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GREG HUTKO: Welcome back to the 14.01 problem-solving videos. Today we're going to work on Fall 2010 Problem Set 4, Problem Number 3.

And this problem is really going to take us through two scenarios. We're dealing with producer decisions.

So now instead of dealing with utilities, we're going to be working with cost functions. And we're going to first go through the short-run scenario. And then we're going to talk about the long-run scenario and the implications of both of these cases.

Problem Number 3 says, suppose the process of producing corn on a farm is described by the function q equals 8k to the 1/3 times quantity L minus 40 raised to the 2/3, where q is the number of units of corn produced, k is the number of machine hours used, and L is the number of person-hours of labor. In addition to capital and labor, the farmer needs to pay a $15 transportation fee to deliver corn to downtown.

So the cost can be written as total cost equals 15 times the quantity produced plus the rental cost of capital plus the wage rate times the quantity of labor.

Part A says, suppose in the short-run the machine hours rented are fixed at k equals 8, and its rental rate equals 64, and the wage rate equals 16. Derive the short-run total cost and the average costs as a function of output level q.

So to start off this problem, we're going to start by working with the short-run scenario. And typically, the only difference between the short-run scenario and the long-run scenario in economics problems is that in the short-run, the amount of capital that a firm can use is going to be fixed. It means that because machines are a fixed cost in the short-run, you can't actually change how many machines or how often you use the machines. So we're going to set that equal to 8 for this scenario.

And we also know that each hour that we use this machine is going to cost us 64. And we know that for each hour that we're using labor, it's going to cost us 16.

We also know that in addition to the cost of the capital and the cost of the labor, which is represented in our total cost function here, for each unit q that we produce, we have to transport it to market. So we also have this 15 times q added into our total cost function, which is something that we might not always see in all of our cost functions. So let's start off by solving for the total cost function.

And to do this, the first thing that we're going to do is we're going to plug in to our production function here what we know the capital is fixed at. And we're going to solve for labor, or L, in terms of q. So plugging in for k we're going to be left with this equation. And from here, we can solve for L in terms of q.

And this is going to be useful for us because what we're going to do is we're going to take this L and we're going to plug it into the total cost function, so that our cost function is no longer in terms of k and L. But it's only going to be in terms of q.

So isolating L in this equation, we're going to have that L equals 40 plus q/16 raised to the 3/2. So now let's go to our total cost function.

We're going to plug in for k and r. So we know that r is 64 and k is 8. We're going to plug in for w 16. And now for L we're going to plug in what we solved for using our production function.

So from this equation when we do the algebraic manipulation, we're going to get the total cost function in terms of only q.

And this problem is really going to take us through two scenarios. We're dealing with producer decisions.

So now instead of dealing with utilities, we're going to be working with cost functions. And we're going to first go through the short-run scenario. And then we're going to talk about the long-run scenario and the implications of both of these cases.

Problem Number 3 says, suppose the process of producing corn on a farm is described by the function q equals 8k to the 1/3 times quantity L minus 40 raised to the 2/3, where q is the number of units of corn produced, k is the number of machine hours used, and L is the number of person-hours of labor. In addition to capital and labor, the farmer needs to pay a $15 transportation fee to deliver corn to downtown.

So the cost can be written as total cost equals 15 times the quantity produced plus the rental cost of capital plus the wage rate times the quantity of labor.

Part A says, suppose in the short-run the machine hours rented are fixed at k equals 8, and its rental rate equals 64, and the wage rate equals 16. Derive the short-run total cost and the average costs as a function of output level q.

So to start off this problem, we're going to start by working with the short-run scenario. And typically, the only difference between the short-run scenario and the long-run scenario in economics problems is that in the short-run, the amount of capital that a firm can use is going to be fixed. It means that because machines are a fixed cost in the short-run, you can't actually change how many machines or how often you use the machines. So we're going to set that equal to 8 for this scenario.

And we also know that each hour that we use this machine is going to cost us 64. And we know that for each hour that we're using labor, it's going to cost us 16.

We also know that in addition to the cost of the capital and the cost of the labor, which is represented in our total cost function here, for each unit q that we produce, we have to transport it to market. So we also have this 15 times q added into our total cost function, which is something that we might not always see in all of our cost functions. So let's start off by solving for the total cost function.

And to do this, the first thing that we're going to do is we're going to plug in to our production function here what we know the capital is fixed at. And we're going to solve for labor, or L, in terms of q. So plugging in for k we're going to be left with this equation. And from here, we can solve for L in terms of q.

And this is going to be useful for us because what we're going to do is we're going to take this L and we're going to plug it into the total cost function, so that our cost function is no longer in terms of k and L. But it's only going to be in terms of q.

So isolating L in this equation, we're going to have that L equals 40 plus q/16 raised to the 3/2. So now let's go to our total cost function.

We're going to plug in for k and r. So we know that r is 64 and k is 8. We're going to plug in for w 16. And now for L we're going to plug in what we solved for using our production function.

So from this equation when we do the algebraic manipulation, we're going to get the total cost function in terms of only q.

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