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[M] The consumption matrix $C$ below is based on input- output data for the U.S. economy in 1958 , with data for 81 sectors grouped into 7 larger sectors: $(1)$ nonmetal household and personal products, $(2)$ final metal products (such as motor vehicles), (3) basic metal products and mining, (4) basic nonmetal products and agriculture, (5) energy, (6) services, and (7) entertainment and miscellaneous products. Find the production levels needed to satisfy the final demand d. Units are in millions of dollars.)
$$
\left[\begin{array}{ccccccc}{.1588} & {.0064} & {.0025} & {.0304} & {.0014} & {.0083} & {.1594} \\ {.0057} & {.2645} & {.0436} & {.0099} & {.0083} & {.0201} & {.3413} \\ {.0264} & {1506} & {.3557} & {.0139} & {.0142} & {.0070} & {.0236} \\ {.3299} & {.0565} & {.0495} & {.3636} & {.0204} & {.0483} & {.0649} \\ {.0089} & {.0081} & {.0333} & {.0295} & {.3412} & {.04837} & {.0020} \\ {.1190} & {.0901} & {.0396} & {.1260} & {.1722} & {.2368} & {.3369} \\ {.0063} & {.0126} & {.0196} & {.0098} & {.0064} & {.0132} & {.0012}\end{array}\right]
$$
$$
\mathbf{d}=\left[\begin{array}{c}{74,000} \\ {56,000} \\ {10,500} \\ {25,000} \\ {17,500} \\ {196,000} \\ {5,000}\end{array}\right]
$$

We were given lists of the universal addresses were asked to determine if believes Oven ordered rooted tree can have this list of universal address and, if so, to construct such an ordered a rooted tree. So in part, A were given a list. 1.1 point one 1.1 point two 1.2 2.11 point 12.1 point two 2.1 point 32.2 3.1 point one 3.1 point 2.1 3.1 point 2.2 and 3.2 So, in fact, it's true that believes of, in order to Tree can have this list of universal addresses. So of course, we left off zero here. But if we ignore that fact, see that we're going to include Vertex zero and will include Children of zero, which include one to and three. Next we see that one is going to have Children 1.1 1.2 So two Children at least we see that 1.1 going to have Children 1.1 point one 1.1 point two So two Children we see that 1.2 is going to have Children. Well, actually won't have any Children. Now, looking at two two is going to have Children. 2.1 2.2 So two Children And here things are going a little bit crowded, so going to draw some curved lines here and use a different color. 2.1 and 2.2. We see that 2.1 is going to have Children two points, 1.1 and 2.12 and 2.13 So three Children again, I'll use different colors. So you have 2.1 point one, 2.1 point two and 2.1 point three. We see that 2.1 point one has Children 2.111 That's the only child. Next we see this 2.2, this is going to have no Children. Now three is going to have Children 3.1 and 3.2. So two Children 3.1 is going to have Children. 3.1 point one, 3.1 point two So two Children 3.12 is going to have Children. 3121 and 3.1 to 2, so two Children and 3.2 is not going to have any Children. So this concludes the construction of such an ordered rooted tree. Now in part B were given the list of Universal addresses 1.11 point 2.11 point 2.21 point 2.32 point 12.2 point 12.3 point 12.3 point to 2.4 point 2.12 point 4.2 point 23.13 point 2.1 and 3.2 point two We see that in fact, a tree with such addresses is impossible for consider the address 2.4 point one. We have that there are addresses for 2.4 points, 2.1, and so it follows that 2.4 point two must be a leaf, but there are no addresses for 2.4 0.1 or any of its Children, and therefore it follows that such a tree cannot exist in Part C were given the list of Universal addresses 1.11 point 2.11 point 2.21 point 2.2 point one 1.31 point 4 to 3.13 point 24.1 point 1.1 We see that this also is not a valid list relieves of in order rooted tree See why notice that 1.2 point two and 1.2 point 2.1 or both leaves. However, this is impossible because a leaf has no Children, so follows that 1.2 point 211.2 point 2.1 cannot both believes otherwise. This means that 1.2 point two has no Children, which is impossible.

Question study Crack Scylla, who lived in ancient Greece, derives utility from reading poems and from eating cucumbers broke silicates. That's the units of marginal utility from a first poem, 27 units of marginally incidents here from a second poem, 24 units of Marginally Excellency from a tab poem and so on, with marginally insanity declining by three units for each additional poem park silicates six units off marginal utility for her first D, cucumbers consumed five units of marginal utility. For each of our next year, cucumbers consumed four units of marginal utility for each of the following three cucumbers consumed and so on with a marginally it's really see declining Buy one for every three cucumbers consumed a poem class three bronze coins, but a cucumber cost only one bronze coin. Saxena has 18 bronc bronze coins, skates Paxil. Its budget sits between poems and cucumbers, placing poems under vertical access and cucumbers on the horizontal axis. So here they just basically asked us to draw the budget constraints King, and these are my points. As you can see, these are more points 0 18 1 15 to 12 and remember that my poem costs three coins. But my cucumbers cost one queen, and since my income is 18 all these choices need to cuss. That's also love. 18 bronze claims, so zero multiplied by three coins, zero plus 18 multiplied by one, um, coin is equal to 18 queens, the same full line and 15 to solve. It's a way for multiply one wants applied by three, which is the Cust off homes. That's three coins plus 15 miles supplied by one one, which is the cost of cucumbers that won't give you on three multiplied by one plus 50 multiplied by one that gives you a toll to love 18. The same applies for too tall, two multiplied by three. You go to six, so multiply by. One is a good 12 plus six is equal to eating. So basically, what I'm trying to say is that all my choices add up to 18 and all these choices are represented on my budget constraints. These choices over here they exhaust my income might also income or prick sinners. Total income off 18 Bronze Queen hold. That was clear, and they continue by saying, um starts off with the choice of zero poems and 18 cucumbers and calculate the changes in marginal utility of moving along the budget line to the next choice of one poem and 15 cucumbers. He was in this step by step process based on marginal utility created table and identify perks enters affinity, maximizing choice compared the marginal utility off the two kids in the relative prices at the optimal choice, see if the expensive relationship holds. All right. So they basically telling us to analyze the marginal game and the marginal last from adding one poem. So they just asking us to analyze are marginal utilities from moving along our budget constraints. So we're gonna move from yeah, and up the graph, and we're gonna analyze the marginal utilities whilst you in there key. And here I just basically have my marginally utilities, which they dictated to us at the start of the question. Okay, so these are my choices. Zero poems and 18 cucumbers. If I add one poem after four feet 32 Cumbers because remember, my budget is is limited at 18. Quaint. So if I want to add poem after sacrifice, three cucumbers and a poem costs three coins and the cucumber only a cost one queen. All right, so if I add one poem, I will gain 30 marginally eternities. And here I have. Ready. And in 543232 numbers. How many marginal utilities will like give up? I'll give up. So I'm here and are forfeiting three cucumbers. How many marginal utilities would like? Give up? Three. Okay, So have three. And I'm gaining 30 but I'm losing threes on my final is 27 all rights. And you thought Add another poll. So I'm moving from 1 to 2. Our game 27 marginal utilities. And after only seven here. Okay, um, sorry, guys. Um, so if I move from 2 to 3, So I'm mourning from 2 to 3. Marginal again. Will be 24 and I have 24 over here, and I'm forfeiting freak. Your compass. So moving from 12,009 moving from 12 to 9 on DDE, I will forfeit nine marginal utilities. Okay. And, Catherine, take six. Let me just check. It's the light. Didn't because I think I explained that. Okay. All right. Let me just explain the six. So you're far more from 15. It's well, cucumbers. So 15 it's Well, I'm basically forfeiting six. Marginal utility. Okay, just what I have six year. So any seven minus six is 21 and now I'm adding my mother. I'm Paul. So from 2 to 3 from 2 to 3 24 Well, I have 24 here, moving from 12 to 9. Tarleton nine on 4 15 9 Margin. The utilities, right. Sorry about that guy. That just got a bit lost. Um, and I'm here by three. So if I had another poem, So from 3 to 4, I'm basically gaining 21 margin. Utilities just lie to anyone, okay? And I'm moving from nine cucumbers. Numbers 96 forfeiting 12 marginally utilities. Which is why I have to anyone minus 12 which gives us nine. And then I move from four poems to five poems. So 4 to 5. And I'm basically gaining 18. Okay? And I'm moving from 6 to 3. 6 to 3. Execute losing Dean. Marginal Utilities key, which is? I have 50 years of 18 and the but look at what happens at this point, moving from 5 to 6 and from 3 to 0. So Brenner, at another poem, 5 to 6. I will gain 15 marshaling utilities. However, if I choose to, because if I start consuming six poems, I'm already exhausting my my income. Okay, as you can see here on my grab, this is the point I'm talking about. And you have poems and I don't have any cucumbers key. Basically, I'm only consumer poems, and I'm not consuming in a cucumber. Okay, so I'm basically forfeiting 18. Marginal utility, as you can see here, 15 gained 18 last. I basically have a net last of minus three. Now that right there tells us that the previous point has to be our optimal point or our utility maximizing point basically held the highest utility out off all the above point. Because remember, from the above points, I was basically adding marginal utilities all the way until a guy here to five on three. But we're not added, um, poems and sacrifice cucumbers. Mine. A cane in utilities decreased by managed three, which proves that this point over here, this points the five poems and D cucumbers is are up two more points and the final question year basically said compared the marshaling its signature of the two kids and the relative prices off the optimal choice, See if they expecting relationship holds. All right, So they basically asking us about this equation over here? Marginal utility off poems divided by the price of poems is equal to the marginal utility of cucumbers divided by the price of cucumbers. So I have to do with that at my optimal point. So I've been glad to prove that this creation over year holds with this poem. Because if this racial off poems is a coincidence, racial off cucumbers, if this equation holds, then it means that this is really my optimal point. So the marginally insanity off poems? Uh, um, 0.5, my 0.5. So he has put five war yet is 18. Hey. Which is why I have 18 year. And I really know that the price of poems is three coins and the marginal utility off cucumbers at 0.3 is is, um shake something. Three marginal utility at the Just look at my paper. So remember that I summarized this table over here was basically, like, wine three, six. And this was my cucumber. This was cool. Camba. Hey, I'm I'm Marshall Utilities. Uh, like this. Thanks. Six all right. Uh, all right. And what I used to keep it is that took these 666 And I added them here by you two. Just summarize my table because it was gonna be too long. Remember that cucumbers had, like, um 18 18. You come largest choses. Summarize it. And then I did receive, for this part's over here. Five. Teen. I mean, I just put the 15 year six. So I'm just doing that to show you that marginal utility at 0.3. It's six and not 18 because I just added all the margin utilities over here to summarize. So that's why I hear from marginal utility off cucumber have six. And the pass of cucumbers is one 18 divided by three of 66 divided by one. Because you got to six. And this over here can cruise at this five and three. Choice is really my optimal choice because this equation holds. Thank you guys. Bye.


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