Intro to Economic analysis

Transcription

1 Intro to Economic analysis Alberto Bisin - NYU 1 The Consumer Problem Consider an agent choosing her consumption of goods 1 and 2 for a given budget. This is the workhorse of microeconomic theory. (Notice that we do not ask where prices come from, nor where the income of the agent comes from.) 1.1 The Budget constraint There are only two goods, X R 2. A consumer who consumes x 1 units of good 1 and x 2 units of good 2 is said to consume the consumption bundle (x 1, x 2 ) X. Any bundle can be represented by a point on a two-dimensional graph. The prices of these goods, p 1 and p 2, are given and known to the consumer who is a price taker. The consumer s income, that is, the amount of money the consumer has to spend, is m. Then the consumer s budget constraint can be written as p 1 x 1 +p 2 x 2 m. The budget set consists of all bundles that are affordable at a given prices and income. More precisely, B = {x X : p 1 x 1 + p 2 x 2 m} Thus, the budget line is the set of bundles that cost exactly m p 1 x 1 + p 2 x 2 = m 1

2 and rearranging, x 2 = p 1 p 2 x 1 + m p 2 Note that this is a linear function with a vertical intercept m/p 2, a horizontal intercept m/p 1 and a slope of p 1 /p 2. The slope of the budget line measures the rate at which the market is willing to exchange good 1 for good 2. Thus, economists say that the slope measures the opportunity cost of consuming good 1. (Why we call it opportunity costs? What is the opportunity cost of consuming 1 unit of good 1? Well, the other opportunity is to sell the unit of good 1, for $p 1, and then use the proceeds to buy good 2. How many units of good 2 could you consume if you did this trade? The answer is p 1 p 2.) Clearly, when prices and income changes, the set of goods that a consumer can afford changes as well. Increasing income shifts the budget line outward. increasing the price of good 1 (good 2) makes the budget line steeper (flatter). A class exercise - What do you think happens to the budget line when both prices are changed at the same time? For example, both prices become t times as large. Give a graphical and an analytical solution. 1.2 Indifference curves Recall that theory of the consumer choice can be formulated by preferences that satisfy the axioms of completeness and transitivity (plus a few more technical assumptions). Now, we will describe preferences graphically by the construction known as indifference curves. At this stage we will describe some more general assumptions - monotonicity and convexity - that we will typically make about preferences and the implications of these assumptions for the associated indifference curves. 2

3 1.2.1 Monotonicity We typically assume that more is better, that is, we consider goods, not bads. More precisely, consider (x 1, x 2 ) as a bundle of goods and let (x 1, x 2) be any other bundle with at least as much of both goods and more of one. That is, x 1 x 1 and x 2 x 2 with at least one strict inequality, or, in short, (x 1, x 2) > (x 1, x 2 ). Monotonicity of preferences implies that if (x 1, x 2) > (x 1, x 2 ) then (x 1, x 2) (x 1, x 2 ). Thus, monotonic preferences imply that more of (less) of both goods is a better (worse) bundle. Define the strictly preferred set to (x 1, x 2 ) and the weakly preferred set to it. The bundles on the boundary of this set - the bundles for which the consumer is just indifferent to (x 1, x 2 ) - form the indifference curve. Monotonicity of preferences implies that indifference curves have a negative slope. Obviously, since the bundle (x 1, x 2 ) was chosen arbitrarily, we can draw an indifference curve through any bundle Convexity A set X is convex if αx + (1 α) x X whenever x, x X and α [0, 1]. In words, A set X is convex if whenever it contains two elements x, x, it contains the entire segment connecting them. Preferences are convex if the set of bundles weakly preferred to any bundle (x 1, x 2 ) is a convex set. 3

4 Well-behaved indifference curves Often, monotonicity and convexity are taken as the defining features for well-behaved indifference curves. Note that if we start at an arbitrary bundle (x 1, x 2 ) and we move up and to the right, by monotonicity we must be at a preferred position. Thus, at a higher indifference curve the consumer is strictly better. Note that if we take two arbitrary bundles on the ( same ) indifference( ) curve, (x 1, x 2 ) and (x 1, x x1 x 2), by convexity the bundle α +(1 α) 1 x 2 x 2 is preferred to each of the initial bundles. Important examples where convexity does not hold include indivisible goods (e.g., shoes) or addictive goods. Can you see why? Class exercise: Can indifference curves representing distinct levels of preference cross? The Marginal Rate of Substitution (M RS) The slope of the indifference curve is known as the marginal rate of substitution (M RS). That is, the rate in which the consumer is just willing to substitute one good for the other. For strictly convex indifference curves, the rate at which the consumer is just willing to trade x 1 for x 2, MRS 1,2, decreases (in absolute value) as we increase the amount of x 1. Thus, we can see that convexity is very natural: the more we have of one good, the more we are willing to give some of it up in exchange for the other good. The marginal utility of good 1, written as Mu 1, measures the change in the consumer s utility as x 1 increases and x 2 is left unchanged. Thus, it measures the rate of change in utility associated with an infinitesimal change in the amount of good 1, keeping the amount of good 2 fixed. Precisely, 4

5 Mu 1 = u u (x 1 + x 1, x 2 ) u (x 1, x 2 ) = lim = u (x 1, x 2 ) x 1 x 1 0 x 1 x 1 We can derive the MRS 1,2, the rate at which the consumer is willing to substitute small amount of good 1 for good 2, from the utility function, u(x 1, x 2 ). 1 More precisely, MRS 1,2 is defined as the infinitesimal change in good 2 that keeps the utility unchanged after a unitary infinitesimal increase in good 1 (the fact that this change is negative is due to monotonicity: an decrease in x 2 is required to keep utility constant if x 1 increases). Formally, consider a change (dx 2, dx 1 ) that keeps the level of utility constant: rearranging, du = u (x 1, x 2 ) x 1 dx 1 + u (x 1, x 2 ) x 2 dx 2 = 0; dx 2 = dx 1 u(x 1,x 2 ) x 1 u(x 1,x 2 ) x 2 But, dx 2 dx 1 is just the rate at which the consumer is willing to substitute an infinitesimal amount of good 2 for good 1 (the MRS 1,2 ), that is, the change of x 2 for x 1 which keeps the agent s utility constant. That is, u(x 1,x 2 ) x MRS 1,2 := 1 u(x 1,x 2 ) x 2 1 What follows goes under the name of Implicit Function Theorem, one of the fundamental theorems of calculus, and one of the most used in economics. A (imprecise) statement of the theorem is as follows. Let f : R 2 R be continuous and smooth (at least twice differentiable). Then there exists a continuous differentiable function g : R R such that, locally, f (x 1, g(x 1 )) = 0 and d g(x 1 ) d x 1 = f(x 1,x 2) x 1 f(x 1,x 2) x 2 5

6 The Cobb-Douglas indifference curves looks just like the well-behaved indifference curves that we previously referred to. Note that at this stage you should be able to point by using algebra why it is so. In fact, the formula of the Cobb-Douglas utility function is about the simplest algebraic expression that generates well-behaved preferences. 1.3 Types of preferences MRS = Mu 1 Mu 2 = u(x 1,x 2 )/ x 1 u(x 1,x 2 )/ x 2 = αxα 1 1 x β 2 βx α 1 xβ 1 2 = αx 2 βx 1 Some examples to get more used with the indifference curve representation of preferences Perfect substitutes Two goods are perfect substitutes if the consumer is willing to substitute one good for the other at a constant rate. Thus, the indifference curves have a constant slope. The utility function in the case x 1 and x 2 are perfect substitutes is of the (linear) form: u(x 1, x 2 ) = αx 1 + βx Perfect complements Two goods are perfect complements if they are always consumed together in fixed proportions. For example, right shoes and left shoes. Thus, the indifference curves are L-shaped. 6

7 The utility function in the case x 1 and x 2 are perfect complements is of the ( form: u(x 1, x 2 ) = min {αx 1, βx 2 } 1.4 Optimal choice We consider three different ways to study the optimization problem of consumer with Cobb-Douglas preferences. 1. Geometric analysis of indifference curves. A consumption bundle (x 1, x 2) is an optimal choice for the consumer if the set of bundles that the consumer prefers to (x 1, x 2) - the set of bundles above the indifference curve through (x 1, x 2) - has an empty intersection with the bundles she can afford - the bundles beneath her budget line. It follows that at (x 1, x 2) the indifference curve is tangent to the budget line: MRS 1,2 = αx 2 βx 1 = p 1 p 2 ; 2 and, naturally, at (x 1, x 2) the budget constraint must be satisfied: p 1 x 1 + p 2 x 2 = m We have now two equations in two unknowns that can be solved for the optimal bundle. Thus, substituting: p 1 x 1 + p 2 βp 1 x 1 αp 2 = m p 1 x 1 + βp 1 α x 1 = m 2 Note that, clearly, the above holds for an interior optimum and not for a boundary optimum. Can you see why a Cobb-Douglas preferences induce an interior solution? How would you deal with preferences implying perfect substitutes? 7

8 αp 1 x 1 + βp 1 x 1 = αm x 1 = α m α + β p 1 This is the demand function for good 1. Similarly, the demand function for good 2: x 2 = β m α + β p 2 2. Algebraic analysis by substituting the budget constraint in to the objective function (can only do with 2 goods). That is, The foc for this problem is By a (very) little algebra, ( m max α ln x 1 + β ln p ) 1 x 1 p 2 p 2 p 2 = α p 1 β = 0 x 1 x 1 m p 1 x 1 p 2 x 1 = α m α + β p 1 x 2 = β m α + β p 2 3. General algebraic analysis of constrained maximization. 8

9 The third way is by using the Lagrangian: L = α ln x 1 + β ln x 2 λ(p 1 x 1 + p 2 x 2 m) Differentiating to get the three foc: L x 1 = α x 1 λp 1 = 0 L x 2 = β x 2 λp 2 = 0 L λ = m p 1x 1 p 2 x 2 = 0 These are three equations with three unknowns. The best way for you to proceed (which you should do!) is to first solve for λ. You will see that you get back to the algebraic component of the first method, the geometric analysis of indifference curves Demand Functions Demand functions give the optimal bundle (amounts of each of the goods) as a function of the prices and income faced by the consumer x(p, m) Note that when two goods are perfect substitutes, the consumer is willing to substitute the goods on a one to one basis, and p 2 > p 1 (p 2 < p 1 ) then the slope of the budget line is flatter (steeper) than of the indifference curve. Thus, the demand function for good 1 will be m/p 1 if p 2 > p 1 x 1 = [0, m/p 1 ] if p 2 = p 1 0 if p 2 < p 1 How is the demand function for good 1 if the goods are perfect complements? [answer this yourself] 9

10 Normal good - a good for which the quantity demanded always changes in the same way as income changes. Precisely, x(p, m) m > 0 Inferior good - a good for which the quantity demanded always decreases as as income increases x(p, m) m < 0 Does the demand of good 1 depend necessarily negatively on its own price? 1.5 Reference H. Varian, intermediate Microeconmics - A Modern Approach, fifth edition, Norton. (2-2.5, ). 10