Consider the following technology F(K,L) = (AK^?)(L^(1??)) where K is the level
ID: 1240435 • Letter: C
Question
Consider the following technology F(K,L) = (AK^?)(L^(1??)) where K is the level of capital, L is the level of labor or hours worked, A is a efficiency parameter and ? ? (0, 1) (a) Write down the firms problem in the long-run. Note the price for capital and labor are respectively, r and w. (Normalize the price of output to be 1 so that w and r are in real terms). (b) Does this production function depict constant return to scale? Prove it. (c) What are the marginal productivities of capital and labor? Are they decreasing in each factor? (d) What is the labor share and capital share of output? Remember the share of a factor is calculate as the compensation of that factor divided by the total output. Show it.Explanation / Answer
We assume a standard two-factor production function, with exogenous productivity growth: Yt = F(Kt,Ht) = F(Kt,egtLt) With: Yt = national income Kt = physical (non-human) capital Ht = human capital = efficient labor supply = egt Lt Lt = labor supply = number of hours of work g = exogenous labor productivity growth rate Yt = YKt + YLt = rtKt + vtLt = capital income + labor income Closed economy ? private wealth Wt = domestic capital stock Kt ? ßt = Wt/Yt = Kt/Yt I.e. wealth-income ratio = domestic capital-output ratio Note 1 : This is an exogenous growth model. If you want you can plug in your favourite endogenous growth model and derives g as a function of innovation, investment in higher education, etc.,… or distance to the world productivity frontier. Note 2 : This is a one-sector growth model, i.e. we assume a homogenous consumption and capital good, i.e. no long run divergence in relative prices; e.g. no divergence in the relative price of land, real estate, oil, services, etc. On multi-sector growth models, see e.g. Baumol “The macroeconomics of unbalanced growth” AER 1967 Note 3: If Lt = L0 (stationary population & labor supply), then in steady-state total output growth = per capita growth = g; if Lt = L0 ent (n = population & labor supply growth rate), then total output growth = g+n, and per capital growth = g (sometime people forget about population growth; but with low total growth the fact that n>0 cannot be neglected: e.g. if g+n=1.5%-2% & n=0.5%-1%, then g=0.5%-1%) 1.2. Linear savings St = s Yt We’re looking for a steady-state growth path with: ßt = Kt/Yt = ß* /Yt = /Kt = g Dynamic equation: = sYt I.e. = [Yt – Kt]/Yt 2 = s - gßt = 0 iff ßt = s/g Harrod-Domar-Solow formula: ß* = s/g I.e. wealth-income ratio (capital-output ratio) = saving rate/growth rate Simple, but powerful Example: If the saving rate s=10% and the growth rate g=2%, then the long-run wealth-income ratio ß*=500%. France 1980-2010: s=9%, g=1.7% ? ß*=550%-600%. If s=10% & g=1%, then ß*=1000%. If s=10% & g=5%, then ß*=200%. If s=25% & g=5%, or s=50% & g=10%, then ß*=500% = China