chapter16_s

chapter16_s - CHAPTER 16 SOLUTIONS TO PROBLEMS 16.1 (i) If...

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Unformatted text preview: CHAPTER 16 SOLUTIONS TO PROBLEMS 16.1 (i) If 1 = 0 then y 1 = 1 z 1 + u 1 , and so the right-hand-side depends only on the exogenous variable z 1 and the error term u 1 . This then is the reduced form for y 1 . If 1 = 0, the reduced form for y 1 is y 1 = 2 z 2 + u 2 . (Note that having both 1 and 2 equal zero is not interesting as it implies the bizarre condition u 2 u 1 = 1 z 1 2 z 2 .) If 1 0 and 2 = 0, we can plug y 1 = 2 z 2 + u 2 into the first equation and solve for y 2 : 2 z 2 + u 2 = 1 y 2 + 1 z 1 + u 1 or 1 y 2 = 1 z 1 2 z 2 + u 1 u 2 . Dividing by 1 (because 1 0) gives y 2 = ( 1 / 1 ) z 1 ( 2 / 1 ) z 2 + ( u 1 u 2 )/ 1 21 z 1 + 22 z 2 + v 2 , where 21 = 1 / 1 , 22 = 2 / 1 , and v 2 = ( u 1 u 2 )/ 1 . Note that the reduced form for y 2 generally depends on z 1 and z 2 (as well as on u 1 and u 2 ). (ii) If we multiply the second structural equation by ( 1 / 2 ) and subtract it from the first structural equation, we obtain y 1 ( 1 / 2 ) y 1 = 1 y 2 1 y 2 + 1 z 1 ( 1 / 2 ) 2 z 2 + u 1 ( 1 / 2 ) u 2 = 1 z 1 ( 1 / 2 ) 2 z 2 + u 1 ( 1 / 2 ) u 2 or [1 ( 1 / 2 )] y 1 = 1 z 1 ( 1 / 2 ) 2 z 2 + u 1 ( 1 / 2 ) u 2 . Because 1 2 , 1 ( 1 / 2 ) 0, and so we can divide the equation by 1 ( 1 / 2 ) to obtain the reduced form for y 1 : y 1 = 11 z 1 + 12 z 2 + v 1 , where 11 = 1 /[1 ( 1 / 2 )], 12 = ( 1 / 2 ) 2 /[1 ( 1 / 2 )], and v 1 = [ u 1 ( 1 / 2 ) u 2 ]/[1 ( 1 / 2 )]. A reduced form does exist for y 2 , as can be seen by subtracting the second equation from the first: 0 = ( 1 2 ) y 2 + 1 z 1 2 z 2 + u 1 u 2 ; because 1 2 , we can rearrange and divide by 1 2 to obtain the reduced form. 92 (iii) In supply and demand examples, 1 2 is very reasonable. If the first equation is the supply function, we generally expect 1 > 0, and if the second equation is the demand function, 2 < 0. The reduced forms can exist even in cases where the supply function is not upward sloping and the demand function is not downward sloping, but we might question the usefulness of such models. 16.3 No. In this example, we are interested in estimating the tradeoff between sleeping and working, controlling for some other factors. OLS is perfectly suited for this, provided we have been able to control for all other relevant factors. While it is true individuals are assumed to optimally allocate their time subject to constraints, this does not result in a system of simultaneous equations. If we wrote down such a system, there is no sense in which each equation could stand on its own; neither would have an interesting ceteris paribus interpretation....
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chapter16_s - CHAPTER 16 SOLUTIONS TO PROBLEMS 16.1 (i) If...

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