Question

The production of a manufacturer is given by the Cobb-Douglas production function

f(x,y)=30x^(4/5)y^(1/5)

where x represents the number of units of labor (in hours) and y represents the number of units of capital (in dollars) invested. Labor costs $10 per hour and there are 8 hours in a working day, and 250 working days in a year. The manufacturer has allocated $4,000,000 this year for labor and capital. How should the money be allocated to labor and capital to maximize productivity this year? Round answers to 2 decimal places, if necessary.

To maximize productivity, they should spend their money on ? hours of labor and invest $?. This leads to a maximum value of ? units. Also, if the number of dollars allocated to labor and capital is increased by 1, the number of units produced will Select an answer (increase/decrease) by approximately ?

Answer 1

According to the given data

f(x,y)=30x^(4/5)y^(1/5) ==>(a)

in subjuct to

y + (10*8*250)x ≤ $4,000,000

y + 20,000x ≤ 4,000,000

lets assume that

g(x,y) ==> y + 20,000x = 4,000,000 ==> (b)

now lets use lagrange multiplier  λ,

fx = λgx

30 * (4/5) * x^(-1/5) * y^(1/5) = λ (20,000)

λ = x^(-1/5) * y^(1/5) * (6/5000) ==>(c)

fx = λgx

30 * (1/5) * x^(3/5) * y^(-3/5) = λ (1)

λ = 6 * x^(3/5) * y^(-3/5)

Now, lets substitute λ from (c)

x^(-1/5) * y^(1/5) * (6/5000) = 6 * x^(3/5) * y^(-3/5)

x^(-1/5) * x^(-3/5) = 5000 * y^(-3/5) * y^(-1/5)

1/x = 5000* 1/y

y = 5000x ==> (d)

From (b) & (d), we get that

5000x + 20,000x ≤ 4,000,000

25,000x = 4,000,00

x = 160

Then, y = 5000*160 = 800,000

x = 160, y = 800,000

therefore, f(x,y)=30x^(4/5)y^(1/5)

==> 30 *(160)^(4/5) * (800,000)^(1/5)

==> 26,365.45

So, to maximize the productivity they have to spend 160 hours of labour & also need to invest $800,000.

This leads to maximum value of 26,365.45 units

If the number of $ allocated to labour & the capital is raised by 1

then we get that,

x = 4,000,001/25,000 = 160

therefore, the number of units provide will not change approximately

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