1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,910 Your support will help MIT OpenCourseWare continue to 4 00:00:06,910 --> 00:00:10,560 offer high-quality educational resources for free. 5 00:00:10,560 --> 00:00:13,460 To make a donation or view additional materials from 6 00:00:13,460 --> 00:00:18,340 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:18,340 --> 00:00:19,590 ocw.mit.edu. 8 00:00:23,060 --> 00:00:24,460 GREG HUTKO: Welcome back to the 14.01 9 00:00:24,460 --> 00:00:26,250 problem-solving videos. 10 00:00:26,250 --> 00:00:29,830 Today we're going to work on Fall 2010 Problem Set 4, 11 00:00:29,830 --> 00:00:31,640 Problem Number 3. 12 00:00:31,640 --> 00:00:33,220 And this problem is really going to take 13 00:00:33,220 --> 00:00:34,490 us through two scenarios. 14 00:00:34,490 --> 00:00:36,660 We're dealing with producer decisions. 15 00:00:36,660 --> 00:00:38,780 So now instead of dealing with utilities, we're going to be 16 00:00:38,780 --> 00:00:40,690 working with cost functions. 17 00:00:40,690 --> 00:00:41,790 And we're going to first go through 18 00:00:41,790 --> 00:00:43,510 the short-run scenario. 19 00:00:43,510 --> 00:00:45,790 And then we're going to talk about the long-run scenario 20 00:00:45,790 --> 00:00:49,440 and the implications of both of these cases. 21 00:00:49,440 --> 00:00:52,670 Problem Number 3 says, suppose the process of producing corn 22 00:00:52,670 --> 00:00:58,050 on a farm is described by the function q equals 8k to the 23 00:00:58,050 --> 00:01:04,230 1/3 times quantity L minus 40 raised to the 2/3, where q is 24 00:01:04,230 --> 00:01:07,430 the number of units of corn produced, k is the number of 25 00:01:07,430 --> 00:01:09,920 machine hours used, and L is the number of 26 00:01:09,920 --> 00:01:12,010 person-hours of labor. 27 00:01:12,010 --> 00:01:14,940 In addition to capital and labor, the farmer needs to pay 28 00:01:14,940 --> 00:01:19,690 a $15 transportation fee to deliver corn to downtown. 29 00:01:19,690 --> 00:01:23,300 So the cost can be written as total cost equals 15 times the 30 00:01:23,300 --> 00:01:28,550 quantity produced plus the rental cost of capital plus 31 00:01:28,550 --> 00:01:32,740 the wage rate times the quantity of labor. 32 00:01:32,740 --> 00:01:35,980 Part A says, suppose in the short-run the machine hours 33 00:01:35,980 --> 00:01:39,670 rented are fixed at k equals 8, and its rental rate equals 34 00:01:39,670 --> 00:01:43,220 64, and the wage rate equals 16. 35 00:01:43,220 --> 00:01:46,510 Derive the short-run total cost and the average costs as 36 00:01:46,510 --> 00:01:50,010 a function of output level q. 37 00:01:50,010 --> 00:01:52,560 So to start off this problem, we're going to start by 38 00:01:52,560 --> 00:01:54,025 working with the short-run scenario. 39 00:02:00,780 --> 00:02:03,110 And typically, the only difference between the 40 00:02:03,110 --> 00:02:06,530 short-run scenario and the long-run scenario in economics 41 00:02:06,530 --> 00:02:09,880 problems is that in the short-run, the amount of 42 00:02:09,880 --> 00:02:13,600 capital that a firm can use is going to be fixed. 43 00:02:13,600 --> 00:02:16,660 It means that because machines are a fixed cost in the 44 00:02:16,660 --> 00:02:19,230 short-run, you can't actually change how many machines or 45 00:02:19,230 --> 00:02:21,340 how often you use the machines. 46 00:02:21,340 --> 00:02:23,850 So we're going to set that equal to 8 for this scenario. 47 00:02:23,850 --> 00:02:27,830 And we also know that each hour that we use this machine 48 00:02:27,830 --> 00:02:31,510 is going to cost us 64. 49 00:02:31,510 --> 00:02:37,210 And we know that for each hour that we're using labor, it's 50 00:02:37,210 --> 00:02:41,600 going to cost us 16. 51 00:02:41,600 --> 00:02:45,140 We also know that in addition to the cost of the capital and 52 00:02:45,140 --> 00:02:48,340 the cost of the labor, which is represented in our total 53 00:02:48,340 --> 00:02:52,500 cost function here, for each unit q that we produce, we 54 00:02:52,500 --> 00:02:54,490 have to transport it to market. 55 00:02:54,490 --> 00:02:58,370 So we also have this 15 times q added into our total cost 56 00:02:58,370 --> 00:03:00,810 function, which is something that we might not always see 57 00:03:00,810 --> 00:03:03,100 in all of our cost functions. 58 00:03:03,100 --> 00:03:08,150 So let's start off by solving for the total cost function. 59 00:03:08,150 --> 00:03:10,090 And to do this, the first thing that we're going to do 60 00:03:10,090 --> 00:03:14,050 is we're going to plug in to our production function here 61 00:03:14,050 --> 00:03:16,680 what we know the capital is fixed at. 62 00:03:16,680 --> 00:03:23,150 And we're going to solve for labor, or L, in terms of q. 63 00:03:23,150 --> 00:03:35,570 So plugging in for k we're going to be 64 00:03:35,570 --> 00:03:37,290 left with this equation. 65 00:03:37,290 --> 00:03:40,370 And from here, we can solve for L in terms of q. 66 00:03:40,370 --> 00:03:43,080 And this is going to be useful for us because what we're 67 00:03:43,080 --> 00:03:45,930 going to do is we're going to take this L and we're going to 68 00:03:45,930 --> 00:03:50,400 plug it into the total cost function, so that our cost 69 00:03:50,400 --> 00:03:54,270 function is no longer in terms of k and L. But it's only 70 00:03:54,270 --> 00:03:56,750 going to be in terms of q. 71 00:03:56,750 --> 00:04:00,060 So isolating L in this equation, we're going to have 72 00:04:00,060 --> 00:04:15,220 that L equals 40 plus q/16 raised to the 3/2. 73 00:04:15,220 --> 00:04:16,935 So now let's go to our total cost function. 74 00:04:20,399 --> 00:04:25,690 We're going to plug in for k and r. 75 00:04:25,690 --> 00:04:32,180 So we know that r is 64 and k is 8. 76 00:04:32,180 --> 00:04:36,970 We're going to plug in for w 16. 77 00:04:36,970 --> 00:04:40,140 And now for L we're going to plug in what we solved for 78 00:04:40,140 --> 00:04:41,870 using our production function. 79 00:04:56,720 --> 00:05:00,500 So from this equation when we do the algebraic manipulation, 80 00:05:00,500 --> 00:05:02,620 we're going to get the total cost function in 81 00:05:02,620 --> 00:05:04,720 terms of only q. 82 00:05:04,720 --> 00:05:07,750 And solving out for this, we find that the total cost-- 83 00:05:07,750 --> 00:05:10,360 and I'm going to denote that this is in the short-run with 84 00:05:10,360 --> 00:05:14,520 the capital fixed by a little sr as a subscript. 85 00:05:14,520 --> 00:05:34,070 The total cost is going to be equal to 1,152 plus 15q plus 86 00:05:34,070 --> 00:05:37,600 16 times quantity q/16 raised to the 3/2. 87 00:05:40,880 --> 00:05:44,370 Now to find the average cost, all the average cost is it's 88 00:05:44,370 --> 00:05:46,750 the total cost divided by q. 89 00:05:46,750 --> 00:05:50,290 So it's per unit, how much on average does the producer have 90 00:05:50,290 --> 00:05:53,500 to spend to actually produce one unit? 91 00:05:53,500 --> 00:05:55,300 So to find the average cost, we're going to divide this 92 00:05:55,300 --> 00:05:58,540 whole thing through by q. 93 00:05:58,540 --> 00:06:07,600 And we're going to find the average cost in the short-run 94 00:06:07,600 --> 00:06:32,460 is going to be equal to 1,152 divided by q plus 15 divided 95 00:06:32,460 --> 00:06:40,370 by q/16 raised to the 1/2. 96 00:06:40,370 --> 00:06:44,130 In part B, what we're going to do is we're going to take the 97 00:06:44,130 --> 00:06:47,120 total cost function, we're going to plug in a fixed 98 00:06:47,120 --> 00:06:51,010 quantity, and we're going to find, what is the actual 99 00:06:51,010 --> 00:06:55,460 amount of money that a producer would have to pay to 100 00:06:55,460 --> 00:06:58,010 produce a fixed quantity? 101 00:06:58,010 --> 00:07:00,500 Part B says, suppose the farm wants to 102 00:07:00,500 --> 00:07:03,300 produce 64 units of corn. 103 00:07:03,300 --> 00:07:05,330 Based on the answer to part A, what is the 104 00:07:05,330 --> 00:07:06,715 total short-run cost? 105 00:07:09,630 --> 00:07:16,080 So using our solution from part A, all you have to do now 106 00:07:16,080 --> 00:07:24,040 for part B is you're going to plug in for q the number 64. 107 00:07:24,040 --> 00:07:52,340 So for part B, plugging in for q, you're going to find that 108 00:07:52,340 --> 00:08:00,670 the total cost in the short-run is going 109 00:08:00,670 --> 00:08:06,140 to be equal to 2,240. 110 00:08:06,140 --> 00:08:08,640 Now the more interesting part of this problem is what 111 00:08:08,640 --> 00:08:11,520 actually happens when instead of having to fix the capital 112 00:08:11,520 --> 00:08:14,650 at 8, what happens when the producer can change the amount 113 00:08:14,650 --> 00:08:15,900 of capital that they're producing? 114 00:08:18,610 --> 00:08:21,820 Part D says, in the long-run, the farm can change its 115 00:08:21,820 --> 00:08:24,810 capital level by minimizing the cost subject to the 116 00:08:24,810 --> 00:08:28,080 production function, derive the cost-minimizing demands 117 00:08:28,080 --> 00:08:34,220 for k and L as a function of output q, the wage rates w, 118 00:08:34,220 --> 00:08:38,820 and the rental rates of machine r. 119 00:08:38,820 --> 00:08:43,470 So now we're going to go back to a similar problem like we 120 00:08:43,470 --> 00:08:46,430 saw with consumer theory where we saw the marginal rate of 121 00:08:46,430 --> 00:08:50,510 substitution had to be equal to the price ratio. 122 00:08:50,510 --> 00:08:53,480 In this case, we're going to use something called the 123 00:08:53,480 --> 00:08:57,690 marginal rate of technical substitution. 124 00:08:57,690 --> 00:09:02,340 Simply put, the marginal rate of technical substitution asks 125 00:09:02,340 --> 00:09:07,440 us, how many machines or how many people would we be 126 00:09:07,440 --> 00:09:12,350 willing to basically lay off for one additional machine? 127 00:09:12,350 --> 00:09:15,540 And we call that the marginal product of capital, or how 128 00:09:15,540 --> 00:09:18,500 much we're actually getting from each unit of capital, 129 00:09:18,500 --> 00:09:22,080 divided by the marginal product of labor. 130 00:09:22,080 --> 00:09:28,070 We're going to set that equal to the price of the capital 131 00:09:28,070 --> 00:09:32,620 and the price of the labor. 132 00:09:32,620 --> 00:09:34,960 To find these marginal products of capital and the 133 00:09:34,960 --> 00:09:38,360 margin marginal product of labor, what we're going to do 134 00:09:38,360 --> 00:09:42,780 is we're going to take the derivative of our production 135 00:09:42,780 --> 00:09:46,510 function with respect to q, or with respect to k, and with 136 00:09:46,510 --> 00:09:49,560 respect to L. And when we do that, we find that the 137 00:09:49,560 --> 00:09:55,300 marginal product of capital is going to be equal to. 138 00:10:06,360 --> 00:10:08,810 And we can also solve for the marginal product of labor. 139 00:10:26,250 --> 00:10:30,170 And we're simply going to divide here and we can find 140 00:10:30,170 --> 00:10:40,490 that L minus 40 over 2k is going to equal r/w. 141 00:10:43,290 --> 00:10:45,400 And finally, the last thing we're going to do here is 142 00:10:45,400 --> 00:10:48,370 we're going to solve for the amount of labor and the amount 143 00:10:48,370 --> 00:10:52,620 of capital rented in terms of the other variables. 144 00:10:52,620 --> 00:10:56,660 We're going to plug this back into our production function 145 00:10:56,660 --> 00:10:59,370 that we have over here in the middle of the board. 146 00:10:59,370 --> 00:11:02,560 And then we can solve for how much capital and how much 147 00:11:02,560 --> 00:11:08,490 labor is demanded in terms of w and in terms of r. 148 00:11:08,490 --> 00:11:11,830 So I'll go through this process first for solving for 149 00:11:11,830 --> 00:11:15,350 the demand function for capital. 150 00:11:15,350 --> 00:11:17,820 To start off, we're going to get this equation in terms of 151 00:11:17,820 --> 00:11:33,990 L. So we have that L is equal to 2rk over w plus 40. 152 00:11:33,990 --> 00:11:36,090 And then we're going to take this L and we're going to plug 153 00:11:36,090 --> 00:11:40,090 it back into our production function. 154 00:11:40,090 --> 00:11:59,630 And when we do this, we're going to have that 8k to the 155 00:11:59,630 --> 00:12:06,170 1/3 times quantity 2rk over w raised to the 2/3. 156 00:12:06,170 --> 00:12:10,570 And now we're just going to solve through for rk. 157 00:12:10,570 --> 00:12:15,670 And when we solve through for k, we're going to find that k 158 00:12:15,670 --> 00:12:29,540 equals q/8 w over 2r raised to the 2/3. 159 00:12:29,540 --> 00:12:33,020 This is our conditional demand for capital. 160 00:12:33,020 --> 00:12:36,030 And we call it the conditional demand because it's dependent 161 00:12:36,030 --> 00:12:39,360 on the price that we're going to pay for labor, the price 162 00:12:39,360 --> 00:12:42,690 that we're going to pay for labor, the price we're going 163 00:12:42,690 --> 00:12:45,050 to pay for capital, and the quantity that 164 00:12:45,050 --> 00:12:47,160 we're going to produce. 165 00:12:47,160 --> 00:12:50,330 Now to do this same process only solving for the 166 00:12:50,330 --> 00:12:53,230 conditional demand for labor, you're going to go back to 167 00:12:53,230 --> 00:12:59,040 this equation right here. 168 00:12:59,040 --> 00:13:02,070 And instead of solving through for L, you're going to solve 169 00:13:02,070 --> 00:13:04,630 through for k. 170 00:13:04,630 --> 00:13:23,200 And when you solve through for k, you're going to find that k 171 00:13:23,200 --> 00:13:27,240 equals quantity L minus 40 times w over 2r. 172 00:13:27,240 --> 00:13:33,930 You're going to take this k, just like we did for labor 173 00:13:33,930 --> 00:13:37,630 you're going to plug it back into the production function. 174 00:13:37,630 --> 00:13:38,900 And you can solve through for the 175 00:13:38,900 --> 00:13:41,710 conditional demand for labor. 176 00:13:41,710 --> 00:13:44,050 And when you do that, you're going to find that labor is 177 00:13:44,050 --> 00:14:06,140 just going to equal q/8 times quantity 2rw raised to 178 00:14:06,140 --> 00:14:09,090 the 1/3 plus 40. 179 00:14:09,090 --> 00:14:12,130 So again, to summarize this problem, we started off with 180 00:14:12,130 --> 00:14:14,700 the short-run scenario. 181 00:14:14,700 --> 00:14:17,240 And what we found with the short-run scenario, we were 182 00:14:17,240 --> 00:14:20,870 able to plug-in for the capital that was fixed at 8. 183 00:14:20,870 --> 00:14:23,250 We were able to find the total cost and the 184 00:14:23,250 --> 00:14:25,590 average cost functions. 185 00:14:25,590 --> 00:14:28,410 Given a fixed quantity that they wanted to produce, we 186 00:14:28,410 --> 00:14:31,510 solved for the total cost in the short-run. 187 00:14:31,510 --> 00:14:33,900 And then finally we said, let's let the producer change 188 00:14:33,900 --> 00:14:36,840 how much capital they are actually using. 189 00:14:36,840 --> 00:14:41,500 And let's figure out based on letting the wage rate and the 190 00:14:41,500 --> 00:14:44,660 rental rate of capital change, let's get a conditional demand 191 00:14:44,660 --> 00:14:47,400 that lets those variables change that would let us then 192 00:14:47,400 --> 00:14:49,700 solve for the amount of capital and the amount of 193 00:14:49,700 --> 00:14:51,690 labor that would be needed to produce a 194 00:14:51,690 --> 00:14:52,940 certain amount of quantity.