# Lecture3 - Lecture Note 3 Deadweight Cost Part 2B Paper 1 Dr T.S Aidt University of Cambridge Michaelmas 2010 1 Introduction A tax is distortionary

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Lecture Note 3: Deadweight Cost Part 2B: Paper 1. & Dr. T.S. Aidt University of Cambridge Michaelmas 2010 1 Introduction A tax is distortionary when taxpayers can change their behavior to avoid pay- ing the tax. An important question is how we can measure the e¢ ciency loss associated with this and equally importantly we would like to inquire into the factors that determine the magnitude of the social loss. The e¢ ciency loss associated with a distortionary tax is called the dead- weight loss of the tax or the excess burden of the tax. It represents the amount that is lost in excess of what the government collects in tax revenue. 2 De&nition There are a number of alternative ways to de&ne the deadweight loss (DWL). The one we shall use can be formulated as follows. De&nition 1 The deadweight loss of a tax is the amount in excess of the tax revenue being collected that the taxpayer would give up in exchange for removal of the tax. Alternatively, we can say that De&nition 2 The deadweight loss of a tax measures how much more could be collected from the taxpayer (and thrown away) than is being collected by the tax under consideration, with no loss in utility, if the collection method was lump sum taxation. To &x ideas, we shall consider a situation with two commodities, x 1 and x 2 and one input labor l 0 . There is one representative consumer and one represen- tative &rm. The consumer prices are q = f q 0 ;q 1 2 g and the producer prices are & Disclaimer: This note may contain mistakes. If you spot any, please bring them to my attention. 1

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p = f p 0 ;p 1 2 g . To &x ideas, we assume that labor supply is &xed and generates income m . The utility function is U ( x 1 ;x 2 ; l 0 ) and the production technology has decreasing returns leaving the possibility of short-run pro&ts open. All mar- kets are perfectly competitive. We shall consider a unit tax levied on consumers of good 1 , t 1 , and investigate what the deadweight loss generated by this tax is and upon which factors its magnitude depends. To study the welfare e/ect of this tax, we shall make use of the indirect utility function and the expenditure function (which should be familiar from last year). The indirect utility function, V ( q;m ) , is simply linking maximized utility to consumer prices and income, i.e., it is de&ned as V ( ) = max f x 1 ;x 2 g U ( x 0 1 ; l 0 ) (1) subject to the budget constraint q 1 x 1 + q 2 x 2 = m & q 0 l 0 : (2) The solution to this problem is the Marshallian (or uncompensated) demand function which records demand for the good for given consumer prices and income, i.e., x i ( ) . The expenditure function is de&ned as the minimum expenditure at given consumer prices needed to obtain a given level of utility, i.e., E ( q; U ) = min f x 1 ;x 2 g 2 X i =1 q i x i (3) subject to U ( x 1 2 ; l 0 ) = U . The solution to this expenditure minimization problem is the so-called Hicksian or compensated demand function, denoted by x c i ( q; U ) . It is called the compensated demand function because it is constructed by varying consumer prices and income so as to keep the consumer±s utility level &xed, i.e., the consumer is compensated for the income (and endowment e/ect)
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## This note was uploaded on 06/04/2011 for the course ECONOMICS paper 1 taught by Professor Aidt during the Spring '11 term at Cambridge.

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Lecture3 - Lecture Note 3 Deadweight Cost Part 2B Paper 1 Dr T.S Aidt University of Cambridge Michaelmas 2010 1 Introduction A tax is distortionary

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