Why It’s Usually Suboptimal for Businesses to Set the Cost Equal to a Fixed Fraction of Revenue

A firm is considering how much to advertise its products to maximize profit, which is the difference between revenue and cost. In this article, we will investigate the profit maximizing conditions and compare it to the convenient method where the advertisement cost is set as a fixed fraction of revenue (fixed ratio method from now on). We can model the revenue (R) as a function of cost (C):

R(C) = aC^{p}

Where a is the scale parameter and p is the shape parameter. When 0 < p < 1, the function is concave and diminishing marginal returns is observed. (Other values of p are uninteresting or uncommon: when p = 0, revenue is independent of cost; when p = 1, cost is directly proportional to revenue; when p > 1, marginal returns increase with cost and expansion is surely needed.)

Analytic Solution

The firm makes positive profit when the function lands above the line R(C) = C. Maximizing profit is equivalent to finding a C* such that the vertical distance between the two curves is greatest. This happens when the derivative of R(C) is equal to that of the R = C line, which is 1:

\frac{\text{d}R}{\text{d}C }=1 when C=C^{*}

Solve for C* and get C^{*}=(ap)^{\frac{1}{1-p}}
Graphically this means finding a 45 degree tangent line on the R(C) curve:

Continue reading “Why It’s Usually Suboptimal for Businesses to Set the Cost Equal to a Fixed Fraction of Revenue”

How AKB48 Can Revive by Balancing Exploitation and Exploration

AKB48 Group is the world’s largest girls idol group based in Tokyo, Japan. It contains six smaller constituent groups called sister groups with over 300 members in total. Naturally the popularity of each member is different which should follow a relatively symmetric unimodal distribution. Since there is no way every member can get equal and adequate opportunities to perform, a dozen or so prominent members are selected to receive half of the attention.  As the histogram below shows, however, the popularity distribution (measured by number of followers on Showroom) is very right skewed due to at least two effects: (1) the distribution of media exposure is extremely right skewed and (2) the number of years since joining the group is right skewed. We know that media exposure can shape member popularity by contrasting the popularity distribution to that of a more egalitarian group named Keyakizaka46, which has very symmetric and concentrated popularity distribution and media exposure distribution. Keyakizaka46 is able to achieve relative equality of media exposure due to its small size of 21 members (not counting under group members).

The distribution of the number of followers on Showroom is very right skewed. Source

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When Should We Replace $5 Bills With $3 Bills?

Almost all currencies in the world have denominations that start with 1, 5, or 0 and the main difference among them is whether $2, $20, $200, etc. bills are used. But prevalence does not prove efficiency. In this post, I will compare the efficiency of different bill denomination systems.

An efficient currency minimizes the cost of transaction. To simplify, we assume that the cost of transaction is proportional to the time spent on it. The cost of transaction can be dichotomized into time spent by the customer (while the cashier waits) and time spent by the cashier (while the customer waits). Each component further divides into the following categories denoted by a, b, and c:

  • Fixed transaction cost (a): time spent on taking out the wallet, opening the cash register, thinking about which bills to use (while doing nothing else), and handing and receiving the money. Avoid double counting if these jobs overlap in time or if the other party is not waiting but doing some other necessary work, such as printing the receipt.
  • Fixed cost for using bills in each denomination (b): time spent on moving the hand to reach a specific slot in the cash register; time spent on finding the place for a specific denomination in the wallet and putting all bills of this denomination on the other hand or the counter before moving on to work on the next denomination.
  • Cost for counting a bill (c).

We acknowledge that cost a can be substantially lower if no change is required from the cashier, which saves the time of passing the changes back to the customer. However, most transactions involve taxes and multiple items that lead to the usage of coins (in the US). Most of the time the customer does not have the exact amount of coins or does not want to pay any coins. Secondly, for the same reason, charges are usually not psychologically convenient numbers (e.g. $5, $20). Nice numbers can make buying decisions easier but they don’t show up in the payment. Therefore we can assume that there are always changes involved and a is constant and can be entirely dropped from this analysis.

Continue reading “When Should We Replace $5 Bills With $3 Bills?”

The Five Goals of Lawmaking and Three Approaches of Combining Them

Making and executing ideal laws with the right penalty is an optimization process with debatable objectives. But at least we can list the five components of the objectives and three components of the costs in an attempt to model lawmaking. Later we can look for the prevailing approaches to combine these goals in ideal lawmaking and determine the origins of these approaches.

Goals of lawmaking

1. Change future behavior of criminals

This is often thought of as the goal of imprisonment. We would like criminals to have limited opportunities to commit another crime while in prison and less likely to commit crimes after they are released, which we attempt to achieve through enrichment programs in prison.

2. Compensate the victims

Victims can be financially compensated through fine or psychologically compensated through apology or by knowing that the criminals will be punished.

3. Deter people from committing crimes

When the punishment is severe enough, it is no longer worth it to commit a crime even when the probability of getting caught is small.

4. Improve the perceived fairness of society

Most people would like to live in a just society where people who harm others without permission are punished.

5. Make the criminals better people

Criminals are people too and many believe that the society should be responsible to help them by pulling them out of the wrong path and teaching them what’s right.

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The Paradoxical Optimal Voting Strategy for the Presidential Election of 2016

Most people agree that when you vote in an election, to maximize your utility, the best voting strategy is to vote for your better preferred candidate of the two most popular candidates assuming  simple plurality is used. This was not the case in 2016’s US presidential election. Both major candidates were deemed so incapable that they could be defeated by any typical candidate in the opposing party. As a result, expecting that the winner will run for reelection and lose in 2020, the best strategy becomes voting for the party you like the least, as the current short loss would trade for a longer victory in the future. This assumes that Trump and Hillary each represent their party well and attract voters who usually vote for their party. We also assume a zero discount rate: a miserable 4 years that starts in 4 years weighs equally as a miserable 4 years that starts immediately.

Created by Christopher Wang. First published on November 8, 2016 on 9gag.com.

Welfare Economics of Labor Migration


Should the United States allow immigrants and to what extent? This is a topic of hot debate because both the economics behind it is unintuitive and everybody’s goal or preference is different.

Are low skill immigrants taking our jobs away?

This is only true for a small portion of the population, the unskilled citizens, or citizens worse skilled than the average low skill immigrants. When low skill immigrants come, the wage in low skill occupations decreases as the labor supply increases (proportionally more than the increase in the size of the economy). Low skill citizens suffer and employers (the producer) benefit from a lower cost of labor. From standard welfare analysis, we know the gain for the producers is definitely greater than the loss of the low skill citizens, resulting in a higher social welfare.

Assume that our country has an excess of higher skill labors, meaning some skilled workers accepted a lower skill job because the high skill job market is saturated or because the low skill job’s salary is attractive. But wait, could a free market be saturated? Because for each high skill job, there has to exist some low skill job to complement it. For example, each company needs customer service (a low skill occupation). If few low skill workers are available, wage will increase, attracting some high skill workers. The customer service will still exist but rather small. This assumption of excess high skill labor is reflected in the high wage of low skill jobs that attracts immigrants.

After the influx of immigrants, many high skilled citizens in low skill occupations will switch to high skill jobs as the wage in low skill jobs decreases (relative to high skill jobs). High skill jobs of course mean higher productivity. When the citizens shift to a higher skilled worker makeup, the country’s GNP (only counting citizens) is bound to increase since the total productivity increases. This resonates with the increasing social surplus I mentioned earlier. A higher productivity benefits both producer and consumer, including low skilled citizens who are worker under a lower wage.

Conclusions: if we allow low skill immigrants

  1. Low skill citizens can be better off or worse off depending on the situation.
  2. High skill citizens are better off
  3. Citizens become richer, on average.

Continue reading “Welfare Economics of Labor Migration”

What does It Really Cost to Buy Any Tangible Goods?

What does it cost when you buy a 300-page paperback from Amazon? The price of the book is only the initial cost. When you store it on your bookshelf, it takes space and renting a space costs money. The price of the book might be only $10; the storage cost for 15 years is probably higher than $5.5.

A 1000 square feet apartment costs around $2100/month in the US. Roughly $300 of the cost is fixed cost independent of the size of usable area, including the cost of transaction, management cost, stairs, mechanical rooms, and partly the costs of outdoor areas, utility bills, and property tax. The rest of the rent should grow linearly with area. This leaves $1800/month for 1000 sq ft, or $1.8/month/sq ft, or $0.0125/month/sq inches.

In the short run, buying a book will not increase your rent, but in the long run you are mobile and you will pick a house that’s about the right size by equating your marginal benefit per sq ft per month to around $1.8.

Assume that a book is 8’’ × 5.5’’ × 1’’, and your bookshelf is 29’’ × 12’’ × 72’’ with 6 shelves plus the top. To calculate the minimal needed space, we squeeze as many bookshelves as possible in a library pattern, with aisles that are only 24’’ wide. Half of the area will be bookshelves and half will be aisles since 2 × bookshelf depth = aisle width. A book would cost 1’’ (width) × 12’’ (depth) of space on one of seven shelves (including the top) and an equal share of aisle space. Therefore, one book occupies 1” × 12” ÷ 7 shelves × 2 = 3.42 square inches of space.

Now the cost of storing one book a month is 0.0125 × 3.42 = $0.04286; or $0.5143 a year. Fifteen years? Let’s use a realistic annual discount rate of 4.5% to calculate the present value: $0.5143 / 0.045 × (1 – 1 / 0.045^15) = $ 5.5232. To store a book forever, the cost is: $0.5143 / 0.045 = $11.4286. Next time you bring home a free book from library, beware that you are signing a contract to pay 51 cents each year (adjust for inflation) until you get rid of the book.

Continue reading “What does It Really Cost to Buy Any Tangible Goods?”

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