Given the function q=D(x)=k/x^n
where k is a positive constant and n is an integer greater than 0.
(a) How would you find the elasticity of the demand function
(b) Is the value of the elasticity dependent on the price per unit
c.) Does the total revenue have a maximum, and if so, when?
I've gotten so far as to find the der. is -knx^-n-1, but that might also not be the right derivative after all.
Thanks
(a) n
(b) no
(c) n=1
Well, well, well, it seems like we've got some elasticity and revenue questions! Don't worry, I'm here to clown around and help you out!
(a) To find the elasticity of the demand function, you need to take the derivative of the function with respect to x (aka the price per unit), and then multiply it by x divided by D(x). So, let's do some math magic!
Taking the derivative of D(x) = k/x^n gives us D'(x) = -knx^(-n-1).
Now, to find the elasticity, we multiply this derivative by x over D(x):
E = x * D'(x) / D(x).
Substituting the values we found earlier, we get:
E = (x * -knx^(-n-1)) / (k/x^n).
Simplifying this expression gives us:
E = -n.
Surprise! The elasticity of the demand function turns out to be just -n. Who knew math could be so hilarious?
(b) The value of the elasticity is not dependent on the price per unit (x). It solely depends on the exponent (n) in the given function. So, no matter the price, the elasticity will always be -n. Isn't that amusing?
(c) Ah, the total revenue! Does it have a maximum? You bet your bottom dollar it does! The total revenue will have a maximum when the price per unit (x) corresponds to the point where the elasticity of demand (E) is equal to -1. Hurray for that sweet spot where revenue reaches its peak!
And there you have it! Elasticity, price per unit, and total revenue - all analyzed with a pinch of clownish charm. I hope it brought a smile to your face!
To find the elasticity of the demand function q = D(x) = k/x^n, where k is a positive constant and n is an integer greater than 0, you will need to follow these steps:
(a) Finding the elasticity of the demand function:
1. Start by taking the derivative of D(x) with respect to x. In this case, you are correct with the derivative -knx^(-n-1).
2. Once you have the derivative, substitute it into the formula for elasticity: E = (x * D'(x))/D(x), where D'(x) represents the derivative of D(x).
3. Substitute the value of D'(x) (-knx^(-n-1)) and D(x) (k/x^n) into the formula and simplify.
(b) Determining if the value of elasticity is dependent on the price per unit:
The value of elasticity in the demand function is not dependent on the price per unit. It solely depends on the variables x and n, as the price per unit is not explicitly mentioned in the function.
(c) Identifying if the total revenue has a maximum and determining when:
To figure out if the total revenue has a maximum, you need to analyze the behavior of the demand function. Total revenue is the product of the quantity demanded (D(x)) and the price per unit (P), given by TR(x) = D(x) * P.
1. Substitute the given demand function into the formula for total revenue: TR(x) = (k/x^n) * P.
2. To maximize the total revenue, you need to find the critical points of the function TR(x), where its derivative is equal to zero or the function is not defined.
3. Take the derivative of TR(x) with respect to x and set it equal to zero: TR'(x) = 0.
4. Solve the resulting equation for x to find the critical point(s).
5. Determine if the critical point(s) are maximum points by performing the second derivative test or by checking the concavity of the curve.
6. If there is at least one maximum point, that will indicate when the total revenue is at a maximum.
Keep in mind that the specific values of k and P are not provided in the given information, so you may have to use additional information or assumptions to calculate the exact maximum point.
To find the elasticity of the demand function q = D(x) = k/x^n, we need to determine its derivative, and then apply the elasticity formula.
To calculate the derivative of D(x) with respect to x, we use the power rule for differentiation.
Taking the derivative of D(x) = k/x^n, we have:
D'(x) = -knx^(-n-1)
Since you arrived at the same derivative, it seems you've done this correctly.
Now, let's proceed to find the elasticity of the demand function. The elasticity of demand measures the responsiveness of quantity demanded to changes in price.
The formula for elasticity of demand is given as:
E = (dq/dx) * (x/q)
where dq/dx is the derivative of the demand function with respect to x, and x/q represents the ratio of the quantity demanded to the demand function.
In this case, dq/dx = D'(x) = -knx^(-n-1), and q = D(x) = k/x^n.
Substituting these values into the elasticity formula, we get:
E = (-knx^(-n-1)) * (x/(k/x^n))
Simplifying this expression, we find:
E = -n
So, the value of the elasticity E is equal to -n.
Now, let's answer the remaining questions:
(b) The value of the elasticity (-n) is not dependent on the price per unit. It is solely determined by the exponent (n) in the demand function. The elasticity shows the percentage change in quantity demanded for a 1% change in price. Hence, it is independent of the actual price level.
(c) To determine whether the total revenue has a maximum, we need to examine the relationship between price and quantity demanded. Total revenue (TR) is given by the product of price (P) and quantity (q), or TR = P * q.
In this case, we have:
TR = (k/x^n) * x
= kx^(1-n)
The total revenue function TR(x) is a decreasing function when n > 1, and an increasing function when 0 < n < 1.
If n = 1, the total revenue is constant.
Therefore, the total revenue has a maximum when n < 1.
I hope this explanation helps you understand how to find the elasticity of the demand function, the independence of elasticity from price per unit, and the conditions for total revenue to have a maximum.