Steve Pischke/Jin Li Labor Economics II Problem Set 2 1 ...

Steve Pischke/Jin LiLabor Economics IIProblem Set 21. Equilibrium with a CES production function (based on Krusell etal, 2000): Assume that production requires only high-skilled labour, K,low-skilled labour, O, and capital, N. Production takes place accordingto the CES function:h i 1\ w = (h ow O w ) +(1 )(N w +(1 )(h kw K w ) ) .Note that > are time-invariant distribution parameters, while h ow >h kw aretime-varying e!ciency parameters.z(a) Derive the wage premium, $ = KzOas a function of the parametersof the production function and the ratios of factor supplies.(b) Log-linearise your expression and take the time derivative. Decomposethe change in the wage premium into the elements due to: (i)the relative growth rate of labor inputs, (ii) the relative growth rateof labor e!ciency, and (iii) the capital-skill complementarity eect.Hints: (1) Remember that ln(1 + [) ' [. (2) Remember log derivativerules:C ln [ 1 C[ [˙= = = j [Cw [ Cw [C ln ¡ ¢ [\ [˙ \˙=Cw [ \ = j [ j \(c) What do we need to assume about > in order for the capital-skillcomplementarity eect to favor skilled labor?(d) Assume now that = . Whatdoesthisimply aboutchanges in theskill premium resulting from changes in capital? In this case, findC ln $³ ´.C lnWhat must be true of the elasticity of substitution in order for inkcreases in h h oh kh oto increase the wage premium?1Cite as: Steve Pischke and Jin Li, course materials for 14.662 Labor Economics II, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloadedon [DD Month YYYY].

(e) Now assume that the relative supply of skilled labor is endogenous,that is, responds to the skill premium. In particular, assumeμ K w= f$ where A 1=O wWhat is the elasticity of supply in this case?ln $hC ln kh o(f) Now solvefor theequilibrium $ and findC ³(g) Now consider the Katz-Murphy regression ofln $ w = + 1 w + 2 ln(K w @O w )+ h w >´ in this case.where 1 estimates the time trend on h k @h o (plus other terms) andˆ2 =1@ˆ> where is the elasticity of substition between K and Olabor. Show that the Katz and Murphy type estimates are inconsistentif skill supply is endogenous. What is the sign of the bias on 2[also explain intuitively]?(h) Katz-Murphy assume that ln(K w @O w ) is quasi-fixed at any given pointin time. How does this help?(i) If we knew , would this allow us to estimate 1 consistently evenwith endogenous labor supply? Explain.2. Technical Progress and New Technology (based on Beaudry andGreen, 2003): Suppose that there are three factors of production: highskilllabor, low-skill labor and capital (K> O> N).Moreover, there aretwo possible production technologies. The old technology uses low-skilllabor, while the new one uses capital and high-skill labor, according tothe following production functions:\ r = D r O r 1\ q = D q K q N q >The economy has endowments of labor, Ō and K¯ .(a) Solve for wages of high and low skill workers as a function of capitalin each sector. Write an expression for $ = z K @z O .(b) Suppose that capital is supplied to the economy inifinitely elasticallyat price ū. Solve for the wage premium in this case. What is theelasticity of the wage premium with respect to relative skill supplies?2Cite as: Steve Pischke and Jin Li, course materials for 14.662 Labor Economics II, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloadedon [DD Month YYYY].

Show that if capital becomes exogenously cheaper (ū falls), the skillpremium rises.(c) Take the time derivative of $ with respect to D r >D q and ū. If D rand D q grow at the same rate, and the interest rate is constant, whathappens to the skill premium over time? What must happen to theinterest rate to maintain a constant wage premium in this case?(d) Now suppose instead that the economy has an endowment of capital,N¯ . Solve for the wage premium in terms of N q . Show that anincrease in N¯ must increase $. What is the eect on $ of neutraltechnical progress in each sector (Ḋ q@D q = Ḋ r@D r )? Of an increasein K? An increase in O?(e) Suppose all workers supply inelastically one unit of O and someamount of K that is directly related to their years of schooling, h (| l ).An increase in the wage premium will have what eect on the slopeof total earnings as a function of years of schooling? What, if anything,does this model imply about changes in the level of earningsover time?3. Linking facts to theory (open ended): List the 3 to 6 most salientfacts that you believe a coherent explanation of rising wage inequality.Think about dierent explanations for rising wage inequality in termsof its ability to convincingly account for the key facts — where it fits,where it misses, where it is silent. What is the greatest weakness of eachhypothesis?3Cite as: Steve Pischke and Jin Li, course materials for 14.662 Labor Economics II, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloadedon [DD Month YYYY].