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Budget Constraint 08/25/2016 Chapter 3 Rational Consumer Choice Chapter Outline • Budget Constraints • Budget Shifts Due to Price or Income Changes • Consumer Preferences and Indifference Curves • Best Feasible Bundle • Appendix: Utility Function Approach to Consumer Choice 2 ©2015 McGraw‐Hill Education. All Rights Reserved. 1 08/25/2016 Budget Limitation • A bundle: a particular combination of two or more goods. • Budget constraint: the set of all bundles that exactly exhaust the consumer’s income at given prices. – Its slope is the negative of the price ratio of the two goods. 3 ©2015 McGraw‐Hill Education. All Rights Reserved. Figure 3.1: Two Bundles of Goods 4 ©2015 McGraw‐Hill Education. All Rights Reserved. 2 08/25/2016 Affordable vs. Unaffordable • Affordable set, or feasible set: bundles on or below the budget constraint; bundles for which the required expenditure at given prices is less than or equal to the income available. • Unaffordable set, or unfeasible set: bundles that lie outside the budget constraint 5 ©2015 McGraw‐Hill Education. All Rights Reserved. Budget Constraint • For income M, quantity of food F, quantity of shelter S, price of food PF, and price of shelter PS, the budget constraint is a straight line PSS + PFF = M or • The food endpoint is the maximum amount of food affordable M/PF, and the shelter endpoint is the maximum amount of shelter affordable M/PS. 6 ©2015 McGraw‐Hill Education. All Rights Reserved. 3 08/25/2016 Budget Constraint • For Figure 3.2, let the price of food be PF = $10/lb, the price of shelter be PS = $5/sq yd. and income be M = $100/week. – What bundles of food and shelter are affordable? • The budget constraint PSS + PFF = M becomes 5S + 10F = 100 or F = 10 – (1/2)S. • Maximum food is M/PF = 100/10=10, Maximum shelter is M/PS = 100/5 = 20. Slope of budget constraint –PS/PF = ‐1/2. 7 ©2015 McGraw‐Hill Education. All Rights Reserved. Budget Constraint • Point D containing 4 pounds of food and 5 square yards of shelter per week is feasible. • F = 4 at PF = 10 will cost $40 and S = 5 at PS = 5 will cost $25 so spend $65 all together, which is less than income per week of $100. 5S + 10 F = $100, 25 + 40 ≤ $100 • Point E containing F = 8 and S = 12 is not feasible as spending $80 on food and $60 on shelter would exceed the $100 weekly budget. 8 ©2015 McGraw‐Hill Education. All Rights Reserved. 4 08/25/2016 Figure 3.2: The Budget Constraint, or Budget Line 9 ©2015 McGraw‐Hill Education. All Rights Reserved. Budget Shifts Due to Price and Income Changes • If the price of ONLY one good changes… – The slope of the budget constraint changes. • If the price of both goods change by the same proportion… – The budget constraint shifts parallel to the original one. • If income changes... – The budget constraint shifts parallel to the original one. 10 ©2015 McGraw‐Hill Education. All Rights Reserved. 5 08/25/2016 Increased Price of Shelter • For Figure 3.3, suppose the price of shelter doubles from PS = $5 to $10, while income and the price of food remain the same. • The new budget constraint becomes 10S + 10 F = $100 or F = 10 ‐ S • Maximum shelter falls in half from M/PS = 100/5 = 20 to 100/10 =10. Maximum food M/PF remains unchanged. More steeply downward sloping (slope now ‐1). 11 ©2015 McGraw‐Hill Education. All Rights Reserved. Figure 3.3: The Effect of a Rise in the Price of Shelter 12 ©2015 McGraw‐Hill Education. All Rights Reserved. 6 08/25/2016 Income Cut in Half • For Figure 3.4, suppose income is cut in half from M = $100 to $50, while the prices of food and shelter remain the same as original case. • The new budget constraint becomes 5S + 10 F = $50 or F = 5 – (1/2)S • Maximum shelter falls in half from M/PS = 100/5 = 20 to 50/5 =10. Maximum food also is halved from M/PF = 100/10 = 10 to 50/10 = 5. Slope unchanged at ‐1/2. 13 ©2015 McGraw‐Hill Education. All Rights Reserved. Figure 3.4: The Effect of Cutting Income by Half 14 ©2015 McGraw‐Hill Education. All Rights Reserved. 7 08/25/2016 Both Prices Double • If the price of shelter doubles from PS = $5 to $10 at the same time as the price of food also doubles from PF = $10 to $20, has same effect on the budget constraint as if income fell in half. • The new budget constraint becomes 10S + 20 F = $100 or F = 5 – (1/2)S • Maximum shelter falls in half from M/PS = 100/5 = 20 to 100/10 =10. Maximum food also is halved from M/PF = 100/10 = 10 to 100/20 = 5. Slope unchanged at ‐1/2. 15 ©2015 McGraw‐Hill Education. All Rights Reserved. Budgets With More Than Two Goods • When we have more than 3 goods, the budget constraint becomes a hyperplane, or multidimensional plane. • In this case, view the consumer’s choice as one between a good, X, and an amalgam of other goods, Y. This amalgam is called the composite good. – The amount of income left after buying good X – The amount the consumer spends on goods other than good X 16 ©2015 McGraw‐Hill Education. All Rights Reserved. 8 08/25/2016 Composite Good • For income M, quantity of composite good Y, quantity of good X, price of X PX, and price of composite PY,=1, the budget constraint is a straight line PXX + Y= M or Y = M –PXX • The maximum composite is always M, the maximum X is M/PX, and the slope is –PX. 17 ©2015 McGraw‐Hill Education. All Rights Reserved. Figure 3.5: The Budget Constraints with the Composite Good 18 ©2015 McGraw‐Hill Education. All Rights Reserved. 9 08/25/2016 Nonlinear Budget Constraints • Budget constraints are typically linear due to prices being constant, but budget constraints can be non‐linear when prices vary with quantity such as with quantity discounts. • For Figure 3.6, M = $400/month. Power costs $0.10/kWh for first 1000 kWh each month, then $0.05 for any additional kWhs. • Slope is ‐0.1 for first 1000kWh, which will cost $100 of the $400 monthly budget. The remaining $300 can buy 6000kWh more with slope ‐0.05. 19 ©2015 McGraw‐Hill Education. All Rights Reserved. Figure 3.6: A Quantity Discount Gives Rise to a Nonlinear Budget Constraint 20 ©2015 McGraw‐Hill Education. All Rights Reserved. 10 08/25/2016 Preference Ordering • Preference ordering: a ranking of all possible consumption bundles in order of preference. – Differ widely among consumers – Four simple properties of preference ordering 21 ©2015 McGraw‐Hill Education. All Rights Reserved. Properties of Preference Orderings • Completeness: the consumer is able to rank all possible combinations of goods and services. • More‐Is‐Better: other things equal, more of a good is preferred to less. • Transitivity: for any three bundles A, B, and C, if he prefers A to B and prefers B to C, then he always prefers A to C. • Convexity: mixtures of goods are preferable to extremes. 22 ©2015 McGraw‐Hill Education. All Rights Reserved. 11 08/25/2016 Indifference Curves • Indifference curve: a set of bundles among which the consumer is indifferent. • Indifference map: a representative sample of the set of a consumer’s indifference curves, used as a graphical summary of her preference ordering. 23 ©2015 McGraw‐Hill Education. All Rights Reserved. Properties of Indifference Curves • Indifference curves … 1. Are Ubiquitous. • Any bundle has an indifference curve passing through it. 2. Are Downward‐sloping. • This comes from the more‐is‐better assumption. 3. Cannot cross. 4. Become less steep as we move downward and to the right along them. • This property is implied by the convexity property of preferences. 24 ©2015 McGraw‐Hill Education. All Rights Reserved. 12 08/25/2016 Figure 3.9: An Indifference Curve 25 ©2015 McGraw‐Hill Education. All Rights Reserved. Figure 3.10: Part of an Indifference Map 26 ©2015 McGraw‐Hill Education. All Rights Reserved. 13 08/25/2016 Figure 3.11: Why Two Indifference Curves Do Not Cross 27 ©2015 McGraw‐Hill Education. All Rights Reserved. Trade‐offs Between Goods • Marginal rate of substitution (MRS): the rate at which the consumer is willing to exchange the good measured along the vertical axis for the good measured along the horizontal axis. – Equal to the absolute value of the slope of the indifference curve. 28 ©2015 McGraw‐Hill Education. All Rights Reserved. 14 08/25/2016 Figure 3.12: The Marginal Rates of Substitution 29 ©2015 McGraw‐Hill Education. All Rights Reserved. Marginal Rate of Substitution • Figure 3.13 depicts three different points on an indifference curve with different MRS. • At S = 3 sq ft of shelter and F = 17 lbs of food each week, MRS = 3 – willing to give up lots of food for an additional sq ft of shelter since eating well but cramped for space. • At S = 7 and F = 9, MRS = 1 • At S = 16 and F = 3, MRS = ¼ – willing to give up little food for more shelter since ample space but hungry. 30 ©2015 McGraw‐Hill Education. All Rights Reserved. 15 08/25/2016 Figure 3.13: Diminishing Marginal Rate of Substitution 31 ©2015 McGraw‐Hill Education. All Rights Reserved. Different Tastes • A consumer’s willingness to exchange goods is indicated by the slope of the indifference curve. • Consumers with different tastes (different MRS at a give point) will have indifference curves with different slopes.
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