What is Arrow’s Impossibility Theorem?

Arrow’s impossibility theorem is a social-choice paradox illustrating the flaws of ranked voting systems. It states that a clear order of preferences cannot be determined while adhering to mandatory principles of fair voting procedures. Arrow’s impossibility theorem, named after economist Kenneth J. Arrow, is also known as the general impossibility theorem.

Understanding Arrow’s Impossibility Theorem

Democracy depends on people’s voices being heard. For example, when it is time for a new government to be formed, an election is called, and people head to the polls to vote. Millions of voting slips are then counted to determine who is the most popular candidate and the next elected official.

According to Arrow’s impossibility theorem, in all cases where preferences are ranked, it is impossible to formulate a social ordering without violating one of the following conditions:

• Nondictatorship: The wishes of multiple voters should be taken into consideration.
• Pareto Efficiency: Unanimous individual preferences must be respected: If every voter prefers candidate A over candidate B, candidate A should win.
• Independence of Irrelevant Alternatives: If a choice is removed, then the others’ order should not change: If candidate A ranks ahead of candidate B, candidate A should still be ahead of candidate B, even if a third candidate, candidate C, is removed from participation. 
• Unrestricted Domain: Voting must account for all individual preferences.
• Social Ordering: Each individual should be able to order the choices in any way and indicate ties.

Arrow’s impossibility theorem, part of social choice theory, an economic theory that considers whether a society can be ordered in a way that reflects individual preferences, was lauded as a major breakthrough. It went on to be widely used for analyzing problems in welfare economics. 

Example of Arrow’s Impossibility Theorem

Let’s look at an example illustrating the type of problems highlighted by Arrow’s impossibility theorem. Consider the following example, where voters are asked to rank their preference of three projects that the country’s annual tax dollars could be used for: A; B; and C. This country has 99 voters who are each asked to rank the order, from best to worst, for which of the three projects should receive the annual funding.

• 33 votes A > B > C (1/3 prefer A over B and prefer B over C)
• 33 votes B > C > A (1/3 prefer B over C and prefer C over A)
• 33 votes C > A > B (1/3 prefer C over A and prefer A over B)

Therefore,

• 66 voters prefer A over B
• 66 voters prefer B over C
• 66 voters prefer C over A

So a two-thirds majority of voters prefer A over B and B over C and C over A—a paradoxical result based on the requirement to rank order the preferences of the  three alternatives.

Arrow’s theorem indicates that if the conditions cited above in this article i.e. Non-dictatorship, Pareto efficiency, independence of irrelevant alternatives, unrestricted domain, and social ordering are to be part of the decision making criteria then it is impossible to formulate a social ordering on a problem such as indicated above without violating one of the following conditions.

History of Arrow’s Impossibility Theorem

The theorem is named after economist Kenneth J. Arrow. Arrow, who had a long teaching career at Harvard University and Stanford University, introduced the theorem in his doctoral thesis and later popularized it in his 1951 book Social Choice and Individual Values. The original paper, titled A Difficulty in the Concept of Social Welfare, earned him the Nobel Memorial Prize in Economic Sciences in 1972.

Arrow’s research has also explored the social choice theory, endogenous growth theory, collective decision making, the economics of information, and the economics of racial discrimination, among other topics.

What Is the Average Age of Inventory?

The average age of inventory is the average number of days it takes for a firm to sell off inventory. It is a metric that analysts use to determine the efficiency of sales. The average age of inventory is also referred to as days’ sales in inventory (DSI).

Formula and Calculation of Average Age of Inventory

The formula to calculate the average age of inventory is:


$\begin{aligned} &\text{Average Age of Inventory}= \frac{ C }{ G } \times 365 \\ &\textbf{where:} \\ &C = \text{The average cost of inventory at its present level} \\ &G = \text{The cost of goods sold (COGS)} \\ \end{aligned}$​Average Age of Inventory=GC​×365where:C=The average cost of inventory at its present levelG=The cost of goods sold (COGS)​

What the Average Age of Inventory Can Tell You

The average age of inventory tells the analyst how fast inventory is turning over at one company compared to another. The faster a company can sell inventory for a profit, the more profitable it is. However, a company could employ a strategy of maintaining higher levels of inventory for discounts or long-term planning efforts. While the metric can be used as a measure of efficiency, it should be confirmed with other measures of efficiency, such as gross profit margin, before making any conclusions.

The average age of inventory is a critical figure in industries with rapid sales and product cycles, such as the technology industry. A high average age of inventory can indicate that a firm is not properly managing its inventory or that it has an inventory that is difficult to sell.

The average age of inventory helps purchasing agents make buying decisions and managers make pricing decisions, such as discounting existing inventory to move products and increase cash flow. As a firm’s average age of inventory increases, its exposure to obsolescence risk also grows. Obsolescence risk is the risk that the value of inventory loses its value over time or in a soft market. If a firm is unable to move inventory, it can take an inventory write-off for some amount less than the stated value on a firm’s balance sheet.

Example of How to Use the Average Age of Inventory

An investor decides to compare two retail companies. Company A owns inventory valued at $100,000 and the COGS is $600,000. The average age of Company A’s inventory is calculated by dividing the average cost of inventory by the COGS and then multiplying the product by 365 days. The calculation is $100,000 divided by $600,000, multiplied by 365 days. The average age of inventory for Company A is 60.8 days. That means it takes the firm approximately two months to sell its inventory.

Conversely, Company B also owns inventory valued at $100,000, but the cost of inventory sold is $1 million, which reduces the average age of inventory to 36.5 days. On the surface, Company B is more efficient than Company A.

What Is Audit Risk?

Audit risk is the risk that financial statements are materially incorrect, even though the audit opinion states that the financial reports are free of any material misstatements.

Understanding Audit Risk

The purpose of an audit is to reduce the audit risk to an appropriately low level through adequate testing and sufficient evidence. Because creditors, investors, and other stakeholders rely on the financial statements, audit risk may carry legal liability for a certified public accountancy (CPA) firm performing audit work.

Over the course of an audit, an auditor makes inquiries and performs tests on the general ledger and supporting documentation. If any errors are caught during the testing, the auditor requests that management propose correcting journal entries.

At the conclusion of an audit, after any corrections are posted, an auditor provides a written opinion as to whether the financial statements are free of material misstatement. Auditing firms carry malpractice insurance to manage audit risk and the potential legal liability.

Types of Audit Risk

The two components of audit risk are the risk of material misstatement and detection risk. Assume, for example, that a large sporting goods store needs an audit performed, and that a CPA firm is assessing the risk of auditing the store’s inventory.

Risk of Material Misstatement

Material misstatement risk is the risk that the financial reports are materially incorrect before the audit is performed. In this case, the word “material” refers to a dollar amount that is large enough to change the opinion of a financial statement reader, and the percentage or dollar amount is subjective. If the sporting goods store’s inventory balance of $1 million is incorrect by $100,000, a stakeholder reading the financial statements may consider that a material amount. The risk of material misstatement is even higher if there is believed to be insufficient internal controls, which is also a fraud risk.

Detection Risk

Detection risk is the risk that the auditor’s procedures do not detect a material misstatement. For example, an auditor needs to perform a physical count of inventory and compare the results to the accounting records. This work is performed to prove the existence of inventory. If the auditor’s test sample for the inventory count is insufficient to extrapolate out to the entire inventory, the detection risk is higher.

What Is AARP?

The American Association of Retired Persons, commonly known by its acronym AARP, is America’s leading organization for people aged fifty and older, providing member benefits, marketing services, and lobbying on their behalf.

Founded in 1958 by retired educator Dr. Ethel Percy Andrus as the American Association of Retired Persons, AARP is a nonprofit, nonpartisan association with a membership of more than 38 million.

How AARP Works

AARP provides information, education, research, advocacy, and community services through a nationwide network of local chapters and experienced volunteers. It focuses its work on consumer issues, economic security, work, health, and independent living issues, and engages in legislative, judicial, and consumer advocacy in these areas.

AARP is considered a powerful lobbying group as well as a successful business, selling life and health insurance, investment products, and other financial and non-financial services. It is also an independent publisher, offering Modern Maturity magazine and the monthly AARP Bulletin. AARP produced $1.70 billion in revenue in 2019, which came from a variety of endeavors, including advertising revenue from its publications, and from royalties for licensing its name and logo.

However, membership fees represent the most significant source of revenue. It is registered as a 501(c)(4) non-profit by the Internal Revenue Service (IRS), which means it is permitted to engage in lobbying. It also administers some 501(c)(3) public charity operations while some of its other operations are for-profit.

AARP Affiliates

There are several AARP-affiliated organizations, and they include the following:

• The AARP Foundation is a non-profit charity that assists people over age 50 who may be at economic and social risk. Within the foundation operates AARP Experience Corps., which encourages tutoring and mentoring of children, and AARP Institute, which holds its gift annuity funds.
• AARP Services develop and manage new products and services and are for-profit.
• Legal Counsel for the Elderly is a non-profit that provides legal services for seniors in Washington, D.C.
• AARP Financial Services holds AARP real estate and is for-profit.
• The AARP Insurance Plan administers some AARP group insurance plans.

AARP also has many other initiatives, including promoting driver safety (AARP Driver Safety), producing television programming that targets seniors, and engaging in sponsorships that support social causes, such as raising awareness of and fighting hunger in America.

AARP manages outreach programs that address housing issues and social isolation among seniors. AARP has also initiated and managed programs that advocate for the strengthening of Social Security and Medicare.

Criticism of the AARP

AARP is one of the strongest lobbying groups in America, and because of its efforts, it often receives attention for exerting its influence in Washington, D.C., and in state capitals. Its non-profit operations also receive millions of dollars per year in the form of federal grants. Some argue that its positions fall into the more liberal part of the political spectrum.