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Find f (2), f (3), f (4), and f (5) if f is defined

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 3E Chapter 5.3

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 3E

Find f (2), f (3), f (4), and f (5) if f is defined recursively by f (0) = ?1, f (l) = 2, and for n = 1,2,…a) f (n+1) = f (n) + 3 f (n?1).b) f (n + 1) = f (n) 2 f (n ?1)c) f (n + 1) = f (n)2 – 4 f (n ?1)2.d) f ( n + 1) = f ( n ?1) f (n).

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Demand and Supply  Demand is the want or desire to possess a good or service with the necessary goods, services, or financial instruments necessary to make a legal transaction for those goods or services.  Supply is the total quantity of a good or service that is available for purchase at a given price, at a given time.  The quantity demanded is the amount of a product people are willing to buy at a certain price.  The quantity supplied refers to the amount of a certain good producers are willing to supply when receiving a certain price.  Equilibrium price is the price at which quantity demand = quantity supply.  Price Elasticity of demand OR supply is the measure of the degree of responsiveness of QD or QS to any change in price of the said commodity. When 01, demand is elastic. Shape of curve is dependent on this.  Goods likely to be more elastic in demand if there are a lot of subsititutes, if the good is not habit­forming/ essential, represents a small proportion of total income e.g. matches.  Goods likely to be elastic in supply if: length of prod. process is short, there are large stocks available etc.  Increase in demand (more demanded at any price) caused by e.g. rise in consumer income, fall in price of substitutes, adverts, fashion etc. Causes shift of demand curve.  Increases in supply caused by e.g. excellent weather conditions (for farm products), new technology, fall in prod. costs etc. Causes shift of supply curve.  Substitutes are goods in competitive demand (e.g. Pepsi and Coke). If the price of one rises, the demand curve of the other shifts to the right.  Complements are goods in joint demand (e.g. camera and film). If the price of one rises, the demand curve of the other shifts to the left.  Some goods are in joint supply (e.g. beef and leather). The increased production of one causes a rise in the supply of another.  Some goods are in competitive supply (e.g. milk and cheese). The increased production of one causes a fall in the supply of another. Market Systems Free market is an economic system that used to ensure the satisfaction of consumers’ demand entirely through the market forces of demand and supply. There is no government intervention in ANY economic feature of the country. The free market has some key features, which are shortly explained below : 1. Private ownership of all FOPs (factors of prod.) – All factors of production are privately owned, controlled and financed. All businesses and economic ventures are undertaken by the general public, termed the entrepreneurs 2. Consumer/Entrepreneur Sovereignty – The freedom of a consumer to demand any good is rightly preserved in the free market, where only the consumer can choose what to buy. Similarly, entrepreneurs are not encumbered by the government in their production lines – they can freely choose what to produce, how to produce and when to produce. 3. Profit motive – Individual sellers are fuelled by profit motive to undertake businesses. 4. Competition – There is competition among sellers to sell their goods to the consumers, which ensures quality and proper pricing of the product. 5. Usage of the price system – The equilibrium price is the de facto means through which demand is met by adequate supply of goods. It serves as an indicator for consumers and suppliers. 6. Economic inequality – The economy is divided by a gap between the rich and the poor, where they are continuously moving apart. On the opposite end of the continuum of market systems, we see the centrally controlled economy, or otherwise known as the command economy. In this system, the government controls all economic processes, and determines both the demand of the consumer and the supply of the producer. Key features are listed below : 1. State ownership of all FOPs ­ All factors of production are owned, controlled and financed by the government. 2. No freedom of choice – It is the government who decides how much of what goods the individual consumer needs, and how much of what goods the producers should produce. 3. Social motive – The government works to ensure that social rights are maintained, and also makes sure that the society is not harmed by any economic activity. 4. Economic Equality – Every single citizen is an equal member of the economy – they consume the same goods of the same quality. 5. Inefficient – As the role of the price system is nullified by the control of the government, the allocation of FOPs and the distribution of goods to meet the consumer demand is very inefficient. Also, bureaucratic procedures and high levels of corruption set back the economic fluency. Market Intervention The government may choose to intervene in the price mechanism largely on the grounds of wanting to change the allocation of resources and achieve what they perceive to be an improvement in economic and social welfare. All governments of every political persuasion intervene in the economy to influence the allocation of scarce resources among competing users. What are the main reasons for government intervention 1. To correct for market failure 2. To achieve a more equitable distribution of income and wealth 3. To improve the performance of the economy Options for government intervention in markets There are many ways in which intervention can take place – some examples are given below. 1. Legislation and regulation – The govt. can pass new laws to alter the economic features, and can regulate the economy on these set grounds, so that the economic functions can be more streamlined, with a more socially minded approach. Examples : employment laws to prohibit trade unions, banning of certain products like narcotics, protection of infant firms from monopoly powers, price control measures. 2. State Provision of Goods and Services – The state provides a portion of the goods and services in an economy with a social motive (e.g. to contain prices, to provide merit and public goods, to ensure supply, to finance huge infrastructure development etc.) 3. Fiscal policy can be used to alter the level of demand for different products and also the pattern of demand within the economy. (a) Indirect taxes can be used to raise the price of de­merit goods and products with negative externalities designed to increase the opportunity cost of consumption and thereby reduce consumer demand towards a socially optimal level (b) Subsidies to consumers will lower the price of merit goods. They are designed to boost consumption and output of products with positive externalities – remember that a subsidy causes an increase in market supply and leads to a lower equilibrium price (c) Tax relief: The government may offer financial assistance such as tax credits for business investment in research and development. Or a reduction in corporation tax (a tax on company profits) designed to promote new capital investment and extra employment (d) Changes to taxation and welfare payments can be used to influence the overall distribution of income and wealth – for example higher direct tax rates on rich households or an increase in the value of welfare benefits for the poor to make the tax and benefit system more progressive. 4. Intervention designed to close the information gap ­ Often market failure results from consumers suffering from a lack of information about the costs and benefits of the products available in the market place. Government action can have a role in improving information to help consumers and producers value the ‘true’ cost and/or benefit of a good or service. Examples might include: (a) Compulsory labeling on cigarette packages with health warnings to reduce smoking. (b) Providing a platform for employers to be able to locate potential employees, who might be facing frictional unemployment. Economic Sectors Primary Sector The primary sector of the economy extracts or harvests products from the earth. The primary sector includes the production of raw material and basic foods. Activities associated with the primary sector include agriculture (both subsistence and commercial), mining, forestry, farming, grazing, hunting and gathering, fishing, and quarrying. The packaging and processing of the raw material associated with this sector is also considered to be part of this sector. In developed and developing countries, a decreasing proportion of workers are involved in the primary sector. About 3% of the U.S. labor force is engaged in primary sector activity today, while more than two­thirds of the labor force were primary sector workers in the mid­nineteenth century. Secondary Sector The secondary sector of the economy manufactures finished goods. All of manufacturing, processing, and construction lies within the secondary sector. Activities associated with the secondary sector include metal working and smelting, automobile production, textile production, chemical and engineering industries, aerospace manufacturing, energy utilities, engineering, breweries and bottlers, construction, and shipbuilding. Tertiary Sector The tertiary sector of the economy is the service industry. This sector provides services to the general population and to businesses. Activities associated with this sector include retail and wholesale sales, transportation and distribution, entertainment (movies, television, radio, music, theater, etc.), restaurants, clerical services, media, tourism, insurance, banking, healthcare, and law. In most developed and developing countries, a growing proportion of workers are devoted to the tertiary sector. In the U.S., more than 80% of the labor force are tertiary workers. In general, as an economy develops, the secondary and tertiary sectors increase in size, and the output from the primary sector declines. However, this does not determine the dominant sector in an economy, as there is only a shift in emphasis of growth in most cases.

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Chapter 5.3, Problem 3E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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Find f (2), f (3), f (4), and f (5) if f is defined