# Homework Help: Min max: how much of the tax to absorb

1. Sep 8, 2018

### Karol

1. The problem statement, all variables and given/known data

2. Relevant equations
Minimum/Maximum occurs when the first derivative=0

3. The attempt at a solution
$$y=20,000+60x,~~P=200-Ax$$
$$N=xP-Y=200x-Ax^2-20,000-60x,~~N'=140-2Ax$$
Two variables

2. Sep 8, 2018

### SammyS

Staff Emeritus
What do your "new" variables represent?

3. Sep 8, 2018

### Karol

x is the same, items sold per week, and A is the new coefficient instead of 0.01.
A<0.01 since P must be higher now.
But maybe P=B-0.01x, i don't know.
Or may be even P=B-Ax

4. Sep 9, 2018

### tnich

Here's the crux of the problem - to pass on some of the tax to the customer, you have to include it in the price. Then you end up selling fewer items per week because "you can sell $x$ items per week at a price of $P = 200 - 0.01x$ cents". So how would you modify that equation to account for the decrease in the number items sold due to the increase in price?

5. Sep 9, 2018

### Karol

Why it's not correct that: $~y=(50+a)x+20,000~$? it's more expensive to make an item.
I don't think you can modify $~P=200-0.01x~$ because it depends only on the customers that are ready to buy x items if the price is P.
The formula $~P=200-0.01x~$ doesn't depend at all on the firm, only on the free market, as i understand it, so i can't change it at all. if P is higher x is lower but according to the same $~P=200-0.01x~$.

6. Sep 9, 2018

### SammyS

Staff Emeritus
That leaves lots of questions (in my mind).

In the Example problem, I see:
$x\$ : number of items sold per week.
$P\$ : price at which each item will be sold.
$y\$ : total cost to put x items on market each week.​
You replied to me mentioning one more variable:
$A\$ : the new coefficient instead of 0.01. (You don't say what $A\$ is in any well defined manner.)​

That leaves at least 2 more, $\ N\$ and $\ Y\$, plus now you mention $\ B\$.

I suppose that $\ N\$ is the profit for selling $\ x\$ items per week and perhaps $\ Y\$ should be $\ y\$.

I think you need to handle $A\$ differently. I see some problems with the way you handle the amount of tax to be passed on in the price.
1) You will be adding $A\$ to the price of each item, not subtracting it.
2) You seem use $A\$ to replace the 0.01 or combine the two. The 0.01 is a factor which reduces the price per item depending on the number of item sold each week. $A\$ should not be multiplied by $x\$ in the formula for $P\$.
3) $A\$ has only a limited range of possible values. This range depends on how you define it (in a detailed way)​

I suggest that $A\$ is simply the actual amount of tax passed on in the price. So it would have a range from 0 to 10 cents.

Also, keep the $\ -0.01x\$ as part of $P \$.

I seems that now you have two variables to use in optimizing, $x\$ and $A\$ .

Last edited: Sep 9, 2018
7. Sep 9, 2018

### tnich

Right. You are raising the price by passing some of the tax on to the customer, so that should lower the number of items you sell in a week. You have to somehow represent that in an equation to solve this problem. What would be the new price on the item if you add some amount of tax T?

8. Sep 10, 2018

### Karol

$$\left\{ \begin{array}{l} P=200-0.01x+a \\ y=(60-a)x+20,000 \end{array} \right.$$
$$N=xP-y=...=2(70+a)x-0.01x^2-20,000$$
$$N'=2(70+a)-0.02x,~~N'=0~\rightarrow~x=100(70+a)$$

9. Sep 10, 2018

### SammyS

Staff Emeritus
Now what should the value of $\ a\$ be?

10. Sep 10, 2018

### tnich

Assuming that $P$ remains constant, will a positive value of $a$ result in more items sold or less? Does your equation make sense in that context?

11. Sep 11, 2018

### Karol

$$\left\{ \begin{array}{l} P=200-0.01x-a \\ y=(60-a)x+20,000 \end{array} \right.$$
$$N=xP-y=...=2(70-a)x-0.01x^2-20,000$$
$$N'=2(70-a)-0.02x,~~N'=0~\rightarrow~x=100(70-a)$$
It still is with 2 variables.
And i see that $~P=200-0.01x~$ represents the ability of the firm to charge the price P if x items are produced and sold. that i understand from the fact you modify this equation. why then is $~y=50x+20,000~$? isn't it doubling the information, about the firm?
I thought $~P=200-0.01x~$ represents the readiness of the market. if it costs P then only x items will be purchased. this interpretation of mine is completely disconnected from the firm. i wouldn't touch P and also the tax doesn't influence this equation, to my interpretation.
Instead i would change the cost, y. it is more expensive now to produce an item.

Last edited: Sep 11, 2018
12. Sep 11, 2018

### tnich

P is the price of one item. y is the cost of making x items. The question is asking how much of the tax you will add to the price and how much of it you will add to the cost. How would you represent that?

13. Sep 11, 2018

### Karol

The answer to the original question, no' 7, is x=7500. If "all other features are unchanged" then x=7500.
$$\left\{ \begin{array}{l} P=200-0.01x-a \\ y=(60-a)x+20,000 \\ x=7500 \end{array} \right.$$
$$N'=0~\rightarrow~x=100(70-a)~\rightarrow~7500=100(70-a)~\rightarrow~a=-5$$

14. Sep 11, 2018

### tnich

I am having a hard time getting from your problem set up to your answer. You can take either P or x as an independent variable. I have been trying to get you to use P as the independent variable, but I think I have only made the problem harder that way. My apologies. It seems simpler to use x. Then
$P = 200 - 0.01x$
$y = (50 + 10 - a)x + 20,000$ (I edited this to correct an error.)

Then the net profit is $N = (P-a)x - y$ because you don't get to count the collected tax as profit.
This seems to be equivalent to your formulation. But how are you getting from there to $N'=0~\rightarrow~x=100(70-a)$?

Last edited: Sep 11, 2018
15. Sep 11, 2018

### SammyS

Staff Emeritus
@Karol ,
I too am having trouble understanding what you are doing and/or trying to do.

Please explain in some detail:What does each and every variable represent and how are you using each?

Well at least, tell us more about you intentions for the variable, $\ a\$.

16. Sep 12, 2018

### Karol

a are the cents that are absorbed by the firm.
$$N = (P-a)x - y=...=140x-0.01x^2+20,000$$
a disappeared

17. Sep 12, 2018

### tnich

Yes. So how much of the tax should be passed on to the customer?

18. Sep 12, 2018

### Karol

$y = (50 + 10 - a)x + 20,000~$ says (10-a) is absorbed by the firm. every item is more expensive to "produce" by (10-a). i shouldn't write $~N=(P-a)x-y~$ since the loss to the firm has already been taken into account in y.
So a is passed to the customer. every item is more expensive by a. if we already want to use $~N = (P-a)x - y~$ then it should be $~N = (P+a)x - y$
But it's not logical. if $~P=200-0.01x~$ represents, as i interpret it, the forces of the market, meaning what are people ready to pay, then $~N=xP-y~$ should be changed in a different way.
I think $~y=60x+20,000~$ only, because each item is more expensive by 10. what we do later with the price we charge is an other thing.
Maybe i should change a to be positive and represent the decrease in the number of people who are buying, so $~P=200-0.01(x+a)~$, but i don't know to do it this way either

Last edited: Sep 12, 2018
19. Sep 12, 2018

### tnich

$P = 200- 0.01x$ is the demand curve, and in micro-economic theory the demand curve is the relationship between the price the firm charges and the number of units it can sell. If the firm decides to pass an amount $a$ of the tax on to the customers in the price of each unit, then it can only keep $P-a$ of the income on each unit because it has to hand the rest over to the government. So your equation in post #16 is correct (except for the sign on 20,000). What it is telling you is that it does not matter how much of the tax you charge to the customer - all values of $a$ give the same result. What does matter is the total price that you charge the customer.

20. Sep 12, 2018

### Karol

$$\left\{ \begin{array}{l} y=60x+20,000 \\ P=200-0.01x \\ N=(P-a)x-y \end{array} \right.$$
$$N=(140-a)x-0.01x^2-20,000$$
$$N'=0~\rightarrow~140-a=0.02x$$
I have to express x in terms of a, x=x(a)