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Consider a two-person (1 and 2) two good (X and Y) one-input (L)production economy. The utility function of person 1 is given byU1 = 208 x1 - 2x12 + y1 - (x2 / 7 ...

Question

Consider a two-person (1 and 2) two good (X and Y) one-input (L)production economy. The utility function of person 1 is given byU1 = 208 x1 - 2x12 + y1 - (x2 / 7 + x22)and the utility function of person 2 is given byU2 = 208 x2 - 2x22 + y2 where xi and yi denoterespectively person i's consumption amount of good X and of goodY, i=1, 2. Each person has Li = 208 units of labour, i =1,2.The production function of good X and that of good Y aregiven respectively by X = 2 Lx and Y =Ly, wh

Consider a two-person (1 and 2) two good (X and Y) one-input (L) production economy. The utility function of person 1 is given by U1 = 208 x1 - 2 x12 + y1 - ( x2 / 7 + x22) and the utility function of person 2 is given by U2 = 208 x2 - 2 x22 + y2 where xi and yi denote respectively person i's consumption amount of good X and of good Y, i=1, 2. Each person has Li = 208 units of labour, i = 1,2. The production function of good X and that of good Y are given respectively by X = 2 Lx and Y = Ly, where LJ denotes the amount of labour used in the production of good J, J=X, Y. Derive the efficient amount of good X for person 1, x1*. Then enter the value of x1* below.



Answers

Suppose two individuals (Smith and Jones) each have 10 hours of labor to devote to producing either ice cream (x) or chicken soup $(y) .$ Smith's utility function is given by \[ U_{s}=x^{0.3} y^{07} \] whereas Jones" is given by \[ U_{I}=x^{0.5} y^{0.5} \] The individuals do not care whether they produce $x$ or $y$, and the production function for each good is given by \[ x=2 l \text { and } y=3 l \] where $l$ is the total labor devoted to production of each good. a. What must the price ratio, $p_{x} / p_{y}$ be? b. Given this price ratio, how much $x$ and $y$ will Smith and Jones demand? Hint: Set the wage equal to 1 here. c. How should labor be allocated between $x$ and $y$ to satisfy the demand calculated in part (b)?

Okay in this equation, the given function and the constrained function Are you in term off X and why equals X Y plus two X and the constraint equation too X plus y equals 30. Do so ingredient derivative or ingredients derived from the original function so ingredient you equals. Why close to in my direction, plus X in G direction. Ingredient G equals two in my direction, plus g direction. So I NJ equation found from the equation ingredient You equals Lunda Ingredient G. We will give us I equals or in either direction why plus two equals longer, too. And in J direction, X equals longer. One putting. Why, in terms in terms off X So why plus two equals longer to then Why plus two equals two X then why equals two X minus two. Now solving for X and y using the constant equation. So two x plus Why equal 30 So two x plus Replace why with this equation So we will have to x minus two equal 30. By solving this, we will have four x equals 32 then X equals eight, and why equals 14 plugging now x and y into the original equation to solve for you, so you will be equals eight times 14 plus two times eight equals $128.

Okay, So there you have it. Mmm. Well, no. Both, uh, with action about the company, your industry, whatever. That's a function Capital B after wearable, sickly. Why Axis? Actually stints labor unions. And why instance Are can we go in the dust? Yeah, the kind of backed a sure protect the year two rounds of Praxis equal prince some dumping of those labor and capital ourselves in doubling production. Mystic, where are the tents? Beans? Just Carrie. Next. There, you curry, right, Three. You know, you see that took to have two parents. So you to your to Paris. Okay, So knowing being have some values of okay, now we're even que nous And what you're told is that you have a coast off when we have our union. He's toying with my wounds. So where one? Excuse me. Let's say we're home. You don't use coast there, Capital, Newt. So there's one more. Okay, so we have to compose cost banks. Excellent units tonight. So if one you think of ex coast than this, you have 250 days. Rex costs for labor and didn't just saying thanks coast, but capital unit eso maximize production reversal working race. That's something that the total cost cannot exit turns in door. So make this construction murderess 100,000. So thank you. Bye. Graze on the status quo So one day you can see but a seven point seriously Couldn't see Yeah. Next you sequel issues. So clay years. Great. It's numb times. Great stuff Cost much they have to compete over. He saw the reality respective acts will be carrying. Okay. Nice. 211. 33 Mmm. Injuries trick workers next. Why? To Minister. Okay, very much missed work stories. I'm cold. 60. And why? So hurry so great in tow. Cost savior This one study 50 experts from the snow. I did not expect Americans. Really? No, it was great genes No routine extreme. I'm sorry. Extinct. I put this on. Okay, so we have to solve for X and wedding Strike the same minds. Never exciting person Second angel and go to their quick sign Clowned Barrett. Nothing. Next. Wait a second amount Three. So we can comparing now. Just Yes. Spirit mix business. We're in 2233. Really Sorry, right. This is ex brains. Fine. Yes. What's doing so excessively? quote. Okay, You can't miss Tonto. Great. Where personally? Under against Asians. Times 24. What's 400? So more And 52 before 1200. No alerts. 1600. Sorry. Like most sir. 16. Why? So where is it going to ground? Very stale cheese. I have a right to identify. Two. 62 point. Okay, Way our ask to thank the material production so direct We're practicing. Xing is curious. Ease won't one and two. So thanks. Which there Your case? No longer pounds, please. Nothing in production. We're actually served. Yeah. Hey, the 64. Really? There is 60 women, right? Lessons Exit doesn't have six here. 1000 units of total production out. Really? That's it.

So thank you. Come. Eso in heartache. You're asked to show that labor. How? The stuff just you. Right one Mr. So what things? Chest art absence. Case went. Voters asked to raise articles. Trained cost doesn't exceed $100 come to coast. Thanks. Tucked you coast. The total cost will be It's ex prize unit it And okay, so to maximize p he's sorry. See? It's thanks. Thanks to you, just he's here. It's a computer. It's No, they're maximized buyers. Which is it? Great. So he has to compute is equal to want steady but respect X 20 times him Water wins once. Okay, Chris. Function hours just so. System Yes. Okay. Experienced. Ditto. He's next, just And okay, so from Hugh. See us so far. Gift relations, Mr Just sure. Next. This Mr please X. Thanks. It's fun. Why? I see eight now we Concepcion. Thanks. Strengths body. So good to 50 times. It's 16 1000 60 cheese. Teens 16. Next, please. And we have to find tucked value is he stations this Terrence six sound. Find out

Okay, so for problem 31 there's two parts parts A and B apartment. It wanted you to show that doubling both labour and capital results of doubling production. T so labour being X and capital being Why so? Basically, I want you to show that p of two x two y is coming equal to two times p of X y. So let's go ahead and write that know sweet have k two X to the A times to why? To one minus a So this condemn be rewritten as que tires to a X A what? Because the X there the first time. And then that could also be written as times two. Why to the first power times two y two negative a power. That's because when two eyes one minus a power, that's the same thing. Is writing is to watch the first times to watch in the debate there for the next set. What we can do. We can go ahead and move that, too from the two wire front because it doesn't have any exporting when it was going to K Times two today, next today times why, and it is too wide to the negative they can down to go to the denominator since it's the negative exponents and we could just make the expletive positive. Once acquitted, this is gonna be all over. I'm also to make that two to a power. And why today? Parts. I split that up, seeing my house, but of dec's. So once you see that you see that right here, to to the A to today cancel out. And then for this next set, I'm gonna go ahead. Bring that. Why today? Back up. This is gonna be to to decay two times k x a. Do you know why? Tells what's it? Negativity. So there you could say that is two k x a Why? To the one minus a power wants the game, and you could see that checks out. That is equal to two p of X y okay, Part me. They said, uh, kz could want 20 and a zygote of three force So Israel I r p of X alive. So it's now nearly equal to 1 20 x to the three force times alive to the one minus exits one minus de force, Or why to me 14 Okay, So that's a P of X Y and G of X y. They said that they every cost $250 for each unit. So that's 250 x then capital cost for in geology for each units or plus 400 Why? And the costs cannot exceed $100,000. So it only minus 100,000 that must be equal to zero. So no tell of p was read was blamed it on the G. So Delle api is gonna be equal to 120 times three Force X. So exit e force of minus lawyer is gonna be exit negative one force. Then you can keep the why to the one for had direction. Waas 120. How's life worth Institute taking this driven Inspector Warren Now exit the three force. Why is his negative G force that would be equal to Well, I am Joe turns 215. No, I had direction plus lander times 400 in tow. J. Edgar No said the components equal to each other and I'm also gonna do some multiplication. So 1 20 times three of our four is gonna be equal to 90 So it's 90. And then, since it's exits to native 14 I'm just put that indigent on Lady. This is 92 y suing for over extra one poor. So I just switched the science and puts it in a nominee that's gonna be equal to Lander Times 200 and they 1 20 before that's gonna be equal to 30. So, so angry. 30 extra tree force Over. Why? To the three force Aston only equal to planned times for Okay, So what I'm gonna do now, I'm just gonna solve you Cycle, ada. So on the right side, I'm gonna get 90. Divided by 2 50 That's just give me 9/25. Why? 214 over. Excellent For is gonna go to Atlanta. 30 over 400. This is gonna be 3/40. So 3/40 extra to be force who are didn't force. It is possible that you could surrender Sinaga cities equal to each other. So when I wanted one for over 25 exit award for is going to be quits, it three Excellent. Three course turns 40 over 40. Watt said if the force now what? I'm gonna do is I'm going to cross multiply both sides. So we're gonna get 40 times and I that's gonna be 300 3 60 mother. So 3 60 and then wide some wide to the 14 times. Watch it. 3/4 we're gonna add the exponents so it's gonna be wide today. 44 So it was a lie is going to be equal to 25 times Street 75. Thanks. And then what we can do? We can divide that d 62 other side and get that wise and equal to 75/3. 60 x, which is just 5 24 Okay, are so let's we're gonna do is we're going to plug our Y into our G of X likely equation. So to 50 X was 400 times 5 24 That is gonna be equal to $100 so we can try. Simplify that out 400 times five divided parts of before. Well, I'm just gonna leave it over 24. Pronounceable is gonna be 2000 over 24. Plus, it's only to treat me over to you for where? So 6000 x over 24 to meet with 100,000 So 1000 X over 3400 and then we can simplify that Excellent legal to 300. And then why is it going 5/24 x so we can just multiply that by five and divide by 24? Why isn't equal to 62 0.5? So that means for maximum production level, um, and not exceeded cost over 100,000. We need brown. Oh x hours of label x 300 hours of labor labor Sorry in 62.5 on UNESCO capital invested.


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