}{\% \Delta P} = \frac{\Delta Q<em>d}{Q</em>d} \div \frac{\Delta P}{P}
    or
    ϵ<em>d=dQ</em>ddP⋅PQd\epsilon<em>d = \frac{dQ</em>d}{dP} \cdot \frac{P}{Q_d}
  • Total Revenue:
    TR=P×QTR = P \times Q
  • Interpretation rules for elasticity:
    • Elastic: |\epsilon_d| > 1
    • Inelastic: |\epsilon_d| < 1
    • Unit elastic: ∣ϵd∣=1|\epsilon_d| = 1
  • Price change example (discrete):
    • %ΔP=P<em>1−P</em>0P0\%\Delta P = \frac{P<em>1 - P</em>0}{P_0}
    • %ΔQ=Q<em>1−Q</em>0Q0\%\Delta Q = \frac{Q<em>1 - Q</em>0}{Q_0}
    • ϵd=%ΔQ%ΔP\epsilon_d = \frac{\%\Delta Q}{\%\Delta P}