INSTRUCTOR: A company has determined that its production level is given by the Cobb-Douglas function f of x comma y equal to 2.5x to the 0.45, end exponent, multiplied by y to the 0.55, end exponent, where x represents the total number of labor hours in one year and y represents the total capital input for the company. Suppose one unit of labor costs $40 and one unit of capital costs $50.

Use the method of Lagrange multipliers to find the maximum value of f of x comma y equal to 2.5x to 0.45, end exponent, multiplied by y to the 0.55, subject to a budgetary constraint of $500,000 per year.

Let's first just note the budget. The budget restriction is that 40x plus 50y is equal to $500,000. Total cost for labor and capital cannot be more than $500,000.

We're going to write another function. It's our restriction function. It's g of x comma y equal to 40x plus 50y minus $500,000. We'll use that budget as our restriction.

Now we have the objective function, f of x comma y. And the method of Lagrange multipliers means that we are going to find the gradient of f and set it equal to lambda multiplied by the gradient of g subject to some restrictions.

So the gradient of f of x comma y is going to be equal to 1.125x to the negative 0.55, end exponent, multiplied by y to the 0.55, end exponent, i plus 1.375 x to the 0.45, end exponent, y to the negative 0.45, end exponent, j.

And this will, as I said earlier, be equal to lambda multiplied by the gradient of g of xy. This will be equal to lambda open parentheses 40i plus 50j close parentheses.

Now, often, it's a good idea to solve or to set the coefficients of i and j equal to one another and then isolate lambda in each of those equations. So if we set these coefficients equal to each other, we have 1.125 x to the negative 0.55, end exponent, y to the 0.55 equal to 40 lambda and 1.375 x to the negative 0.4-- pardon me, 1.375x to the 0.45, end exponent, y to the negative 0.45, end exponent, is equal to 50 lambda.

Now we're going to solve for lambda in both of these and set them equal to each other. So that means that we have lambda is equal to 9 over-- 9 over 320x to the negative 0.55, end exponent, y to the positive 0.55 is going to be equal to the second equation solve for lambda.

So, in fact, I'll just erase the lambda in this. That expression will be equal to 11 over 400 times x to the 0.45, end exponent, y to the negative 0.45, which, rearranging some terms here using exponent properties and whatnot, we see that y is going to be equal to 44 over 45 multiplied by x.

Now we can substitute this into our budget restriction, the g of x comma y. And, when we do that, we should see fairly quickly that x is equal to-- x is equal to 5,625.

Now, taking that x value, putting it back into y equals 44/45x, y is equal to 5,500. And, finally, we can evaluate that point, the point 5,625 comma 5,500, into f to find the production level, the maximum value here.

So f of 5,625 comma 5,500 is approximately 13,889.8. So, as a production level, we would say the maximum production level is 13,890 units.