Lecture Microeconomics (5th edition): Chapter 4 - Consumer choice. This chapter presents the following content: The budget constraint, consumer choice, duality, some applications, revealed preference.
Chapter Copyright (c)2014 John Consumer Choice Chapter Four Overview The Budget Constraint Consumer Choice Copyright (c)2014 John Duality Some Applications Revealed Preference Chapter Four Key Definitions Budget Set: • The set of baskets that are affordable Budget Line: • The set of baskets that one can purchase when spending all available income PxX + PyY = I Y = I/Py – (Px/Py)X Chapter Four Copyright (c)2014 John Budget Constraint: • The set of baskets that the consumer may purchase given the limits of the available income Assume only two goods available: X and Y • • Price of x: Px ; Price of y: Py Income: I Total expenditure on basket (X,Y): PxX + PyY The Basket is Affordable if total expenditure does not exceed total Income: PXX + PYY ≤ I Chapter Four Copyright (c)2014 John The Budget Constraint A Budget Constraint Example I = $10 Px = $1 Py = $2 All income spent on X → I/Px units of X bought All income spent on Y → I/Py units of X bought Budget Line 1: 1X + 2Y = 10 Or Y = – X/2 Slope of Budget Line = -Px/Py = -1/2 Chapter Four Copyright (c)2014 John Two goods available: X and Y A Budget Constraint Example Y • -PX/PY = -1/2 B •C • I/PX = 10 Chapter Four X Copyright (c)2014 John I/PY= A Budget line = BL1 • • Location of budget line shows what the income level is Increase in Income will shift the budget line to the right – • More of each product becomes affordable Decrease in Income will shift the budget line to the left – less of each product becomes affordable Chapter Four Copyright (c)2014 John Budget Constraint A Budget Constraint Example I = $12 PX = $1 PY = $2 Shift of a budget line If income rises, the budget line shifts parallel to the right (shifts out) Y = - X/2 … BL2 If income falls, the budget line shifts parallel to the left (shifts in) BL2 BL1 10 Chapter Four 12 X Copyright (c)2014 John Y • • Location of budget line shows what the income level is Increase in Income will shift the budget line to the right – • More of each product becomes affordable Decrease in Income will shift the budget line to the left – less of each product becomes affordable Chapter Four Copyright (c)2014 John Budget Constraint A Budget Constraint Example Y Rotation of a budget line If the price of X falls, the budget line gets flatter and the horizontal intercept shifts out Copyright (c)2014 John I = $10 PX = $1 BL1PY = $3 If the price of X rises, the budget line gets steeper and the horizontal intercept shifts in Y = 3.33 - X/3 … BL2 3.3 BL2 10 Chapter Four 10 X Interior Consumer Optimum Y Example 2.5 • Copyright (c)2014 John 50X + 200Y = I U = 25 10 Chapter Four X 18 Equal Slope Condition “At the optimal basket, each good gives equal bang for the buck” Now, we have two equations to solve for two unknowns (quantities of X and Y in the optimal basket): MUx/Px = MUY/PY PxX + PyY = I Chapter Four 19 Copyright (c)2014 John MUx/Px = MUy/Py What are the equations that the optimal consumption basket must fulfill if we want to represent the consumer’s choice among three goods? • MU /X P X= MU Y/ P Yis an example of “marginal reasoning” to maximize • P XX + P YY = I reflects the “constraint” Chapter Four 20 Copyright (c)2014 John Contained Optimization Contained Optimization Copyright (c)2014 John U(F,C) = FC PF = $1/unit PC = $2/unit I = $12 Solve for optimal choice of food and clothing Chapter Four 21 Some Concepts Composite Goods: A good that represents the collective expenditure on every other good except the commodity being considered Chapter Four 22 Copyright (c)2014 John Corner Points: One good is not being consumed at all – Optimal basket lies on the axis Copyright (c)2014 John Some Concepts Chapter Four 23 Copyright (c)2014 John Some Concepts Chapter Four 24 Copyright (c)2014 John Some Concepts Chapter Four 25 Copyright (c)2014 John Some Concepts Chapter Four 26 Copyright (c)2014 John Some Concepts Chapter Four 27 Duality The mirror image of the original (primal) constrained optimization problem is called the dual problem where: U* is a target level of utility If U* is the level of utility that solves the primal problem, then an interior optimum, if it exists, of the dual problem also solves the primal problem Chapter Four 28 Copyright (c)2014 John Min PxX + PyY (X,Y) subject to: U(X,Y) = U* Optimal Choice Y Optimal Choice (interior solution) • U = U* Decreases in expenditure level PXX + PYY = E* X Chapter Four 29 Copyright (c)2014 John Example: Expenditure Minimization Optimal Choice Y Example: Expenditure Minimization 25 = XY (constraint) Y/X = 1/4 (tangency condition) 2.5 • U = 25 X 10 Chapter Four 30 Copyright (c)2014 John 50X + 200Y = E Revealed Preference ð If A purchased, it must be preferred to all other affordable bundles Chapter Four 31 Copyright (c)2014 John Suppose that preferences are not known Can we infer them from purchasing behavior? Revealed Preference ð ð ð All baskets to the Northeast of A must be preferred to A This gives us a narrower range over which indifference curve must lie This type of analysis is called revealed preference analysis Chapter Four 32 Copyright (c)2014 John Suppose that preferences are “standard” – then: ... (c)20 14 John Some Concepts Chapter Four 23 Copyright (c)20 14 John Some Concepts Chapter Four 24 Copyright (c)20 14 John Some Concepts Chapter Four 25 Copyright (c)20 14 John Some Concepts Chapter. .. Budget Line = -Px/Py = -1 /2 Chapter Four Copyright (c)20 14 John Two goods available: X and Y A Budget Constraint Example Y • -PX/PY = -1 /2 B •C • I/PX = 10 Chapter Four X Copyright (c)20 14 John I/PY=... contains (X =4, Y =4) Basket B contains (X=10, Y=2.5) Question: • Is either basket the optimal choice for the consumer? Basket A: MRSx,y = MUx/MUy = Y/X = 4/ 4 = Slope of budget line = -Px/Py = -1 /4 Basket