Question

142 143 144 The Cobb Douglas production function for a product is 145 N(X) - 101X0.8)(y^0.2) 147 148 149 150 where is the number of units of labor and y is the number of units of capital required to produce N units of the product, 151 What is the marginal productivity of labor and the marginal productivity of capital? What are they when there are 40 units of labor and 50 units of capital? NxIx,y) - Bly/X)^0.2 Nx(40, SO) 8.4 Ny(x,y) - 2/x/y)^0.8 Ny(40, 50) 1.7 152 153 154 155 156 157 158 159 160 161 162 163 of each unit of labor costs $100, each unit of capital costs $50, and $10,000 is budgeted for production of this product, what is the constraint equation? What is F(x, y, A)? Use k instead of 2. What are F(x,y,h), y(x.y.h), and F(x,y,h)? 164 g(x) = Flx.y.h) - Fx(xy.h) Fylx.y.h) = 165 166 167 168 169 170 171 72 173 274 175 176 77 78 79 80 81 82 83 Fhfx.y.h) - What are the units of labor and the units of capital that will maximize production? What is the maximum production? What is the marginal productivity of money? What is the approximate increase in production that would result from an increase of $2,000 in the amount budgeted for production? Use exact formulas, do not round. Y- hr N Approximate increase: 35 36 37 BB 39 30

Answer 1

Production function X labor te 2(308 & At Nix,y) = 10x00402 y capital. * Marginal productivity of labor. MP, F aN(x,y) 10 10:8)x-6-2y0.2 IMPx 8(1)°2] * Marginal productivity of capital Miry- IN(X,Y)= 10 10.2)xo-By-0-8 sy MPy = x=40 y=so MPx = 8(& J = 8.36 MPy = 2(3208 2(3)0.2 - 1.67 Px= $100 Py Pya $50 Budget = $10,000 Constraint function. Soy = * F(x, y, h) = 10 x0 Byº 2+ th (lovoo - 100x-soy). * Fx (x, y, h) - 1010.8)x-? 0.2 - 100h * Fy (x, y, h) = 10 (0.2) x 0.84 0.8 – 5oh. * Fh (x, y, h)= 2010 \000o - loox-soy. Jogos for optimal labor and capital equate x = 0, Fyzo 8x02yo.2 loooo S 10oX + bort tope loon