PS14+exercise+solutions

# PS14+exercise+solutions - Intermediate Microeconomics, 2010...

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Intermediate Microeconomics, 2010 Problem Set 14: Exercise Solutions In this topic we see two more very common examples of situations in which the First Welfare Theorem is often violated. In the presence of externalities and public goods the competitive allocation is not necesserily e±cient. In the presence of an externality the actions of one part of the economy directly in²uence the welfare of another part. A typical example is a negative externality in which the production of a good directly harms the consumer. So there is some el- ement e ( q S ) which enters the consumer’s utility funciton. e ( q S ) must be taken into consideration when evaluating total surplus, but the ³rm does not take it into consid- eration when choosing q S . So, while the competitive allocation is still determined by v 0 ( q ± ) = c 0 ( q ± ), surplus is now S ( q ) = v ( q ) ± c ( q ) ± e ( q ) so total surplus is maximized when v 0 ( q b ) = c 0 ( q b ) + e 0 ( q b ). One way to achieve this e±cient outcome is to tax the ³rm so that it internalizes the costs of the externality: t = e 0 ( q b ). It may be e±cient for a government to tax the consumer’s consumption in order to provide a public good. This would be the case if the net bene³ts from producing the public good ( B ± C ) outweigh the losses in surplus caused by the tax (( v ( q ± ) ± c ( q ± )) ± ( v ( q t ) ± c ( q t ))). We explore examples in Problem 2 in which this is and is not the case. 1. Positive Externalities. Production of medicine often requires research that pro- duces new general insights, i.e., production of medicine has a positive externality. Consider the following quasilinear economy. A ±rm can produce medicine at costs c ( q s ) = 5 q 2 . Production of q s units of medicine leads to an increase in our gen- eral knowledge k ( q s ) = 2 : 5 ( q s ) 2 . The consumer has preferences over medicine, knowledge, and money: U ± q s ;q d ;m ² = m + v ± q d ² + k ( q s ) = m + ³ 90 q d ± 5 ± q d ² 2 ´ + 2 : 5 ( q s ) 2 (a) Find the demand and supply functions, D ( p ) and S ( p ) . Derive the competitive allocation x ± . What is the surplus in the competitive allocation? (The surplus with the externaility is S ( q ) = v ( q ) ± c ( q ) + k ( q ) ). v 0 ( q ) = 90 ± 10 q = P D ( q ) is the inverse demand function (so the demand function is D ( p ) = 9 ± 1 10 p ) and c 0 ( q ) = 10 q = P S ( q ) is the inverse supply function (so the supply function is S ( p ) = 1 10 p ). Setting these equal, we ³nd q ± = 4 : 5. Plugging this into the supply or demand functions we ³nd that p ± = 45. And now we calculate m ± = Y ± p ± q ± = Y ± 202 : 5 and ± ± = p ± q ± ± c ( q ± ) = 101 : 25.

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## PS14+exercise+solutions - Intermediate Microeconomics, 2010...

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