Question

The Cobb Douglas production function for a product is N(XY)-10[0.8)(y^0.2) 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, 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 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, 50) 8.4 Ny(x,y) = 2/x/y)^0.8 Ny(40, 50) 1.7 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 a. What are F(x,y,h). R(x.y.h), and F(x,y.h)? 164 g(x,y) = Flx.y.h) Fx[xy.h) Fyxy.h) Fhix.y.h) 165 166 167 168 169 170 171 172 173 274 175 176 77 78 79 80 81 82 83 84 35 36 37 BB 89 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 NE Approximate increase:

please also find approximate increase!

Answer 1

The cobb-douglas production function is: N(x,y) = 10 0.8 0.2 x ५ N number of y number of units units of labor of capital. MPA = JN(ky) da oli 0-8 x10x0-230-2 Nx(x,y) 0.2 2 MP y a ON(x, y) dy 0.2 xloxo.8 ,-0.8 y IN 2(4)08. Ny(x, y) At x=40 and y 250 MPx = 8/50 (50)02 = 8.365 ~ 8 4 = N₂(40,50) MPy 2(40) 1.673 • Ny (40,50] 0.8 = 1.7 5o Labor costs $100 each Capital costs $50 each Budget for production $10000 a. Constraint equation is 100x + 50 y 21oooo We need to maximise. N(2,4) - 10x0.8 yo.2 g (mig): 100xt x+ 50y >g(y) 2 lovoo Subject to

The Lagrange is F (a, g, h) = loro 0-2 th[10000-100x-50 y first-order conditions: -0-2 Fx (x, y, h) = 10x0.8x2 xy 0-2 - 10oh zo 10x0.8x202 0.2 y -10oh = 0 0.2 a 100 h x h = 8 (4) 0:2 8 100 Fylkiy, h) 0.8 -0.8 - 50h = 0 10X 0.2xx y. 0.8 250h h & Fn(xiy, h) = 10000 100x-50y zo = 10000 2) 100x + 5 oy (3) 2 from (1) and (2), we get equating h 0.8 2.12 (u) 50 ly (4) 2 X 100 2) y 1 cos n 02 0.8 (4) 8 2 x=2yong

Putting, n=2y in equation (3) 100% +50 y =10000 - 100x2y + 5 Oy = 10000 2007 +50 y = 10000 250 y 2 10000 =) y = 40 x = 2840 = 80 10x x0.8 Production will be maximised labor = be = 807 Capital = y = 40 Production: =N(x,y) = x 4.0-2 Na tox(8000-8 X (40)0-2 IN = 696.44 moml), h = &(302 h= 2 = 0.0696=h 25 80/0 0.2 X Marginal productivity of money gives the appro ximate increase in production for each additional dollar spent on produchun Kamar h is the marginal productivity of money whom we can multiply h' by change in production budget to find approximate increase in production.

When production budget is increased by $2000 I production increases by 2000 x 0.0696 (1 h = 0.069 139.2 units (approximately) Approximate increase! 139-2 units