Intermediate Microeconomic Theory Demand

Demand Analysis In analyzing individual’s behavior regarding a given good, we start with a consumer’s demand function for that good. q1(p1,p2,m) Recall that a demand function was derived from “first principles”, or the explicit model of preferences and choice given a budget constraint. However, in the end, it is this derived demand function that tells us all we need to know about consumer behavior.

Demand Analysis The first thing we will characterize is how a consumer’s demand for a given good changes with his or her income. Normal Good - demand for good rises with income, or How would we show this graphically? Examples of normal goods?

Demand Analysis Alternatively, consider a good where demand decreases as income increases, or Inferior Good How would we show this graphically? Examples of inferior goods?

Demand Analysis Would linear preferences (U(q1,q2) = q1 + q2) imply q1 is a normal good? How about quasi-linear preferences (U(q1,q2) = aq10.5 + q2) ?

Income Expansion Paths and Engel Curves We often want to describe how demand changes as income changes. To do so, we use: Income Expansion Path - traces out optimal bundle as income increases, holding prices constant. Engel Curve – shows how demand for a given good changes as income changes, holding prices constant. How do we show these graphically?

Deriving Engel Curve Analytically Straightforward given a utility function and prices. Example: Suppose prices are p1 = 5, p2 = 10 and utility function given by U(q1,q2) = q10.5q20.5 . What will be the equations for his Engel Curve for good 1 and good 2? Do these make sense?

The Slope of the Engel Curve The slope of the Engel Curve is informative. Consider the Engel Curve shown below. What does linearity imply (sometimes referred to as homothesticity)? Do Cobb-Douglas preferences imply such a linear Engel curve? m q1

The Slope of the Engel Curve The slope of the Engel Curve don’t have to be constant. Consider the Engel Curve shown below. What does concave shape imply? What might be examples of goods with such an Engel curve? What is a good term to describe such goods? m q1

The Slope of the Engel Curve Consider the Engel Curve with convex shape. What does convex shape imply (hint: consider Engel curve for quasi-linear preferences)? What might be examples of goods with such an Engel curve? What is a good term to describe such goods? m q1

The Slope of the Engel Curve What will be the shape of the Engel Curve for an inferior good?

Demand Curves We also often want to describe how demand changes as prices change. To do so, we use: Price Offer Curve - traces out optimal bundle as price of one good changes, holding other prices and income constant. Demand Curve – shows how demand for a given good changes as its price changes, holding other prices and income constant. How do we show these graphically?

Deriving a Demand Curve Analytically Also straightforward given a utility function, prices of other goods and income. Example: Suppose prices are m = 24, p2 = 10 and an individual has a utility function U(q1,q2) = q10.5q20.5 . What will be the equation for his Demand Curve for good 1? Does this make sense?

Demand Curves Slope of demand curve indicates how much demand reacts to price. Generally, demand curves will be downward sloping, meaning as price rises demand falls, or We saw this will be the case with Cobb-Douglas specification of preferences. How about with linear utility function? Given our assumptions, is it possible for a demand curve to slope upward?

Gross Substitutes and Complements We have already discussed perfect substitutes and perfect complements. We can now consider more nuanced definitions of substitutes and complements, with perfect versions being subsets.

Gross Substitutes and Complements Beer and pizza. Generally, I like to consume them together, one beer with every slice. However, if the price of a slice went to $10, I might behave a little differently. Some degree of complementarity, but not perfect. Pizza and Chicken Wings. Clearly they aren’t perfect substitutes. Even if pizza was more expensive, I still might order a few chicken wings. A raise in the price of one would definitely cause me to consume less of it an more of the other. Some degree of substitutability between them but not perfect.

Gross Substitutes and Complements Gross substitutes - a rise in the price of one increases the demand for the other, or if How would we show this graphically? Consider skiing and golf. Suppose the price of lift tickets increased. How will the demand for golf course time be affected? How about the demand curve for golf course time?

Gross Substitutes and Complements Gross complements - a rise in the price of one decreases the demand for the other, or if How would we show this graphically? Consider skiing and plane tickets. Suppose the price of lift tickets increased. How will the demand for a plane ticket to SLC be affected? How about the demand curve?

Gross Substitutes and Complements If someone’s preferences over two goods are captured by a Cobb-Douglas Utility function, will the two goods be gross substitutes, gross compliments, or neither?

Demand Curve and changes in income Consider the Demand Curve for good 1 for an individual with Cobb-Douglas Utility U(q1,q2) = q10.6q20.4 , who has m = 20 and p2 = 4. How would this Demand Curve be affected by a change in endowment from m = 20 to m = 30? What if preferences were such that good 1 was an inferior good?

Summary Engel curves and Demand curves are derived from demand function (which is in turn derived from underlying preferences). A Demand curve for a given good describes how the demand for that good changes as its own price changes, holding other prices and endowment fixed. Changes in prices of other goods or endowment may shift a Demand curve. An Engel curve for a given good describes how the demand for that good changes as income changes, holding other prices fixed. Changes in prices may shift an Engel curve.

Summary This means that: Similarly, The change in demand for good i in reaction to a change in the price of good i is captured by a movement along the demand curve for good i. The change in demand for good i in reaction to change in price of some other good j or change in income is captured by a shift in the demand curve for good i. Similarly, The change in demand for good i in reaction to a change in income is captured by a movement along the Engel curve for good i. The change in demand for good i in reaction to change in price of good i or another good j is captured by a shift in the Engel curve for good i.