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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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In Example 2, suppose that each labor unit costs $200 and each capital unit costs $250. Find the maximum production level when labor and capital cannot exceed $50,000.

To determine

To calculate: The maximum production level when labour and capital cannot exceed $50,000 if manufacture production function is f(x,y)=100x34y14.

Explanation

Given Information:

The cob-Douglas function of the manufacture production is,

f(x,y)=100x34y14

x represents the units of labour and each labour unit cost is $200 and y represents the units of labour and each labour unit cost is $250 also Given that the labour and the capital cannot exceed $50000.

Formula used:

Step 1: Make the constraints sing the labour unit cost and the capital unit cost.

Step 2: If f(x,y) be the function which have to maximize and the constant g(x,y) then find a new function.

F(x,y,λ)=f(x,y)λg(x,y)

Where λ is the langrage multiplier

Step3: Finding the partial derivatives of f(x,y,λ) of 1st order with respect to x, y and λ and equating them with zero to find the value of x,y and λ.

Step 4: Putting that values of x and y in the given equation f(x,y) find the maximum production level

Calculation:

Consider the given equation,

f(x,y)=100x34y14

Since each labour costs 200 and each capital costs $250 then the total labour and capital is 200x+250y.

As the labour and capital does not exceed $50,000.

Hence, the constant is,

200x+250y=50,000200x+250y50,000=0

And, g(x,y)=200x+250y50,000.

Consider the standard equation,

F(x,y)=100x34y14λ(250x+250y50000)

Now, the partial derivatives of F(x,y,λ) with respect to x, y, λ

Fx(x,y,λ)=100×34x341y14200λFx(x,y,λ)=75x14y14200λ

And,

Fy(x,y,λ)=100×14x34y14250λFy(x,y,λ)=25x34y34250λ

And,

Fλ(x,y,λ)=(200x+250y50000)Fλ(x,y,λ)=200x250y+50000

Now, equate all above equation to zero,

Fx(x,y,λ)=0Fy(x,y,λ)=0Fλ(x,y,λ)=0

Then substituting the values,

75x14y14200λ=025x34

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