Lecture 2 - 2 2.1 Welfare measurement Introduction Last...

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Unformatted text preview: 2 2.1 Welfare measurement Introduction Last class we introduced the idea of when the market economy does a good job of allocating resources, and when and how government inerventions may be appropriate (equity, market failures, etc.). But how can the planner make decisions? We need some tools for measuring wellbeing of individuals and comparing among individuals. 2.2 Social welfare measurement Public economics involves making social choices: choosing an allocation of resources among individuals in society, and an associated distribution of income and utility for all. To do so, we require a representation of feasible allocations (utility possibilities frontier) and of social preferences over allocations (social welfare function). Utility possibilities frontier. Combination of maximal utility levels attainable, given individual utility functions and the production possibilities frontier of the economy. (See Figure 2.1, constructed from the Edgeworth box for an exchange economy.) The UPF corresponds to the set of Pareto efficient allocations. Formally, the UPF in a 2 consumer, 2 good exchange economy is the set that solves ¯ max u A ( x A , y A ) s.t. u B ( x B , y B ) ≥ u B x A + xB ≤ X y A + yB ≤ Y This problem can be solved by Lagrangian or by direct substitution. Some examples: Exercise 1. 1. Society has 100 units of a private good available to share between two consumers with utility functions u A (x A ) = uB (xB ) = √ 10x A 20x B Find an equation for the utility possibilities frontier. 2. There are two private goods which are in fixed supplies X and Y , and two consumers A and B with utility functions u A (x A , y A ) = x A u B ( x B , y B ) = min{ x B , y B } Find an equation for the utility possibilities frontier and graph it. Copyright c 2008 by Michael Smart 1 Printed on: September 11, 2009 LECTURE 2. WELFARE MEASUREMENT 0 B U B C E F D D E F W W C UPF 0 A A U Figure 2.1: The Utility Possibilities Frontier As we move along the UPF, utility for A falls and B rises or vice versa (why must this be so?). This correspond to moves along the contract curve in the Edgeworth box. Second-best frontier. In defining the UPF we implicitly assumed the planner could engage in lump sum redistribution. We noted before this may be hard in practice, so that redistribution will be confined to inteventions that change prices (through taxes/subsidies) and so take us off the contract curve. Economists talk about lump-sum redistribution as the first-best case, and the any case in ehivch the planner’s instruments for intevention are limited as being the secondbest case. So we could derive a first-best UPF (we just did) and a second-best UPF. Consider our earlier example of a subsidy for food. We know that the second-best UPF must lie inside the first-best UPF, and that thy must coincide at at least one point. (How?) See Figure 2.2. Pareto principle. How should society rank points on or inside the UPF? The Pareto principle states society should prefer an allocation at which some people are better off but no-one is worse off. (In Figure 2.1, which points are Pareto efficient and inefficient? Which points are Pareto improvements over W ?) Note the distinction between Pareto improvements and Pareto efficient allocations. All agree on a Pareto improvement. But not all agree that a move from an inefficient to an efficient allo2 LECTURE 2. WELFARE MEASUREMENT P U FB UPF UPF SB E R U Figure 2.2: The Second-Best UPF cation is desirable. Social welfare functions. A better approach is to make explicit value judgments about tradeoff between welfare of individuals, captured by social welfare function W (u P , u R ) (analogous to individual utility functions). The choice of social welfare function depends on society’s preferences over inequality. Two polar cases 1. utilitarian or Benthamite : W (u P , u R ) = u P + u R (See Figure 2.3.) 2. Rawlsian: W (u P , u R ) = (See Figure 2.3.) uP uR if u P ≤ u R if u R < u P 2.3 Consumer surplus We wish to have a monetary measure of the change in an individual’s utility caused by a tax policy, provision of a public good, or other government program. Measuring welfare change in income units allows us to compare gains to an individual to the cost of the program to government, or to compare gains and losses experienced by different individuals. A key question in this course (examined in more detail next lecture) is: What is the cost to an individual consumer of a tax on a specific good? An economist’s answer to this question is not the amount of tax that is paid; it is generally a higher number. Heuristically, this is because each unit of a good consumed confers a benefit to the consumer that is at least as high as the amount paid for it. One measure of this additional benefit is known as consumer’s surplus. Therefore a tax that discourages consumption of a good creates losses in welfare pver and above the amount of tax paid; this additional amount is known in turn as the excess burden of the tax. 3 LECTURE 2. WELFARE MEASUREMENT uR uR W Figure 2.3: Utilitarian and Rawlsian social welfare functions. An individual’s inverse demand curve measures marginal willingness to pay for the commodity, given current consumption level. The area under the inverse demand curve measures consumer surplus : the consumer’s total willingness to pay for the amount consumed in excess of the amount actually paid. Consider a price increase caused by taxation of a single good. (See Figure 2.4.) The reduction in consumer surplus for a single consumer has two components: the tax revenue raised by government (rectangular area) and the excess burden of the tax (triangular area). Excess burden, also known as the Harberger triangle, is the loss in consumer surplus that is not compensated by an increase in tax revenue. Example: T.he inverse demand curve is the linear function p( x ) = b − ax Consumer surplus at any price p is therefore the area of a triangle of height b − p and base x = (b − p)/ a, so ( b − p )2 CS( p) = 2a ¤£¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¡ ¡¡ ¡¡¢ ¢¡ ¢¡¡ ¢¡¡ ¢¡¢¡¡¢¡¡ ¢ ¡ ¡¡ ¡¡ ¢¡¢¡¡¢¡¡¢ ¢ ¢ ¢ ¢ ¡ ¡¡ ¡¡ £ ¡¡¡¡¡ ¢ ¢ ¢¡ ¢¡¡ ¢¡¡ ¢ ¢ uP = uR uP = uR W(u P , u R) = W UPF uP W(u P ,u R = W ) UPF uP 4 LECTURE 2. WELFARE MEASUREMENT P c+t c Figure 2.4: Revenue and excess burden of a price change There are some conceptual problems with the consumer surplus measure, however. For example, suppose that we wish to calculate the willingness of a consumer to pay for the simultaneous change in the prices of two goods. How would you draw the demand curves for the two goods and calculate the consumer surplus change? Aggregating consumer surplus. Many allocations are not ranked by Pareto principle (see Figure 2.1, points W and F). Social choices involve real trade-offs, as policy changes create some winners and some losers. According to one criterion, allocation F is socially preferred to allocation W if the sum of willingnesses to pay to move from W to F is positive. Thus, winners could in principle comepnsate losers for the change and still be better off. For a single price change, this can be measured by the area under the market demand curve. Problems with the compensation principle: • Boadway paradox: Sum of willingnesses to pay for a move between two points on UPF (such as F and D in Figure 2.1) is never negative and is usually positive. 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WELFARE MEASUREMENT • Distributional insensitivity: Assumes that monetary gains are equivalent, no matter who receives them. Equivalent to a social welfare function (see below) that weights individual gains by the inverse of individual marginal utilities of income. 6 ...
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