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In dollars per unit, that consumers are willing t0 pay ior units of an item_ and S(x) is the price D(x) is the price that producers are willing to accept for X unit...

Question

In dollars per unit, that consumers are willing t0 pay ior units of an item_ and S(x) is the price D(x) is the price that producers are willing to accept for X units Find (a) the equilibrium point; (b) the consumer dollars per unit; surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point: D(x) = 2000 - 1Ox, S(x) = 1100

in dollars per unit, that consumers are willing t0 pay ior units of an item_ and S(x) is the price D(x) is the price that producers are willing to accept for X units Find (a) the equilibrium point; (b) the consumer dollars per unit; surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point: D(x) = 2000 - 1Ox, S(x) = 1100



Answers

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. $$D(x)=\frac{100}{\sqrt{x}}, \quad S(x)=\sqrt{x}$$

In problem it. We have the demand and supply functions for the same number of in six. If we have here X number of units and we have here why which is in dollars? The end price, that the demand starts at 800 8800 here and goes something like that. And for the supply function it starts at seven thousands and goes up something like that. This is SSA ploy and this is at the moment for body. We want to get the Caribbean Point. This point is the Caribbean point eventually intersect, which has the same X and y values, which means we can equate both functions and solve the equation. To get the ex valley, we have it. 1008 100 minus 30 x equals seven thousands plus 15 x. We can subtract seven Southern from both sides. Then we have 100 and 1800 minus 30 x equals 15 x. We can at 30 x for both sides. Then we have 1000 and 800 equals 45 x. This means we have X equals 40. The Caribbean point is at 40 units and for the unit price we can substitute in the supply or the demand function plus 15. Multiply it by 40. It equals 13 equals 7000 and 600 dollars. Here we have 7000 and 600. This means the Caribbean Point is at 40 number of units and 760 600 7600 dollars for the any price we're about to be. We want to calculate the consumer surplus, which is the area below the demand function. Until the Caribbean boy this area, we can calculate it either by integration or by geometry. It's easier to calculate it by geometry equals. The area of the triangle is half by base, which is 40 the blyde. By the height, which is 808 100 minus 7600 it equals 20 multiplied boy, once other than 200 equals 24 thousands dollars for birth. See, we went, we want to calculate the supply, or the producer the producer surplus, which equals the area above the supply. Until the Caribbean point, this area by geometry equals half by base. By height minus seven, thousands equals 20 multiplied by 600 equals 12 thousands dollars. This is often answer of birth C, an answer of bar to be the final answer off party

In problem six. We have the demon and the supply function for the same number of when it's X and we have here. Why the any price in dollars? If this, for example, is the demand, and this is the supply we have here, the equilibrium point to get the Caribbean point for birdie, we can equip it at the moment and the supply function because they have the same X and Y values we have X minus eight squared equals X squared. Then, by getting the square root off both sides, we have X minus eight equals positive or minus X. We have here to solutions. X minus eight equals X, which is trivial because it gives us minus eight equals zero. She's wrong. The other solution is X minus eight equals minus X, which gives two X equals eight. This means X equals four and this is the right solution. To get the price we substitute in the supply function we get y equals four squared equals 16, which means the equilibrium point and four and 16 dollars four units on. The only price is $16 for birth. We we want to calculate the consumer surplus which is the area above the Caribbean Point under the demand curve can be calculated as consumer surplus equals the integration off. The demand function from zero until the equilibrium point, which has X equals four from 0 to 4, minus the area off this rectangular, which is the number of quantities multiplied by the price of the Caribbean boy equals X minus eight Cube divided by three. We substitute from 0 to 4 minus the quantity multiplied by the only price four multiplied by 16 equals. We suggest you first by X equals four. We have minus for cube minus four cube divided by three minus minus eight. Cube, divided by three minus 64 equals minus four cube plus a cube divided by three minus 64 equals 85.33 dollars. We can calculate the producers surplus by the same way, which is the area off this. The area off this region can be calculated as the area of district angular, which is the quantity want to buy by the price for the Caribbean Point minus the integration off the supply function which he is X squared. The X we integrate from zero to the clear point at X equals four equals 64 minus execute divided by three with substitute from 0 to 4 equal 64 minus. We substitute by X equals four. First we have 64 divided by three minus with super ex equals zero, which gives you it equals 64 multiplied by two divided by three or it equals 42 0.67 dollars, which is a final answer of party is the final answer off part B and finally this final answer off party.

In problem food. We want to get the Caribbean Point for this demand function and this supply function as X number of units. And why is the price the unit price? If we have here the domain, the demand and if we have here, that's a ploy. This is the month, and this is supplied the point at which the Intersect is the equilibrium point. Let's get the craven point by equating the domain function, the demand function and the supply function we have X minus three. All squared equals X squared plus two x last one. We can rewrite the the right hand side as it equals X plus one all the square. And then we get the square root off the both sides. We get X minus three equals posted or minus exit plus one. Then we have two routes. If X minus three equals X plus one, this means we have X equals two and the solution we have X minus three equals minus x. No, this gives us a trivial solution. This needs us to make minus three equals own, which is wrong. And we have here minus X minus one for the other solution. X minus three equals minus X minus one, which gives two X equals two all X equals one, which is the correct solution. This means the Caribbean Point is at X equals one one quantity. And to get the price we substitute back in the demand. In the demand function we have, the white value equals one minus. City square equals four. This means the Cadavra point is at the 0.1 and for dollars for bar TV. We want to calculate the consumer surplus, which is this area. This area can be calculated as the integration for the demand function from zero to the equilibrium point one minus. We suggested we subtract the area of this rectangle which equals to you multiply it by B for the equilibrium boy equals X minus City cube divided by three minus one. But the blow it by four we subsided from 0 to 1 equals we start by the upper limit. We have minus two cube divided by three minus minus three. Cubed, divided by three minus four equals we have minus eight, divided by three plus 27 divided by three minus four equals seven divided by three or it equals $2.33 for the producer surplus. We calculate the area above. It's a supply liquor we can create this area can create. This area can be calculated as the rectangular you multiply by B for the Caribbean Point, minus the integration off the supply function, which is X plus one all the square The X we substitute from zero to the program point at X equals one equals four minus exit plus one cube divided by three. The substitute from 0 to 1 equals four minus. We start by X equals one. We have to cube divided by three plus we service to buy X equals zero we have one thing equals we have a divided by three plus one third equals the 34 minus three is one, and the answer is $1 for part C. And this is the final answer off part B. And this is the final answer off party

In problem 12. We have the demand and supply function for the same number of in its X. We have here the number of minutes and for why is the only price in dollars we have. The demand function starts at 100 1800 and it goes something like that and we have the supply function starts at two and it goes something like that and they intersect at they could've reemploy for party. We want to calculate the Caribbean Point. They have the same X and y values, which means we can equate the two functions and then sold of the equation toe. Get the value of X. We have 101,800 voice but square root of X Plus one, which equals two multiplied by square root of X blossom. By multiplying both sides by square root of X plus one, we get 101,000 and 800 equals to multiply by X plus form by dividing both sides By two, we have 900 equals X plus one, which means that X equals my 100 minus one, which is it 100 on 90 mine. The Taliban point is at 899 units and to get the unit price if you substitute in the domain in the demand with the supply functions to multiply it by a school route 899 plus one equals to multiplied by 30 equals 60 dollars. This means a Caribbean point is ed 899 units and 60 so $60 for both me. We want to calculate the consumer surplus, which is the area under the demand, until the Caribbean point. This area can be calculated as yeah, consumer surplus equals the integration off the demand function. To get the area under the demand from zero to the Caribbean point multiply by the X minus. We subtract the area off this rectangular, the dotted rectangle, which is the quantity multiplied. But the price for the equilibrium boy equals 1800 deployed by the integration from 0 to 899 for exit plus warm toes about off minus off the X minus, the quantity multiplied by the price. This is a country, and this is the price. 899 multiplied by 60 equals 100 1800 multiplied by integration. No, we integrate. The integration of X plus one is X plus. One is about off off. We add one is about and then divide by the new board. The substitute, from 0 to 899 minus 899 matter by by 600 equals 539 thousands and 40 400. It equals one soda on 800 multiplied by we substitute by X equals 899 1st, which is the square root of 900 which is 30 divided by half or multiplied by two. Minus three is still by X equals zero, which gives to minus 539 for on equals 60 minus two multiplied boy 1800 which equals 144 400 we have here and we have here a zero by mistake buying us 539 and foot 53 and 940 equals 50 thousands, 46 $460 for board. See, we want to calculate the producer surplus, which equals this area. She can be calculated as the area of this rectangle minus integration off. That's a bloody function. If you multiply by B for the Caribbean Point points integration from zero to the Caribbean point off the supply function which is to multiply boy square root off X plus one. The X equals 53 thousands 140 minus the integration from zero off. 2, 899. We get two out of the integration for X plus one toe the bar off off the X equals 53 by 14 minus two, multiplied by the integration of experts. Want to the bar off? We add one to the power and then divide by the newborn. We substitute from 0 to 899 equals 53 thousands 140 minus two multiplied boy. The service Uber X equals 899 1st, which gives squared off to cube. But the boy boy two thirds Twitter Tau 930 cube minus two thirds. It's obscured by X equals zero gives two thirds these equals. We have two thirds multiplied by three 30 cube my institutions but the boy by two then we have the producer surplus equals 17,941 $0.33 which is a final answer off Parsi, and this is the final answer off bar to be on. This is a fine answer off party.


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