Ec 11 Caltech Lecture 1 Notes (revised: version 2.0)

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1 Ec 11 Caltech Lecture 1 Notes (revised: version 2.) Prof. Antonio Rangel Spring 212 Contact rangel@hss.caltech.edu with corrections, comments, and suggestions. Lecture notes are organized around the following pedagogical principles. Principle 1. Most figures are not included with the notes, but will be discussed in detaile during lecture. This is done to encourage active learning. Principle 2. The notes contain small exercises (labeled Think) designed to test and expand your understanding of the material. We urge the students to work through all of them. Contents 1 Introduction to Part I Three basic economic questions Basic assumptions of model Consumer demand The consumer problem and the demand function Consumer demand Market demand Individual and market consumer surplus Application: Demand analysis in practice Application: measuring the value of new products

2 1 Introduction to Part I This part of the course provides an in-depth and calculus based introduction to the basic theory of competitive markets. This is the most basic, widely used, and important model in all economics: learn it well! 1.1 Three basic economic questions Every society needs to provide an answer to the following basic questions: 1) WHAT is produced?, 2) HOW is it produced?, and 3) For WHOM is it produced? Different economic systems solve these questions in different ways. In centralized economies, a small group of individuals chooses the answers. In pure (or idealized) market economies, the outcome is determined through an initial system of comprehensive and fully enforceable property rights and volutary trade. In mixed market economies, the outcome is determined by a mix of property rights, voluntary trade, and government intervention (e.g., redistributive taxation, legal limitations on what can be traded,...). Mixed market economies can be described by a continuous parameter q that indexes the degree of government intervention, with q = denoting pure market economies, and q = 1 denoting pure centralized economies. Most real economies are in the middle, since it is nearly impossible to eliminate all volutary trade, and all economies have some degree of government intervention. In Part I we develop a simple model of the allocations that arise when consumers and producers interact in an idealized and highly stylized market.: a single good is traded in a market that is independent from the rest of the economy. We will proceed in two steps: Step 1. Develop a theory of pure competitive markets. Step 2. Extend the theory to the case of government intervention. We proceed in this order because the analysis of the dealized case provides critical insights for understaning the effects of government intervention. 1.2 Basic assumptions of model The model considers a market for a specific good (e.g., restaurant meals), in a particular location (e.g., LA), and in a particular period (e.g., Spring 212). 2

3 The identity of participants is fixed and exogeonously given Consumers are labeled c = 1,..., C. Producers (or firms) lare abeled f = 1,..., F. In this setting the three basic economic questions take the following form: WHAT: total quantity Q produced in the market HOW: amount produced (i.e., supplied) by each firm, denoted by s 1,..., s F FOR WHOM: amount purchased and consumed by each consumer, denoted by d 1,..., d C In addition, the model also predicts the price p at which everybody trades in the market. Terminology and notation: > Variables with a * super-script are called equilibrium variables. > These are the endogenous variables predicted by the model. > Ex: p is the equilibrium price. Additional assumptions: > The good can be traded in infinitesimal amounts (i.e., it is assumed to be perfectly divisible) > All units traded in the market are identical from the point of view and consumers (i.e., they do not care about their differences) and producers (i.e., they are equally costly to produce). 2 Consumer demand A key element of the model are the demand functions d c (p,i) that characterizes the behavior of each consumer c: I is an exogeneous variable denoting the consumer s income (measured in $) p is the price at which the consumer gets to buy the good (in $/unit) d c (p,i) is a function specifying the amount the consumer wants to buy given p and I. The demand function provides a complete description of the consumer s behavior As we will see below, it is possible to compute the equilibrium of markets without knowing more about where it comes from. However, a much more powerful theory of competitive markets can be developed if we assume that the demand function is the result of optimizing behavior. Note: To simplify the notation, whenever there is no risk of confusion we drop the subscripts denoting specific consumers or firms. 3

4 2.1 The consumer problem and the demand function Each consumer derives utility from consumption given by U(q, e) = v(q)+e where: q denotes the amount of the market good consumed (in units of the good) e denotes the amount spent in all other goods and services. (in $). U() is often called experienced utility, since it describes the quality of the allocation at the time of consumption. When U(.) takes the form assumed here, it measures the utility of consumption in $. Note that v(q) denotes the utility derived from consuming q units of the market good (measured in $). Assumptions on the utility function: q is non-negative (i.e., q ) e is allowed to be negative. This is not critical but simplifies the model. (If it helps, think of e < as a case in which the consumer borrows to consume more in the market good). v > v <. The function v describes the marginal utility (MU) of the market good, and it measures the increase in utility obtained from consuming an additional unit. The assumptions imply that: v > : every additional unit of q increaes utility. v < : it does so by a decreasing amount (i.e., each subsequent unit of q increases utility by less than the preceeding ones). v is a concave function Fig 1 depicts the shape of typical v and MU functions. Think Does v(q) = log(q + 1) satisfy the assumptions of the model? Think What function v leads to a decreasing linear MU function? Is such a v consistent with all of the assumptions of the model? The budget constraint describes the set of feasible consumption bundles (q, e): It is given by pq + e apple I See Fig. 2 for a graphical description Key assumption: Consumer makes decisions in order to maximize their experienced utility, subject to feasibility constraints. Consumer problem (CP) : Pick q and e in the budget constraint to maximize U(q, e). d c (p,i) denotes the solution to the CP for p and I. 4

5 Simplication of the CP: Since utility increases in both q and e, at the optimal consumption bundle the budget constraint holds with equality Then CP can then be rewriten as: Pick q to maximize U(q, I pq). This version of the CP problem is esier to solve. Characterization of the solution:. Two types of solutions: 1) interior (with q >,) and 2) corner, with q =. Interior solution characterized by first order condition (FOC): v (q) =p Cornder soltution characterized by FOC: v () apple p. v < implies that the second-order-conditions for maximization are satisfied. Note that only one of these conditions can hold at the optimum, and that the solution to the CP is unique under the model s assumptions. Fig 3 provides the intuition for the interior case: At the depicted price, v () > p, and thus it is worth it to buy the first unit. In fact, it is worth it to keep buying as long as v (q) > p, since this increases consumption utility, but not after the MU has decreased to the point where v (q) < p. Fig 4 provides intuition for the corner case: it is not worth it to buy the first unit since v () < p. Tip To compute the demand function for a consumer with a specific utility function, simply compute v and verify the FOCs for every possible p >. Think. A) Does the CP has a solution (i.e., the solution is non-infinite) when p <? B) What about when p =? 2.2 Consumer demand Given the arguments in the previous section, the demand curve is as depicted in Fig. 5 Note that the diagram places the independent variable p in the vertical axis, and the dependent variable q in the horizontal axis. Thus, the axes are flipped from normal math usage. This axes convention is wide-spread in economics because it is useful in market analyses (as we will see throughout the rest of this course). Think Examine Fig 5 very carefully and make sure that you understand why it is drawn exactly as it is). Think Is it possible for a demand function to have d(p, I) > for all p >? Claim. In the basic model, p d apple. Why? Obvious for prices above v () which generate corner solutions. For prices below v (), demand satisfies the FOC: v (d(p, I)) = p 5

6 Taking the derivate with respect to p in both sides we get: v d p = 1. Thus, p d = 1 v < (by the properties of the utility function). This result is often called the Law of Demand. Think. In the basic model, demand curves are indpendent of income: d I =. Prove it. Would this be true without the assumption the e can be negative. Why? 2.3 Market demand Market demand at any price p is given by D(p) =Â d c (p). We can omit the incomes from the market demand given the absence of income effects in the basic model. Since all consumers buy goods at the same price, market demand can be characterized by the process of horizontal addiction depicted in Fig. 6. Think. Suppose that the market demand depicted in Fig. 7 was generated by two groups of 1 identical consumers each. What do the individual demand curves look like? 2.4 Individual and market consumer surplus Economists are interested in both positive and normative analysis. Positive analysis seeks to characterize the outcomes generated in markets as a function of the exogenous variables (e.g., What happens to Q and p if another firm enters the market?). Normative analysis is about measuing the impact of these changes in consumer s well-being (e.g., Does the entrance of an additional firm improve consumers well-being?). In this section we develop one of the most remarkable and powerful features of the basic model: It is possible to measure the impact of market changes in the consumer s well-being, even when his actually utility function is unknown to the analyst by using only observable information about his demand function! These analyses are based on a concept known as consumer surplus (CS). In order to understand CS it is useful to first look at another object called the indirect utility function: Denoted by V(p, I) V(p, I) =U(d(p, I), I pd(pm, I)) It measures the level of consumption utility when the consumers chooses the optimal allocation at p and I. 6

7 Given the assumptions of the model, we have that: V(p, I) =[v(d(p, I))] + [I =[v()+ d(p,i) =[I + v()] + [ pd(p, I)] v (q)dq]+[i d(p,i) d(p,i) (v (q) p)dq] pdq] Note: The term I + v() is equal to the consumption utility of a subject who has no access to the market, and thus consumes unis of the market good, and $I in all other goods. Denote this level of utility by Vno market The term d(p,i) (v (q) p)dq is equal to the increase in utility that results from being able to buy the optimal amount in the market at price p. Using the indirect utility function it is then easy to define consumer surplus: CS(p) =V(p, I) Vno market Thus, it measures the value of market participation for the consumer at price p (in $s). By the previous argument, we see that CS(p) = d(p,i) (v (q) p)dq As shown in Fig. 8, CS(p) has a nice graphical interpretation: it is equal to the area under the demand curve between the quantities of and d(p) Think. A) Can CS(p) =? Under what conditions. B) Can you prove that CS(p),with strict inequality if d(p) >, without relying on the CS formula above. (Hint: It is a one line proof that relies on the basic properties of the consumer problem). C) Is it possible that CS(p) > pd(p)? Illustrate with an example/graph. D) Is it possible that CS(p) > I? Illustrate with an example/graph. Another useful construct is the change in consumer surplus: 4CS(p! p 1 )=CS(p 1 ) CS(p ). Measures change in CS when prices change from p to p 1. It is also equal to the change in consumer well-being between the two situations, measured in $s As before, it measures the amount of $ that would have to give (if 4CS 7

8 is positive) or take (if 4CS is negative) to leave consumer indifferent between the two situations Fig. 9 illustrates. Think Show that 4CS(p! p 1 )=(p 1 p )d(p )+ d(p 1 ) (v (q) p 1 )dq d(p ) From the individual to the market: consumer surplus and consumer surplus changes can be added across consumers to produce measures of market consumer surplus and market changes in consumer surplus. NOTATION: We omit subscripts with individual identifiers whenever there is no risk of confusion. But whenever necessary we index individuals by their labels and the market aggregates by mkt. Important. 1) The normative analysis and interpretation behind CS is valied only if consumers truly make decisions that are consistent with maximizing their experienced utility. 2) There is no requirement that they do this consciously and explictly, only that whatever processes they used lead to choices compatible with experience utility maximization. 3) If this is not the case, the demand function is not linked to the underlying experienced utility, and the CS measure of well-being is not valid. 4) See Problem set 1 for an example of the problems that can arise when consumers make choices incompatible with optimizing their utility function. 2.5 Application: Demand analysis in practice In practical applications, economists apply these tools by: a) Collecting data that allows them to estimate the individual or market demand functions. b) Assuming that the assumptions of the consumer model approximately hold, so that CS is a good measure of consumer well-being Here is an outline of how this type of exercise looks like: Suppose that careful observation has generated data about market demand that is depicted in Fig. 1 (each dat point is represented by a cross) Using statistical and econometric methods (that you will learn in more advanced courses), the researchers have estimated that the marked demand function is given by D(p) =max{, 1 2p}. Given this, the 4CS generated by an exogenous shock that reduces 8

9 prices from p = 3 to p 1 = 1 is given by area shown in Fig. 1, which equals 2[$/unit] 4[units]+.5 2[$unit] 4[units] =$12, 2.6 Application: measuring the value of new products (?)... Intuition suggests that the invention and development of new products (e.g., Prof. Rangel loves his Macs) is a great source of increases in wellbeing over time. The value for consumers of having access to a new product that sells at price p is given by 4CS(p be f int! p), where p be f int is any price sufficiently high to that the consumer buys zero units of the product, and p is the current market price. Problem set 1 asks you to apply this tool to estimate the value of introducing the Ipad. 9