The production function for a company is given by fx y...

The production function for a company is given by f(x, y) = 100x^0.6y^0.4 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. 1. Find the maximum production level for this manufacturer. (Round your answer to the nearest integer.) 2. Find the marginal productivity of money. (Round your answer to three decimal places.) 3. Use the marginal productivity of money to find the maximum number of units that can be produced when $145,000 is available for labor and capital. 4. Use the marginal productivity of money to find the maximum number of units that can be produced when $380,000 is available for labor and capital.

Transcript

00:02 Hello, let's have a look at the question.

00:04 So we have been given fxy is equals to 100x to the power 0 .6 and y to the power 0 .4.

00:13 Here, x is the number of unit of labors and here y is the number of unit of capital.

00:32 Now the total cost of the labor and capital is, so now this can be written as cxy is equals to 72x plus 36y equals to 1 .000000 and 0.

00:57 So now this is 1 lakh.

00:58 Now we get 72x plus 36y minus 1 lakh is equal to 0.

01:09 Now using lagrangee multiplier, we have that fx comma fy is equals to lambda cx and cy.

01:24 So let us calculate the differentiation first of fx with respect to fx and fy.

01:31 So we will get that d by dx of 100 x to the power 0 .6 into y to the power 0 .4.

01:42 And comma differentiation with respect to y of 100 x to the power 0 .6 and y to the power 0 .4.

01:54 So now this is equals to lambda d y d x of 72x plus 36y minus 1 lakh.

02:10 Comma d by d y of 72x plus 36y minus 1 lakh now here we can write this as here we will get 100 into 0 .6 and x to the power minus 0 .4 and y to the power 0 .4 is equal to lambda into 72 and 100.

02:43 0 .4 x to the power 0 .6 and y to the power minus 0 .6 is equal to lambda into 36.

02:53 Now let us eliminate lambda from the above equation.

02:59 So we will get 100 into 0 .6 into x to the power minus 0 .4 and y to the power 0 .4 divided by 100 into 0 .4 and x to the power 0 .4 and x to the power 0 .6 and x to the power 0 .6 and y to the power minus 0 .6 which is equals to 72 lambda divided by 36 lambda.

03:21 So on canceling the terms here we will get 3y divided by 2x is equal to 2...

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