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10) The production level of a factory during nta time period is modeled by Plx,Y) = Kxlh 10) yl2 where K is positive integer, is the number of units of labor schedu...

Question

10) The production level of a factory during nta time period is modeled by Plx,Y) = Kxlh 10) yl2 where K is positive integer, is the number of units of labor scheduled and y is the number of units of capital invested. If labor costs s42Ounit, capital costs $70O/unit and the ar has 51,800,000 available for one time period, what amount of labor would maximize prodt and cpital luction?

10) The production level of a factory during nta time period is modeled by Plx,Y) = Kxlh 10) yl2 where K is positive integer, is the number of units of labor scheduled and y is the number of units of capital invested. If labor costs s42Ounit, capital costs $70O/unit and the ar has 51,800,000 available for one time period, what amount of labor would maximize prodt and cpital luction?



Answers

Suppose that the daily output of a manufacturing plant is modeled by $Q=1000 K^{1 / 2} L^{1 / 3},$ where $K$ is the capital investment in thousands of dollars and $L$ is the size of the labor force in worker-hours. Assume that $L$ stays constant and think of output as a function of capital investment, $Q(x)=1000 L^{1 / 3} x^{1 / 2}$ Find and interpret $Q^{\prime}(40)$.

Hey guys lived through problem two In this problem the production function Q is given as 5-plus 40. The capital is fixed at 500. We need to find out what is the short run production function and what is the marginal product of labour? The production function is given as chemical to 5-plus 4 K. And capitalists fixed and equal to 500. Therefore the production function Q is going to 5-plus four multiplied by developed a which is 500 Which gives us q equal to five plus 2000. Which is a short run for Russian function. You equal to five L plus 200. 2000. Next we need to find out the marginal product of labour. We know the marginal product of labour is the change in output due to changing labour input. Therefore mpl it is the Kanto. Delta Q divided by delta L. Change in production. Changing the output. Little change in the labor. If L increases by delta L. Units then Q is going to be five and in the bracket L plus delta L last 2000. Or we can say Q. Is we can't do five L plus five delta L. Last 2000. And the change in our food therefore will be delta Q. Is going to the new the new cube minus the initial one. The final output minus initial output. Just this one. If we insult these values here. Five plus five del 10 plus 2000 -5 -2000 which we got here. We're using this value over here. Therefore the marginal part of flavor is equal to Delta. Cute. Very by Delta L, delta Q. We found five delta L. Because 55 bills will be gone. 2000s will be gone will be left with five dental, Then five telltale divided radial. Tell gives us five. Therefore the marginal product of labour is going to fight. Yeah.

The production function for the personal computers of Dicks. Inc is given by Q. Is equal to 10 key Rest Apart 0.5 L. rest apart 7.5. Now, what Q. Is the number of computers produced per day? Kids number of hours of machine time and L. Is a um hours of labor unit impute. Now digs competitors floppy inc. So this is what gigs inc now flopping. Corporation is using the production function. Q is equal to 10-K. 1st of all. 0.6. L. A Enterprise 0.4. Now first question is if both companies use the same amount of capital and labor which will generate more output. So cute. One is equal to So Q one is equal to 10 x. What I used to express about five doing X rays apart five 0.5. And this would be called suits hand x rays to paul. 0.5 plus 0.5. And I let this equal to turn X and Q. to recall to 10 X. Let's just be cute too. Right? 10 X ray supply 0.6 x rays to part 0.4. And this would be called to 10 X. Resupply 0.6 plus for 3.4. And then this is the cost of 10 x. Right? So because Q one this is Q one is he called Q two. So the question says a book cos you seem amount of capital and labor which will generate more output. So Q one is equal to Kill two. Moving on to the same court to the next question. Um This days are seen that capital is limited tonight. Machina was but labour is unlimited supply in which companies marginal product of liberal Greta top of times leave me tonight. I was LeBeau is only limited. So to find this Q one is he called Sue. If you go back to the formula you want to be called to. That's the l reciprocity 0.5 And cute to be equal to 37.37 L. rest a par four. So clearly as we can see the difference. Um The marginal in which companies marginally above um marginal product of liberal Greta is going to be floppy. Copernicus floppy is cute, too. So this before pay, inc. Yeah.

Okay. This question asks us about how productivity would change if we added 1000 workers given the starting conditions. So what we'll do is use the fact that if we fix the capital, then the change in productivity would be approximately equal to the partial with respect, the labour times, the change in labor. So we're just gonna calculate DPD l at our given point and use that. So DPD l is just equal to 20 k to the 1/3 because that isn't affected. And then we have a 1/2 times l to the negative 1/2 from the power rule here. Or we could simplify this as DPD. L is equal to 10 times Kate of the 1/3 divided by the square root of l. So now we're looking at the point where we're estimating from which is 1 25 900. So we get 10 times que brute of 1 25 divided by the square root of 900. And then this will just punch into our calculators real quick to get 1.66 seven. So that is our derivative. So using that, we see that Delta P is approximately 1.667 times Delta L, which is 1000. So Delta P is gonna be somewhere in the ballpark of 1666

Okay, This question asks us what the effect would be of adding 1000 workers to this productivity equation. So we're going to use the fact that if we fix the capital, Delta P is approximately given by the partial with respect to labour times, the change in labor and in our case, it says Delta L is 1000 workers. But since cells and thousands Delta P is approximately just gonna be given by the partial respect l at the point we want. So DPD l is just 20 k to the 1/3 because that doesn't change. And then we get 1/2 times l to the negative 1/2. Or we could simplify this as 10 case of 1/3 over lt with 1/2. And the point we want is CPL. At 1 25 comma 900. And that's equal to 10 times 1 25 to the 1/3 divided by 900 to the 1/2. And doing this, we get an answer of 5/3 so that would be to change and productivity. And again, that's only because Delta L is equal to one. In this case,


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