The production function for a company is given by f(x, y) = 100x^0.25y^0.75 wher
ID: 2880797 • Letter: T
Question
The production function for a company is given by f(x, y) = 100x^0.25y^0.75 where x is the number of units of labor (at $72 per unit) and y is the number of units of capital (at $36 per unit). The total cost for labor and capital cannot exceed $100,000. (a) Find the maximum production level for this manufacturer. (Round your answer to the nearest integer.) units (b) Find the marginal productivity of money. (Round your answer to four decimal places.) (c) Use the marginal productivity of money to find the maximum number of units that can be produced when $125,000 is available for labor and capital. (Round your answer to the nearest integer.) units (d) Use the marginal productivity of money to find the maximum number of units that can be produced when $340,000 is available for labor and capital. (Round your answer to the nearest integer.) unitsExplanation / Answer
a>
we could do this problem using the Lagranges Multipliers
the constraint here are the labour units and the capital units and their sum cannot exceed 100000
=> 72x + 36y = 100000
we write this as a Lagranges multiplier problem as follows :
F(x,y,lamda) = 100x^(.25)y^(.75) - lamda(72x +36y - 100000)
now we'll find the partial derivatives of the function F with respect to x , y and lamda and equate them to 0
=> Fx= 25x^(-.75)y^(.75) - 72lamda= 0 --------(1)
Fy = 75x^(.25)y^(-.25) - 36lamda = 0 ---------(2)
Flamda = -(72x + 36y - 100000)= 0 ---------(3)
now we'll solve the above three equations and we'll find the valve of lamda
from the above we get Lamda = 1.33113
x= 347.22 and y = 2083.33
after solveing the equations we get just 1 critical point
that is (x , y , lamda) = (347 , 2083 , 1.33113)
Since F(x,y,lamda) has only one critical point, we conclude that maximum productivity
occurs when 347 units of labor and 2083 units of capital are used .
f(x,y) = 100x^(.25)y^(.75)
=> at x = 347 and y = 2083
f = 100*(347)^(.25)*(2083)^(.75) = 133076 units
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