Question

Need help on some of my calculus homework

The Cobb-Douglas production function for a product is N(x,y) = 10(x^0.8)(y^0.2) where x is the number of units of labor and y is the number of units of capital required to produce N units of the product.

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?

Nx(x,y) =

Nx(40, 50) =

Ny(x,y) =

Ny(40, 50) =

If 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,λ)? Use h instead of λ.

What are ??(?,?,h)F_x (x,y,h), ??(?,?,h)F_y (x,y,h), and ?h(?,?,h)F_h (x,y,h)?

g(x,y) =

F(x,y,h) =

Fx(x,y,h) =

Fy(x,y,h) =

Fh(x,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.

x =

h =

Approximate increase:

y =

N =

Answer 1

Kindly go through the solution provided below.

we are given the multivariable function N(x,y) = lo xos you? We need to Nx , Ny Nx = and 10 30*20.3 x*-+)- s (*)** And 0.8 Ny ។ = = ON ду 10 x®®[0.2 you-a)= 2 Nowo, No (+0,50) = *(*.*- * (2) ** Ny ( 40, 50) = 2 (75) **={? () 0.8 Constraint equation is given by! 100 x + 50y = 1 0,000 OY 2x + y = 200 Now; g (x,y) = 2x + y - 200 = 0 F(x, y, n) = N(x,y) hg(x,y) fx = df dx ON дух h 29 = 84 дх = 8(*)**- h Similarly, Fy = 2(*)** - 24 Fu = 2x + y - 200 the critical points for F(x,y,h) occurs at Fx = 0 , Fy = 0', fo = 8 (*)=h; 2x+y=200 Now, - h í (3)" a 8 y = x 16y + y = 200 í T y = loo and x = 1600 17 17 h = (8)° And Nao, we similarly find out the new valuss of x, y and a when the budget is increased by ļooo New constraint 50 y = 12000 [100x + + 50 22 + y = 540 = g(x,y) = 2x + y -5t0 = 0 Now, the critical points are: y = 5 to and x = 160 20 1구