# Chapter 4: Individual Demand

## Price Changes

### Increased Price of a Good

• Budget line rotates inward (steeper) and maximum utility is found at a new point on a lower indifference curve.
• Lower purchasing power and utility.

### Decreased Price of a Good

• Budget line rotates outward (flatter) and maximum utility is found at a new point on a higher indifference curve.
• Higher purchasing power and utility.

## Individual Demand Curve

### Price-Consumption Curve

• Curve tracing the utility maximizing combinations of 2 goods as the price of one changes.
• If one price falls, attainable utility increases and consumer buys more of the other.

### Individual Demand Curve

• Curve relating the quantity of a good that a single consumer will buy to its price.
• The level of utility that an be attained changes as we move along the curve.
• At every point on the demand curve, the consumer maximizes utility by satisfying the condition that the MRS of food for clothing equals the ratio of the prices of food and clothing.

## Income Changes

### Income Increase

• Budget line shifts outward parallel to the original line, allowing a utility level at a higher indifference curve.

### Income Decrease

• Budget line shifts inward parallel to the original line, forcing a utility level at a lower indifference curve.

### Income Consumption Curve

• Traces the utility maximizing combinations of 2 goods as a consumer’s income changes.
• It slopes upward as consumption of both food and clothing increase as income increase.
• Each demand curve is measured for a particular level of income, so any change in income leads to a shift in the demand curve itself.

## Normal VS. Inferior Goods

### Normal Goods

• At a positively sloped income-consumption curve, quantity demanded increases with income.
• Income elasticity of demand is positive. (Consumers want more if they have higher income.)

### Inferior Goods

• Quantity demanded falls as income increases.
• Income elasticity of demand is negative. (Consumers want less if they have higher income)
• Graphically→ As income rises, the income-consumption curve bends backwards towards the normal good and away from the inferior good.

## Engel Curves

• Curve relating the quantity of a good consumed to income.
• Upward sloping if the goods are normal goods.
• If one good is a normal good for income less than an amount, the Engel curve slopes downward in the income range when it is classified as inferior.

## Substitutes and Complements

• Substitutes – An increase in the price of one leads an increase in the quantity demanded for the other.
• Complements – An increase in the price of one leads to a decrease in the quantity demanded for the other.
• Independent – Changes in price have no effect on quantity demanded of another.

### Price-Consumption Curve

• Downward Sloping – Goods are substitutes.
• Upward Sloping – Goods are complements.
• Decisions to classify goods is empirical → we must analyze the change in demand for the first good in response to price changes.

## Income and Substitution Effects

• A fall in price of a good has 2 effects

### Substitution Effect

• Consumers will tend to buy more of the good that has become cheaper and less of the goods that are now relatively more expensive.
• Change in consumption of a good associated with a change in its price, with the level of utility held constant.
• Involves movement along an indifference curve.
• Instructions → Draw a budget line parallel to the new budget line tangent to the original indifference curve. The new lower imaginary BL reflects the fact that nominal income was reduced to show the substitution effect.
• Indifference curves are convex, so an increase in one good results in a decrease in another.

### Income Effect

• Because one good is now cheaper, consumers have higher real purchasing power. They are better off and buy more of normal goods because they can afford it.
• Effects occur simultaneously but it is useful to distinguish between them.
• Graphical Example- A decrease in price of food causes consumption of food to increase to point B. Draw a new BL to reflect price increases, tangent to the Indifference Curve (substitution). Then shift the BL2 to match the intercept of the BL1 and shift the indifference curve to a new tangent point.
• The substitution effect (A to D) changes relative prices of food and clothing but keeps real income constant.
• The income effect (D to B) keeps relative prices constant but increases purchasing power. (Where food is a normal good with a positive income effect.)
• Change in consumption of a good resulting from an increase in purchasing power, with relative prices held constant.
• Instructions- Move from the imaginary BL to the parallel BL constructed – find the tangent of the parallel BL to the parallel indifference curve.

## Total Effect & Inferior Goods

### Total Effect ﻿$\left (F_{1}, F_{2} \right )$﻿﻿$=$﻿ Substitution Effect ﻿$+$﻿ Income Effect ﻿$\left ( EF_{2} \right )$﻿

• Direction of substitution effect is always the same, but income effect moves demand in either direction depending on the good.

### Inferior Good

• A good that has a negative income effect.
• Graphically- If food is an inferior good, the income effect will be negative, causing a shift back towards clothing on the parallel BL. (If exceeded by the substitution effect, the total effect will still be an increase in food consumption.)

## A Special Case → The Giffen Good

• Good whose demand curve slopes upward because the (negative) income effect is larger than the substitution effect.
• In practice → Price of food declines, freeing income so that the consumer desires to buy more clothing and less food → the consumer is better off here despite the fact that less food is consumed.
• Rarely of practical interest → it requires a large income effect but the income effect is usually small.

## Demand Theory → A Mathematical Treatment

### Utility Maximization

#### Utility Function

• Attaches a level of utility to each market basket.

#### Marginal Utility

• Change in utility associated with a one-unit increase.
• Marginal Utility ﻿$MU_{x}$﻿ is found by finding the partial derivative of the utility function with respect to good X.

#### Constrained Optimization

• Maximize U(X,Y)
• Subject to the constraint that all income is spent on 2 goods.

• Where ﻿$P_{x}$﻿ and ﻿$P_{y}$﻿ are the prices of the goods, X and Y are the quantities of the goods, and I is income.
• Individual Consumers Demand for 2 goods → choose values X and Y to maximize U(X,Y) given the constraint that all income is spent.

### Lagrange Multipliers Method

• Technique to maximize or minimize a function subject to one or more constraints.

#### A. Stating the Problem

• Write the Lagrangian for the problem, this is the function to be maximized. Allow the variable λ to be multiplied by the constraint.

• Where budget constraint is represented by-

#### B. Differentiating the Lagrangian

• Choose values of X and Y that satisfy the budget constraint, then the second term in equation will be zero.

#### C. Solving the Resulting Equations

• Rewrite the three equations.

• Solving for the three unknowns yield resulting X and Y are the solution to the consumer’s optimization problem – they are the utility-maximizing quantities.

### The Equal Marginal Principle

• Third equation above is consumer’s budget constraint. First 2 tell us that each good will be consumed up the point at which the marginal utility from consumption is a multiple λ of the price of the good.
• The marginal utility of each good divided by its price is the same.
• Optimized- consumer must get the same utility from last dollar spent by consuming either X or Y.
• If not, consuming one more of one good and less of the other increases utility.

• Rewritten as a ratio

• Ratio of the marginal utilities is equal the ratio of the prices.

### Marginal Rate of Substitution

• An indifference curve represents all market baskets that give the consumer same amount of utility. (Where U* is a fixed utility level)

• As market baskets are changed by adding small amounts of X and subtracting Y, total utility must equal zero.

• Or rearranged

• Where ﻿$MRS_{xy}$﻿ represent individual’s marginal rate of substitution of X for Y.
• Left hand side represents the negative slope of the indifference curve.
• At the point of tangency, MRS ﻿$=$﻿ ratio of marginal utilities ﻿$=$﻿ ratio of prices between two goods.
• When individual indifference curves are convex, the tangency of the curve to the BL solves the optimization problem.

### Marginal Utility of Income

• Lagrange multiplier λ represents extra utility generated when the budget constraint is relaxed.
• Shown by differentiating U(X,Y) with respect to I

• Any increment in income must be divided between 2 goods.

• Substitute the Equal Marginal Principle into the “differentiated utility function with respect to I” equation.

• Substitute the “increment division between 2 goods equation” into the equation we just found.

• Therefore, the Lagrange multiplier is the extra utility that results from an extra dollar of income.
• Utility maximization requires utility obtained from the consumption of every good, per dollar spent on that good, to be equal to the marginal utility of an additional dollar of income.

## Duality in Consumer Theory

#### Duality

• Alternative way of analyzing the consumer’s utility maximization decision → rather than choosing the highest indifference curve given a budget constraint, the consumer chooses the lowest budget line that touches a given indifference curve.
• Goal

• Subject to Constraint
• Corresponding Lagrangian

• Where μ is the Lagrange multiplier.
• Differentiating the Lagrangian with respect to X, Y, and μ and setting derivatives equal to zero reveals 3 conditions for expenditure minimization.

• First Two Equations Reveal

• Also True

• Cost minimizing choice of X and Y must occur at the point of tangency of the BL and IC that generates utility U*
• Dual Approach with Cobb-Douglas U(X,Y) ﻿$= X^{a}Y^{1-a}$﻿
• Lagrangian given as
• Differentiate with respect to X, Y, and μ and equating to zero obtains 2 conditions.

• Multiply ﻿$P_{x}$﻿ equation by X and second by Y and adding

• Let I be the cost-minimizing expenditure. It follows that ﻿$\mu=I/U^{*}$﻿ ﻿$\mu=I/U*$﻿
• Substituting in the equations above

### Income and Substitution Effects

• Any portion of a price change involves movement along an indifference curve from that portion which involves movement to a different indifference curve. (Changing purchasing power)
• Denote change in X that results from unit change in price of X with constant utility

• Total change in quantity demanded of X resulting from a unit change in price of X

• First term on the right side (𝛿X/ 𝛿I) is the substitution effect as utility is fixed.
• Second term on the right side (𝛿I/ 𝛿PX) is the income effect as income increases.
• Budget Constraint → from differentiation we get

#### The Slutsky Equation

• Formula for decomposing the effects of a price change into substitution and income effects.

• First Term – Substitution effect- change in demand for good X obtained by keeping fixed utility.
• Second Term – Income effect- change in demand resulting from a change in purchasing power.

#### Hicksian Substitution Effect

• Alternative way to decompose a price change into substitution and income effects.
• Consumer consumes a market basket A. Decrease in price of food shifts the budget line out to higher food consumption.
• If a sufficient enough of income is taken away, there are 2 conditions to be met
• The new market basket chosen must lie on line segment BT’ of budget line R’T’.
• Quantity of food consumed must be greater than at A.

## Market Demand

### Market Demand Curve

• Curve relating the quantity of a good that all consumers in a market will buy to its price.

### From Individual to Market Demand

• Market demand curve represents the horizontal summation of the demands of each consumer.
• Find the total amount that the consumers demand at any given price.
• Slopes downward but does not need to be a straight line. It may kink if one consumer makes no purchases at a certain price.

### 2 Important Points

• The market demand curve shifts right as more consumers enter the market.
• Factors that influence the demands of consumers will also affect market demand.
• Important in practice with demands of different demographic groups or from consumers in different areas. (i.e. households with or without children, single individuals, etc.)

### Elasticity of Demand

• Price Elasticity of Demand measures percentage change in Quantity Demanded resulting from a 1-percentage increase in price.

#### Inelastic Demand

• ﻿$E_{p}< 1$﻿ in absolute value.
• Quantity demanded is relatively unresponsive to changes in price.
• Total expenditure on the product increases when the price increases.

#### Elastic Demand

• ﻿$E_{p}< 1$﻿ in absolute value.
• Quantity demanded is relatively responsive to changes in price.
• Total expenditure on the product decreases when the price increases.

#### Isoelastic Demand

• Price elasticity of demand is constant all along the demand curve.
• Price elasticity of demand has a downward slope as it increases in magnitude until it becomes infinite when the price is high enough for quantity demanded to be zero.
• Unit-Elastic Demand Curve – demand curve with ﻿$EP= -1$﻿ at all points. Total expenditure is the same after a price change.

### Speculative Demand

• Demand driven not by the direct benefits one obtains from owning or consuming a good but instead by an expectation that the price of the good will increase.
• It may be profitable to buy the good and resell it later at a higher price.
• Can be irrational → it is typically little more than “gambling.”

## Consumer Surplus

• Difference between what a consumer is willing to pay for a good and the amount actually paid.
• It measures how much better individuals are because they can buy those goods.

### Consumer Surplus and Demand

• Measure total benefit from product consumption by drawing the demand curve as a staircase than a straight line.
• Individuals prefer to purchase products with surplus, so find the max amount they are willing to pay and sum the excess values or surpluses.
• In a Market, find the area below the market demand curve and above the price line.
• Find the triangle area- ½ x (Highest price – price line) x quantity of goods at price line

For this example, CS is calculated by the yellow shaded triangle- ½ x ($20 -$14) x 6500 = \$19 500.

### Applying Consumer Surplus

• It measures the aggregate benefit that consumers receive from buying goods in a market.
• Combining CS with aggregate profits for producers, we can evaluate costs and benefits of alternative market structures and public policies that alter consumer and firm behaviour.