Could someone pls. help me out on the following question:
find the producers surplus (to the nearest dollar ) at a price level of p=$26 for the price-supply equation p = s(x)= 5ln(x+1) .
An early response would be greatly appreciated pls.
Thank you.
To find the producer's surplus at a price level of p=$26 for the price-supply equation p = s(x) = 5ln(x+1), we need to determine the producer surplus at that specific price level.
Producer's surplus represents the difference between the price at which producers are willing to supply a product and the actual price they receive. It can be calculated by finding the area under the supply curve and above the price line.
Here's how we can find the producer's surplus:
1. Set up the equation by equating the supply equation to the price level:
5ln(x+1) = 26
2. Solve for x by isolating it:
ln(x+1) = 26/5
x+1 = e^(26/5)
x = e^(26/5) - 1
3. Calculate the producer's surplus by integrating the supply equation from 0 to x:
∫[0 to x] 5ln(t+1) dt
Since we already found the value of x in step 2, we can substitute it into the integral.
4. Calculate the definite integral, which represents the area under the supply curve and above the price line. You can use calculus or an online integral calculator:
∫[0 to e^(26/5)-1] 5ln(t+1) dt
5. Evaluate the integral to get the producer's surplus, which will be in terms of dollars. Rounding to the nearest dollar:
PS = ∫[0 to e^(26/5)-1] 5ln(t+1) dt
Note: The calculated value will represent the producer's surplus at a price level of p=$26 for the given supply equation.