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Elasticity refers to the percent change to which one value changes, quantity of , when another does. Supply and demand change with respect to price; investment and savings change with respect to interest rate. The name is "X elasticity of Y" where a change in X causes a change of magnitude (the elasticity * Y).
Price elasticity of Demand
[edit | edit source]Let's begin with some definitions. If we want to determine the percent change in price for the demand for a good, we need to understand what we are doing. Let denote the quantity demanded at some arbitrary initial point, and let denote the quantity demand at some point close to our initial point that is the final value we get to. To determine the percent change, we have to find a change in the values that we have chosen divided by our old value: if you want to put it in terms of what we learned, it is
- , where is simply a letter that denotes a very small change in (quantity demand).
We learn the above information for one very special reason. That means something: percentage change. We are looking at the percentage change in quantity demanded. However, we cannot find the percentage in quantity demanded without knowing the prices of goods that correspond to those values. Therefore, let be the price of and be the price of . To determine the percent change, we have to find a change in the values that we have chosen divided by our old value: if you want to put it in terms of what we learned, it is
- , where is for price corresponding to .
For any ordered pair and that corresponds to a demand curve, the price elasticity of demand is
(1)
Before talking about applications, it is important to get an intuitive understanding of elasticity. Please note the use of elasticity. It is not an arbitrary term. Rather, it brings to mind the rubber band. The maximum stretching distance of the rubber band can be assigned a value or , where . As a collective, we would say that the rubber band with is relatively more elastic than . Similarly, the "stretchiness" of a demand curve represents this fundamental relationship in the form of a percentage change in both quantity demanded and price using the greek letter eta ().
Because the demand curve comes in all types of curves (that is downward-sloping), the elasticity must be negative. This does not change anything fundamental to how we interpret the value, yet the absolute value of the price elasticity of demand is often used. Keep in mind that the change is not some constant either but a percentage. This means we are not finding the slope. Instead, we are looking at percentage change. As a result, some strange properties come true. One of them is that moving down along a linear demand curve implies a non-constant elasticity. In fact, the change in the elasticity decreases further and further. A proof for this concept is given below:
Price elasticity of demand decreases along a linear demand curve (Proof)
Knowing is the reciprocal slope, multiplying by , which is the point that is variable, makes the expression equal to the elasticity of demand . The reason for this is because . That is, As you move down the demand curve, and . Ergo, . However, given is a reciprocal slope of the linear demand curve, the value for and are both constant. Hence, the slope must be constant. Summary: Because decreases per every point to point , and is constant, the elasticity of demand, , along a demand curve must be falling as the price decreases. |
Notice that we were able to derive one more equation relating to the elasticity of demand, . However, we did not this proof to show them. By the very fact that
we can further show one more new truth, simply through rearrangement. Here is the work that leads to the final equation:
(2)
Remember (
) and ( ) for later, especially equation 1.Interpreting the Price Elasticity of Demand
[edit | edit source]We will look at the following demand curve above for this little exercise in price elasticity of demand. Keep note that the demand function, , is the solid, black, downward-sloping line. We will talk about the other two functions later.
Use of Price Elasticity of Demand
[edit | edit source]Just by knowing a numerical value, one can know more about the type of good that is possibly given in the market. However, before an honest analysis can be given for one particular good, the effects need to be given first before we can move on.
Price elasticity of Supply
[edit | edit source]By using the same logic for supply, the price elasticity of supply is
where denotes the quantity supplied, denotes the quantity supplied at some arbitrary initial point along the demand curve, and denotes the quantity supplied at some final point close to .
To reiterate why we determine price elasticity, the percent change can help us determine by how much our good has increased in quantity demanded or quantity supplied and can determine not only the slope but also the iterative change of a good. Say that an ordered pair exists in which and corresponds to a supply curve.
Note:
- is initial price
- is final price
- is initial supply
- is final supply
- is initial demand
- is final demand
- is initial interest
- is final interest
- is initial investment
- is final investment
- is initial savings
- is final savings
Price elasticity of demand
Price elasticity of supply
Interest elasticity of investment
Interest elasticity of savings
Cross elasticities
[edit | edit source]The elasticities mentioned above refer to one object. Cross elasticities refer to the effects of something's price, interest, etc. on something else. This comes into play with substitute and complementary goods and services for the consumer
Rational and Absolute Value Problem
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For Exploration 3-1 through Exploration 3-4, is an absolute value function, and g(x) is any arbitrary function.
Logarithm and Summation Problem
[edit | edit source]2. Look at the function below. After, do items (a)-(c).
- (a) Simplify the function .
- (b) Find the value of and the corresponding variable associated with that value of in each situation below for the function :
- (i). where is the y-intercept;
- (ii). ;
- (i). where is the y-intercept;
- (c) Find such that .
(a) First, look if you can simplify the inside of the expression in some way. Do not start that very dramatically if you cannot do it. Here, the book will start slowly so that all students can understand what is going on. First, notice that each expression inside the first sigma has an multiplied to it: , then , etcetera. Therefore, factor an from each term.
Footnote 1
[edit | edit source]Allow either or to equal something. We allowed to equal something by adding to both sides of the equation.
From there, substitute back into the system of equations, equation (1), to get:
After, solve for and substitute that into equation (2):
- (1):
- (1):
- (2):
- (2):
In an attempt to solve for , the resultant answer tells us an identity (5 always equals 5). Therefore, b has infinitely many solutions. The same thing also happens with . Finally, because is equal to itself, the following is also true:
- (4): .
The following truth reveals itself. As long , equation (3), and , equation (4), the following must be true:
However, the same situation results when trying to find one solution set : there are infinitely many solutions. We learn one important lesson: what mathematics means by factor is make it so that all terms can be multiplied by another term. There is no specific term you need to care about.
Physics 1 Problem
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Calculus Problems
[edit | edit source]Generalizing the Can Optimization
[edit | edit source]Never thought you would see a proof? Well, here you go.
Small Problems
[edit | edit source]Only the most simplified answers will be accepted for "gapfill" responses. Use a calculator when needed.