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Solved: In Exercises 15 to 20, find the extrema of the

Vector Calculus | 6th Edition | ISBN: 9781429215084 | Authors: Jerrold E. Marsden; Anthony Tromba ISBN: 9781429215084 87

Solution for problem 20 Chapter 3

Vector Calculus | 6th Edition

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Vector Calculus | 6th Edition | ISBN: 9781429215084 | Authors: Jerrold E. Marsden; Anthony Tromba

Vector Calculus | 6th Edition

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Problem 20

In Exercises 15 to 20, find the extrema of the given functions subject to the given constraints. z = cos x + cos y, subject to the condition x + y = /4

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Chapter 7: Consumers, Producers, and Efficiency of Markets Welfare estimates: • Allocation of Resources • Affects economic wellbeing ➔ Consumers ➔ Producers ➔ Government Consumer Surplus Willingness to pay: maximum amount buyers will pay for a good. Illustration 1: Consumer Surplus The illustration above displays that the maximum willingness to pay for a good is at 50 dollars. The amount that a consumer pays for that good is at 25 dollars. Thus we can say that the consumer surplus of this item is 25 dollars (Max. willingness to pay – actual amount paid). To find the total area of consumer surplus is ½ times base time height. In our illustration it is 45. The consumer surplus is also below the demand curve. The illustration above displays the Market Demand of multiple people, for instance we have two people that are willing to pay 70 dollars, one person willing to pay 60 dollars, and 4 more people willing to pay 50 dollars for an action figure. The lines that mark quantity and price for each person(s) is considered a step down market demand curve. Producer Surplus Cost is the value of everything a seller (firm) gives up to produce a good. The producer surplus is the difference of price received and the minimum willings to pay (cost). Illustration 3: Producer Surplus The producer surplus is found the same way as consumer surplus and the area is above the supply curve. Total Surplus (Consumer and Producer Surplus Combined) The total surplus is consumer surplus and producer surplus combined. The illustration below displays both the demand curve and supply curve which allows for the consumer surplus and producer surplus to be found. The equilibrium point divides the total surplus into the two. Consumer and Producer Surplus with Government As we recall, the government can implement price ceilings and price floors on a market. The following illustration displays what happens when the government places a price floor on a market. Illustration 5: Consumer and Producer Surplus with Price Floor At equilibrium, consumer surplus wasA+ B +C and producer surplus was E + D.After the price floor, the consumer surplus changes toAand the producer surplus changes to B + E. C and D become dead weight loss and is not included in the total surplus.As we can see that this not efficient for the market due to the dead weight loss that does not benefit society as a whole. Market Efficiency Consumer Surplus is the value to buyers minus amount paid by buyers. Producer Surplus is the amount received by buyers minus cost to suppliers. Efficiency: property of resource allocation of maximizing total surplus received by all members of society. The illustration below shows the market efficiency when producing a points that are not at equilibrium. Illustration 6: Market Efficiency The value to buyer (Vb) at Q1 is larger that the cost that the seller pays to create the product (Cs).At Q2 the cost of producing the good is greater than the value to buyers. For both the Supplier and the Consumer to have an equal cost to benefit the firm has to produce at equilibrium. Chapter 8: Application: Cost of Taxation Effects of Tax: When a tax is in effect that supply of a product decreases. Thus the price paid by the buyer increases and the price the supplier receives decreases. The government gains a tax revenue as well. The following illustration displays the effects of taxation. Illustration 1: Effects of Tax The length between Pe and Pb is the tax that the consumer pays and the length between Pe and Ps is the tax that the producer pays. Welfare with and without Tax The following illustration displays how the total surplus is divided up with tax and without tax. Illustration 2: Total Market Surplus The total surplus of a market is divided up into sections –A, B, C, D, E, and F – the table below displays where these sections are allocated before and after tax. If you are wanting to know why there is a Q2, Q3 and additional lines on the graph, it is to prepare you for a graph that has these extra things because the area that you want to focus on is between Pb and Ps. Without Tax With Tax Change Consumer Surplus A+ B + C A - B – C Producer Surplus D + E + F F - D – E Tax Revenue 0 B + D + B + D Total Surplus A+ B + C+ D + E + F A+ F - B – C – D – E Dead Weight 0 C + E + C + E At this point we can safely say that consumers are worse off because the price of that good goes up and the consumer surplus decreases. Producers are worse off also because the price revived for a good decreases and the the producer surplus decreases also. The Government is better off because it now gains a tax revenue. However, because of the dead weight loss the whole economy is worse off. Determinants of Dead weight Loss Elastic Responsiveness of quantity demanded with change in price, keeping everything else constant.A demand curve that more steep is inelastic, where the demand curve that is close to being flat is elastic. The following illustration displays the two on a graph. Illustration 3: Elasticity of Demand As you can see D1 is more elastic because it is more responsive to change, where D2 is inelastic because is less responsive to change. The same concept can also be applied to the Supply curve. The following figures illustrate how the dead weight loss can be larger when the supply or demand curves are elastic. Figure 2: Elastic Supply Figure 1: Inelastic Supply The dead weight loss is less when the supply is inelastic then it is when it is elastic as shown in Figure 1 and Figure 2. Figure 3: Inelastic Demand The dead weight loss is less when the demand curve is inelastic when compared to Figure 2. Dead weight Loss and Tax Revenue as Tax Varies The following illustrates show how the dead weight loss and tax revenue change when the tax is small, medium, and large. Illustration 4: Small Tax Illustration 5: Medium Tax Illustration 6: Large Tax In Illustration 4, the tax revenue is larger than the dead weight loss. If the government increases the tax to where it is medium (See Illustration 5) the dead weight loss increases as well as the tax revenue. However, if the tax is large the tax revenue is small and the dead weight loss is larger. Thus governments have to look at the dead weight loss and the tax revenue when deciding how large of a tax they want to place on a market. The next two illustrates show that the dead weight loss increases at an increasing rate as the tax increases and that the tax revenue will increase at a increasing rate to a specific point before it starts to decrease. Illustration 7: Tax and Dead weight Loss As you can see that the dead weight loss increase as the tax size gets bigger. However, it plateaus at a specific point because the tax cannot get any bigger. Illustration 8: Tax and Tax Revenue As you can see with the increase in the tax size, the tax revenue increase until it reaches a point that it cannot no longer increase but decrease.

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Chapter 3, Problem 20 is Solved
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Textbook: Vector Calculus
Edition: 6
Author: Jerrold E. Marsden; Anthony Tromba
ISBN: 9781429215084

The full step-by-step solution to problem: 20 from chapter: 3 was answered by , our top Calculus solution expert on 09/09/17, 04:03AM. The answer to “In Exercises 15 to 20, find the extrema of the given functions subject to the given constraints. z = cos x + cos y, subject to the condition x + y = /4” is broken down into a number of easy to follow steps, and 33 words. Vector Calculus was written by and is associated to the ISBN: 9781429215084. Since the solution to 20 from 3 chapter was answered, more than 245 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Vector Calculus, edition: 6. This full solution covers the following key subjects: . This expansive textbook survival guide covers 8 chapters, and 305 solutions.

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