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PS5Soln

# PS5Soln - Problem Set 5 ECON 231W Spring 2010 Suggested...

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Problem Set 5 ECON 231W Spring 2010 Suggested Solutions 1. a)You cannot use OLS, because the equation under consideration is not of the form αX 1 + βX 2 + ε, where ε enters additively and the coefficients, which we estimate (in power 1), are multiplying the explanatory variables. b) You cannot compute E [ ε i | L i K i ] using the fact that E [ln ε i | L i K i ] = 0, because expectation is a linear operator and we are not given information about the whole conditional distribution of ln ε i (but just its first moment). In this case, you can only conclude that E [ln ε i | L i K i ] = 0 < ln E [ ε i | L i K i ] (as ln x is a concave function of x ), which implies that E [ ε i | L i K i ] > 1. c) You can use take the log of the right hand side and left hand side of the equation to get: ln y i = α ln L i + β ln K i + ln ε i d) In the equation above let e y ln y, e L ln L, e K ln K , e ε ln ε, so we have a convenient linear representation e y i = α e L i + β e K i + e ε i , with E [ e ε i | L i , K i ] = E [ e ε i | e L i , e K i ] = 0, which altogether satisfies OLS Assumption 1. e) In economic terms, α is the elasticity of output with respect to labor labor input, and β is the elasticity of output with respect to capital. f) One way is to use a simple t -test, where: H 0 : α + β = 1 H 1 : α + β 6 = 1 Where the standard error of b α + b β is q d V ar ( b α ) + d V ar ( b β ) + 2 d Cov ( b α, b β ) . An alternative method would be to re-write the equation as: ln y i = α ln L i + β ln K i + u i = α ln L i + β ln L i - β ln L i + β ln K i + u i = ( α + β ) ln L i + β (ln K i - ln L i ) + u i = γ ln L i + β (ln K i - ln L i ) + u i And run a simple t -test, where: H 0 : γ = 1 H 1 : γ 6 = 1 . 1

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2. a.) Here the RHS variable is measured in levels, and the LHS in logs so the relationship is log- linear for Fiveret . Thus, the coefficient of .0016 on Fiveret implies that a 1% change (because the units are in %) in Fiveret is expected to increase Salbon by .16% (a pretty small effect). Moreover,
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PS5Soln - Problem Set 5 ECON 231W Spring 2010 Suggested...

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