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1.5 — Demand

ECON 306 • Microeconomic Analysis • Fall 2022

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microF22
microF22.classes.ryansafner.com

Outline

Income Effect

Digression: Measuring Change

Cross-Price Effects

The consumer’s Problem: Review

  • We now can explore the dynamics of how individuals optimally respond to changes in their constraints

  • We know the (static) problem is:

  1. Choose: < a consumption bundle >

  2. In order to maximize: < utility >

  3. Subject to: < income and market prices >

A Demand Function (for Good X)

  • A consumer’s demand (for good x) depends on current prices & income:

$$q_x^D = q_x^D(m, p_x, p_y)$$

  • How does demand (for x) change?

A Demand Function (for Good X)

  • A consumer’s demand (for good x) depends on current prices & income:

$$q_x^D = q_x^D(m, p_x, p_y)$$

  • How does demand (for x) change?
  1. Income effects \(\left(\frac{\Delta q_x^D}{\Delta m}\right)\): how \(q_x^D\) changes with changes in income

A Demand Function (for Good X)

  • A consumer’s demand (for good x) depends on current prices & income:

$$q_x^D = q_x^D(m, p_x, p_y)$$

  • How does demand (for x) change?
  1. Income effects \(\left(\frac{\Delta q_x^D}{\Delta m}\right)\): how \(q_x^D\) changes with changes in income
  2. Cross-price effects \(\left(\frac{\Delta q_x^D}{\Delta p_y}\right)\): how \(q_x^D\) changes with changes in prices of other goods (e.g. \(y)\)

A Demand Function (for Good X)

  • A consumer’s demand (for good x) depends on current prices & income:

$$q_x^D = q_x^D(m, p_x, p_y)$$

  • How does demand (for x) change?
  1. Income effects \(\left(\frac{\Delta q_x^D}{\Delta m}\right)\): how \(q_x^D\) changes with changes in income
  2. Cross-price effects \(\left(\frac{\Delta q_x^D}{\Delta p_y}\right)\): how \(q_x^D\) changes with changes in prices of other goods (e.g. \(y)\)
  3. (Own) Price effects \(\left(\frac{\Delta q_x^D}{\Delta p_x}\right)\): how \(q_x^D\) changes with changes in price (of \(x)\)

Income Effect

Income Effect

  • Income effect: change in optimal consumption of a good associated with a change in (nominal) income, holding relative prices constant

$$\frac{\Delta q_D}{\Delta m} >^{?}< 0$$

Income Effect (Normal)

  • Consider football tickets and vacation days

Income Effect (Normal)

  • Consider football tickets and vacation days

  • Suppose income \((m)\) increases

Income Effect (Normal)

  • Consider football tickets and vacation days

  • Suppose income \((m)\) increases

  • At new optimum \((B)\), consumes more of both

  • Then both goods are normal goods

Income Effect (Inferior)

  • Consider ramen and steak

Income Effect (Inferior)

  • Consider ramen and steak

  • Suppose income \((m)\) increases

Income Effect (Inferior)

  • Consider ramen and steak

  • Suppose income \((m)\) increases

  • At new optimum \((B)\), consumes more steak, less ramen

  • Steak is a normal good, ramen is an inferior good

Income Effect

$$\frac{\Delta q_D}{\Delta m} >^{?}< 0$$

  • Normal goods: consumption increases with more income (and vice versa)

  • Inferior goods: consumption decreases with more income (and vice versa)

Digression: Measuring Change

Quantifying Changes I

  • Several ways we can talk about how a measure changes over time, from time \(t_1 \rightarrow t_2\)

  • Difference \((\Delta)\): the difference between the value at time \(t_1\) and time \(t_2\) $$\Delta t=t_2-t_1$$

Quantifying Changes II

  • Several ways we can talk about how a measure changes over time, from time \(t_1 \rightarrow t_2\)

  • Difference \((\Delta)\): the difference between the value at time \(t_1\) and time \(t_2\) $$\Delta t=t_2-t_1$$

  • Relative Difference: the difference expressed in terms of the original value $$\frac{\Delta t}{t_1} = \frac{t_2-t_1}{t_1}$$ this becomes a proportion (a decimal)

Quantifying Changes III

  • Percentage Change (Growth Rate): relative difference expressed as a percentage

$$\begin{align*} \% \Delta &= \frac{\Delta t}{t_1} \times 100\%\\ &=\frac{t_2-t_1}{t_1} \times 100\% \\ \end{align*}$$

A Simple Example Growth Rate

Example: A country's GDP is $100bn in 2021, and $120bn in 2022 Calculate the country's GDP growth rate for 2022:

Elasticity, in General

$$\epsilon_{y,x} = \frac{\% \Delta y}{\% \Delta x}=\cfrac{\frac{\Delta y}{y}}{\frac{\Delta x}{x}}$$

  • An elasticity between any two variables \(y\) and \(x\) describes the responsiveness of a variable \((y)\) to a change in another \((x)\).
    • (relative change in \(y\)) over (relative change in \(x\))

Elasticity, in General

$$\epsilon_{y,x} = \frac{\% \Delta y}{\% \Delta x}=\cfrac{\frac{\Delta y}{y}}{\frac{\Delta x}{x}}$$

  • An elasticity between any two variables \(y\) and \(x\) describes the responsiveness of a variable \((y)\) to a change in another \((x)\).

    • (relative change in \(y\)) over (relative change in \(x\))

  • Interpretation: \(\epsilon_{y,x}=\) the percentage change in \(y\) from a 1% change in \(x\)

Elasticity, in General

$$\epsilon_{y,x} = \frac{\% \Delta y}{\% \Delta x}=\cfrac{\frac{\Delta y}{y}}{\frac{\Delta x}{x}}$$

  • An elasticity between any two variables \(y\) and \(x\) describes the responsiveness of a variable \((y)\) to a change in another \((x)\).

    • (relative change in \(y\)) over (relative change in \(x\))

  • Interpretation: \(\epsilon_{y,x}=\) the percentage change in \(y\) from a 1% change in \(x\)

  • Unitless: easy comparisons between any 2 variables

    • e.g. crime rates and police, GDP and gov't spending, inequality and corruption

Income Elasticity of Demand I

  • The income elasticity of demand measures how much quantity demanded \((q_D)\) changes in response to a change in income \((m)\)

$$\epsilon_{q,m}= \frac{\% \Delta q_D}{\% \Delta m}$$

Income Elasticity of Demand I

  • The income elasticity of demand measures how much quantity demanded \((q_D)\) changes in response to a change in income \((m)\)

$$\epsilon_{q,m}= \frac{\% \Delta q_D}{\% \Delta m}$$

  • If \(\epsilon_{q,m}\) is negative: an inferior good

Income Elasticity of Demand I

  • The income elasticity of demand measures how much quantity demanded \((q_D)\) changes in response to a change in income \((m)\)

$$\epsilon_{q,m}= \frac{\% \Delta q_D}{\% \Delta m}$$

  • If \(\epsilon_{q,m}\) is negative: an inferior good
  • If \(\epsilon_{q,m}\) is positive: a normal good

Income Elasticity of Demand I

  • The income elasticity of demand measures how much quantity demanded \((q_D)\) changes in response to a change in income \((m)\)

$$\epsilon_{q,m}= \frac{\% \Delta q_D}{\% \Delta m}$$

  • If \(\epsilon_{q,m}\) is negative: an inferior good
  • If \(\epsilon_{q,m}\) is positive: a normal good
  • Two subtypes of normal goods:
    • Necessity: \(0 \leq \epsilon_{q,m} \leq 1\)
      • \(\uparrow\) quantity demanded as \(\uparrow \uparrow\) income (water, clothing)

Income Elasticity of Demand I

  • The income elasticity of demand measures how much quantity demanded \((q_D)\) changes in response to a change in income \((m)\)

$$\epsilon_{q,m}= \frac{\% \Delta q_D}{\% \Delta m}$$

  • If \(\epsilon_{q,m}\) is negative: an inferior good
  • If \(\epsilon_{q,m}\) is positive: a normal good
  • Two subtypes of normal goods:
    • Necessity: \(0 \leq \epsilon_{q,m} \leq 1\)
      • \(\uparrow\) quantity demanded as \(\uparrow \uparrow\) income (water, clothing)
    • Luxury: \(\epsilon_{q,m} > 1\)
      • \(\uparrow \uparrow\) quantity demanded as \(\uparrow\) income (jewelry, vacations)

Income Elasticity of Demand II

  • For now, we can calculate the income elasticity of demand simply by calculating the relative changes:

$$\frac{\% \Delta q}{\% \Delta m}= \cfrac{\left(\frac{\Delta q}{q_1}\right)}{\left(\frac{\Delta m}{m_1}\right)}$$

  • We'll use some fancier methods when we talk about the only elasticity you've probably seen before: price elasticity of demand

Income Elasticity of Demand: Example

Example: You can spend your income on golf and pancakes. Green fees at a local golf course are $10 per round and pancake mix is $2 per box. When your income is $100, you buy 5 boxes of pancake mix and 9 rounds of golf. When your income increases to $120, you buy 10 boxes of pancake mix and 10 rounds of golf.

  1. What type of good is golf (inferior, necessity, luxury)?

  2. What type of good are pancakes (inferior, necessity, or luxury)?

Income Effects: Example

Example: Is the environment a normal good?

Income Effects: Example

Example: Is the environment a normal good?

Income Effects: Example

Example: Is the environment a normal good?

Engel Curves

  • Engel curve visualizes income effects: shows how consumption of one good changes when income increases

  • When positively sloped: normal good

  • When negatively sloped: inferior good

Cross-Price Effects

Cross-Price Effects

  • Cross-price effect: change in optimal consumption of a good associated with a change in price of another good income, holding the good's own price (and income) constant

$$\frac{\Delta q_x}{\Delta p_y} >^?< 0$$

Cross-Price Elasticity of Demand I

  • The cross-price elasticity of demand measures how much quantity demanded of one good \((q_x)\) changes in response to a change in price of another good \((p_y)\)

$$\epsilon_{q_x,p_y}= \frac{\% \Delta q_x}{\% \Delta p_y}$$

Cross-Price Elasticity of Demand I

  • The cross-price elasticity of demand measures how much quantity demanded of one good \((q_x)\) changes in response to a change in price of another good \((p_y)\)

$$\epsilon_{q_x,p_y}= \frac{\% \Delta q_x}{\% \Delta p_y} = \cfrac{\frac{\Delta q_x}{q_x}}{\frac{\Delta p_y}{p_y}}$$

Cross-Price Elasticity of Demand II

$$\epsilon_{q_x,p_y}= \frac{\% \Delta q_x}{\% \Delta p_y}$$

  • If \(\epsilon_{q_x,p_y}\) is positive: goods \(x\) and \(y\) are substitutes

  • An rise (fall) in price of \(y\) causes more (less) consumption of \(x\)

    • Consumption of \(x\) moves in same direction as price of \(y\)

Cross-Price Elasticity of Demand III

$$\epsilon_{q_x,p_y}= \frac{\% \Delta q_x}{\% \Delta p_y}$$

  • If \(\epsilon_{q_x,p_y}\) is negative: goods \(x\) and \(y\) are complements

  • Goods \(x\) and \(y\) consumed in a bundle, concern about overall price of bundle

  • A rise (fall) in price of \(y\) causes less (more) consumption of \(x\)

    • Consumption of \(x\) moves in opposite direction as price of \(y\)

Cross-Price Elasticity: Example I

Example: You can travel into the city every week on Lyft rides and Uber rides. When Lyft is $20/ride, you ride 10 Uber rides. When Lyft raises prices to $25/ride, you ride 15 Uber rides.

  1. What is the relationship between these two goods?

  2. What is the cross-price elasticity?

Outline

Income Effect

Digression: Measuring Change

Cross-Price Effects

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