Note 1: Indifference Curves, Budget Lines, and Demand Curves

Transcription

1 Note 1: Indifference Curves, Budget Lines, and Demand Curves Jeff Hicks September 19, 2017 Vancouver School of Economics, University of British Columbia

2 In this note, I show how indifference curves and budget constraints relate to demand curves. 1 Budget Constraint 1. Let s assume there are two products: x 1 and x The price of x 1 is P x1, and the price of x 2 is P x2. 3. An individual has income I, to split between the two goods. The budget constraint is: We assume all income is spent. P x1 x 1 }{{} Expenditure on x 1 + P x2 x 2 }{{} Expenditure x 2 = Y }{{} Income (1) The budget constraint is visually represented below. On the horizontal axis is units of x 1, and on the vertical axis is units of x 2. The blue line represents all combinations of x 1 and x 2 that together cost the same amount - exactly equal to income. 1. If the individual spends all of their income (Y) on x 1, then P x1 x 1 = Y. This equals x 1 = Y/P x1 units of x 1. This is the horizontal axis intercept. 2. If the individual spends all of their income (Y) on x 2, then P x2 x 2 = Y. This equals x 2 = Y/P x2 units of x 2. This is the vertical axis intercept. 1

3 3. To find the slope of the budget line, simply rearrange the budget equation: P x1 x 1 + P x2 x 2 = Y P x2 x 2 = Y P x1 x 1 x 2 = Y P x 1 x 1 P x2 x 2 = Y P x2 }{{} Vertical Axis Intercept P x 1 P x2 }{{} Slope x 1 2

4 What happens if the price of X 1 decreases? 3

5 2 Indifference Curves People have preferences over different bundles of stuff. An indifference curve is simply a graphical representation of preferences. Definition: Every point on an indifference curves represents one bundle of products (e.g. 4 units of x 1, 2 units of x 2 ). The individual is indifferent between every bundle on the indifference curve. Indifference curves are graphically represented below. In general, people prefer more stuff. So an indifference curve further out from the origin is better : it contains more stuff. 1. Very importantly, the slope of an indifference curve represents the amount of one good that an individual is willing to trade for a single unit of the other product. 2. What determines the shape of an indifference curve? We typically draw them that as inwardly-bent curves. This captures the intuition that people prefer variety - for instance, I prefer both food and clothing, rather than just food or just clothing. 3. There are two extreme cases: perfect substitutes, and perfect compliments, represented below. Neither are very common. 4. In general, the more curved an indifference curve is, the more complimentary the two goods are. 4

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9 3 Bringing Budgets and Indifference Curves Together We have established the two components of choices: budget constraints and preferences (represented by indifference curves). The individual needs to choose a consumption bundle: the amount of x 1 and x 2 that she most prefers, given her budget constraint. Since higher indifference curves are more preferable, she will choose the highest indifference curve possible. Typically, the individual s choice of consumption bundle is where the indifference curve is tangent to the budget line, as illustrated below. Convince yourself that any consumption bundle on indifference curve B is not attainable. The consumer does not have enough money. Also convince yourself that any indifference curve below A is not preferred because people like more stuff. Point of Tangency: When the budget constraint and the indifference curve are tangent, this by definition means their slopes are the same. So, Px 1 P x2 = X2 X1. Let s say x 1 is food, and x 2 is clothing. What happens if the price of food decreases? Food consumption definitely increases, because it has become cheaper. Clothing consumption might also increase, because the lower price of food means I have extra cash to spend on clothing, if I want to. We call two goods substitutes if they price of one is 8

10 positively correlated with the quantity of the other. Pepsi and Coca-Cola are substitutes. if the price of pepsi goes up, people shift their soda consumption to Coca-Cola. Two goods are compliments if the price of one is negatively correlated with the quantity of the other. Skis and ski boots are compliments. If the price of ski boots goes up, people buy fewer skis. 4 Individual Demand Curves So, when the price of food (x 1 ) changed, the quantity of food (x 1 ) consumed also changed. That should be intuitive. A demand curve captures the relationship between food choices and the price of food. This is visually represented below: 9

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13 Demand curves are fundamental to economics. They capture the simple notion that the price of something influences that amount people want to buy. What we often care about are market demand curves, not the demand curve for a single individual. Intuitively, the market demand curve is derived from all the underlying individual ones. For any given price, what is the sum of quantity demanded across all consumers. In the exercises, there is an example of adding up individuals curves to obtain the market curve. Demand curves also have a lot of intuition. Let s think about some aspects. The first is the price elasticity of demand: the percentage change in quantity divided by the percentage change in price. Let s call the elasticity η, and play with it a little bit: η = % Q Q % P = Q P P = Q P P Q (2) Now, remember P is the slope of the demand curve. This will always be negative, Q because when the price increases, the quantity demanded (Q) decreases. If the elasticity is large, lets say -2, this means people are very responsive to prices. A product with a large elasticity is called elastic. If the elasticity is small, lets sat -0.1, we call the food inelastic: demand is not very responsive to price. Can you think of some good examples for both? The table below lists price elasticity estimates for various products. Note, often people report the elasticity as positive, but don t be fooled, the price elasticity of demand will (practically) always be negative. People will (almost) always consume less of something if you raise the price. 12

14 Table 1: Elasticity Estimates: Source Inelastic Price Elasticity of Demand Salt 0.1 Airline travel, short-run 0.1 Gasoline, short-run 0.2 Gasoline, long-run 0.7 Coffee 0.25 Unitary Elastic Movies 0.9 Private Education 1.1 Radio and television receivers 1.2 Elastic Restaurant meals 2.3 Automobiles, short-run 1.5 Fresh Tomatoes

15 5 Exercises 1. The price of a pound of rice is 1 dollar, and the price of a pound of quinoa is 3 dollars. Jeff only buys rice and quinoa, and has 50 dollars of income to spend. He spends it all on either rice or quinoa. Draw the budget constraint with quinoa on the horizontal axis and rice on the vertical axis. What is the slope of the budget line? What are the Y and X axis intercepts? Without knowing what Jeff s optimal choice is, can we say what the slope of the indifference curve is at the optimal choice? What is it? If the price of quinoa rises to 4 dollars, what are the new slope, and intercepts? Draw the new budget curve on the same plot. Assume that before the price change, Jeff bought 6 pounds of quinoa, and after the price change, Jeff bought 4 pounds of quinoa. What is the slope of Jeff s demand curve - at least around those points? What is Jeff s price elasticity of demand, at least around those points? Hint: Use equation 2. You ll notice that there are two possible elasticity numbers you could arrive at. Why? 2. The demand curve for electricity is represented by the following equation: Q e = P e. If the price of electricity is 25, what is the quantity demanded? At that point, what is the price elasticity of demand? If the price changes to 15, what is the quantity demanded, and what is the new price elasticity of demand? Does the elasticity differ at the two points? The previous question implies that the price elasticity is different depending on where we are on the demand curve. Can you think why? 14