Tag Archives: formula

## Managerial #Economics ~ Understanding #MRTS the Fun Way!

12 Jun

As always, this lesson is not intended to be professional advice. This is simply lesson material for ESL students in a Managerial Economics and Finance class. Posted here for their use or for helping other students.

## PART 1 – Key Words

1. Quantity (量) ~ How many products = Q (# of  🚗)
2. Labor (员工) ~ The Number of Workers = L (👱)
3. Capital (资金) ~ The Money () we need = K (💲)
4. Change (变化) ~ How much did the # change? = Δ(🔺)
5. Marginal (边际成本) ~ Result if you add ONE MORE (+1) Q
6. Rate (比率) ~ Ratio
7. Substitution (取代) ~ XK = 1L (Substitution asks “what is X?”)
8. Input (输入) ~ All the resources you put into a product.
1. Ice Cream 🍦has many inputs:
1. Milk🥛
2. Eggs🍳
3. Sugar
4. Ice
5. Salt
6. Chocolate Sauce
9. Output (产量) ~ The product you create
1. Ice Cream 🍨🍦is the output!

## Part 2 – The Relationship Between L, K, and Q

Every product (产量) can have lots of inputs (输入), just like the Ice Cream 🍦 or a Car 🚗.

Input + Input + Input + Input = Output (🚗)

But in our class, we focus on TWO inputs: Labor (👱) and Money (💲)

👱+💲=🚗
Labor (👱) + Money (💲) = Quantity (🚗)
L (👱)+ K (💲)= Q(🚗)

Example:
Justin sells 200 cars 🚗every day. Not 201 cars. Not 199 cars. He sells EXACTLY 200 cars 🚗every day.

L (👱) + K (💲) = 200 cars (🚗)

Justin knows that in ONE DAY🔆:

• 1 worker 👱 can create 50% of a car 🚗~ 2 workers 👱👱can create 100% of a car 🚗(one car)
• 1L = 0.5Q 🚗
• 2L = 1Q 🚗
• $5 💲 can pay for 20% of a car 🚗~$25 💲can pay for 100% of a car 🚗(one car)
• 1K = 0.2Q 🚗
• 5K = 1Q 🚗

Rule #1 ~ If Land K ⬆, then the # of 🚗 cars will also ⬆

Rule #2 ~ If Land K ⬇, then the # of 🚗 cars will also ⬇

Rule #3 ~ If L ⬇  and the # of 🚗 cars is still 200 (stay the same), K must ⬆

Rule #4 ~ If K ⬇  and the # of 🚗 cars is still 200 (stay the same), L must ⬆

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Today, Justin looks 🤔at his Money 💲and is NOT happy😭. He thinks he spends TOO MUCH money😡!  He wants to buy a new bicycle🚲, so he decides to SAVE $100 💲 This means today: 👱L + (💲K – 💲100K) = ? Q🚗 WHAT IS THE NEW Q (number of cars🚗) that Justin Makes Today? Day 1 (Yesterday): 👱L + 💲K = 🚗200Q Day 2 (Today): 👱L + (💲K – 💲100K) = 🚗200Q – all the 🚗cars$100K would pay for.

Remember!  💲1K = 🚗0.20Q (one dollar pays for 0.20 cars in a day)

If Justin does not spend $100 today, he will lose the money for 20 cars! 1K = 0.20 cars 💲-100K = -20 cars🚗🚗🚗🚗🚗🚗🚗🚗🚗🚗🚗🚗🚗🚗🚗🚗🚗🚗🚗🚗 ANSWER: Justin makes 180 cars today! L + (K – 100K) = 200 cars – 20 cars = 180 cars🚗 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ OK?!?🤔 NO!!! 😡Remember –> “Justin sells 200 cars 🚗every day. Not 201 cars. Not 199 cars. He sells EXACTLY 200 cars every day. ” PROBLEM: How can Justin still make and sell 200 cars tomorrow if he still saves the$100 (-100K) as today.

### Rule #4 ~ If 💲K ⬇  and the # of 🚗 cars is 200 (the same), 👱L must ⬆

QUESTION: HOW MUCH should 👱L  go up (⬆)?

• Step 1 ~ How many extra cars 🚗does Justin need to make? ~ Justin can make 180 cars right now if he saves $100 (-100K) but L stays the same as yesterday. 200 🚗 – 180 🚗= 20🚗 Justin needs to make 👱L ⬆ enough to make 20 extra cars🚗 tomorrow. • Step 2 ~ How much L does Justin have to add (+) to make 20 more cars tomorrow? ?L + (K – 100K) = 200Q Remember, 👱1L = 0.5Q 🚗 | 👱👱2L = 1Q 🚗 20Q 🚗= 40L ANSWER: Justin will have to hire 40 workers (+40L) in order to make 20 more cars tomorrow. 👫👫👫👫👫👫👫👫👫👫👫👫👫👫👫👫👫👫👫👫 # FINAL SOLUTION *Substitution = Adding L to Decrease K (👱L + 👱40L) + (💲K – 💲100K) = 200 Q (🚗) ## Part 3 ~ MRTS MRTS = Marginal Rate of Technical Substitution Go Back to Part 2. • 🔺K💲 • Yesterday, Justin had K💲 • Tomorrow, Justin has -100K💲 • 🔺K = -100K💲 • 🔺L👱 • Yesterday, Justin had L👱 • Tomorrow, Justin has +40L👱 • 🔺L = +40L👱   MRTS = 5K : 2L |5💲 : 2👫 This just tells us: 1. For every 2 👫workers Justin has, he spends$5💲💲💲💲💲.
2. For every 1 👱worker Just has, he spends $2.50💲💲½ 3. If Justin wants to hire 1 worker (+1L) , he will save$2.50 (-2.50K)

$100,000 = Principal 7% = Interest ## SIMPLE INTEREST (单利) Problem! ~ 7% of what? Answer! ~ It depends 🙂 It depends on how Company A decides to count it. There are two separate mathematical formulas (数学公式) you can use to figure out the Interest. The first one is called Simple Interest (单利). Simple Interest says that each payment period Company B is going to pay an additional 7% of the original principal ($100,000).  The formula for Simple Interest is:

I = Interest
P = Present Value
R = Interest Rate
T = Number of Years Involved

Company A (a large global corporation) invests $100,000 in a small new business called Company B. Company B has 10 years to pay it back. The interest rate is 7% per year. What is the Total Interest (利) you will pay over 10 Years? ## Calculating FUTURE VALUE and PRESENT VALUE using SIMPLE INTEREST The total interest is of course important to both Company A and Company B, there are two other important numbers that the financial managers of Company A want to know–the Present Value of their money and the Future Value of their money. ### Future Value FV = Future Value (how much money you will make in total) PV = Present Value R = Interest Rate T = Number of years involved Using our example above with Company A & B, the Future Value is calculated like this: That means Company A will make a total of$70,000 if they invest their $100,000 in Company B now. Over 10 years, their$100,000 will change into $170,000. 🙂 We like this plan! ### Present Value Sometimes, for example with bonds (债券), we know the Future Value (how much money will be paid to us in the end). But we don’t know how much money has to be invested (Present Value) to get that future result. So Present Value is calculated by the formula: PV = Present Value PV = Future Value R = Interest Rate T = Number of years involved Example: Mary Jane knows that in 4 years, she needs to have a total of$150,000 to pay her college tuition. She has an interest-generating account that gives her a 4% interest rate on everything she puts in. How much money should she invest today (Present Value) in order to have $150,000 in 4 years? That means that Mary Jane needs to put$129,310.35 in her bank account now in order to get \$150,000 in the future. ## KEY TERMS TO REMEMBER

1. Value (值)
2. Present Value (现值)
3. Future Value (未来价值)
4. Interest (利)
5. Simple Interest (单利)
6. Principal (本息)

## How to Use Mathematical Equations in your Blogging

27 Apr If you’re like me, you might need the help of mathematical equations in your blogging or professional writing.  But writings and copying an equation from Word into your website simply won’t work.

Research kept taking me to sites that recommended MathJax, LaTeX, MathML, etc. But to be honest, without an idea of how those work it just got all sorts of confusing. Certainly not as simple as I needed.

But I did finally find a website that worked for me! I’ve tested this method and it allowed me to add Mathematical Equations to:

1. WordPress
2. Weebly