Given a worker with 8hours shift, where each hour of effort produce 3 widgets
The price rate of one widget produced is $5 , also given the persians
Utility(s, m)=s^1/3 m^2/3. Where s is shirking and m is salary
Find the hours of shirking and income
To find the hours of shirking and income, we need to calculate the number of widgets produced and the salary earned first.
Given:
- Worker's shift: 8 hours
- Hourly productivity: 3 widgets/hour
- Price rate per widget: $5
- Utility function: Utility(s, m) = s^(1/3) * m^(2/3), where s is shirking and m is salary
Step 1: Calculate the number of widgets produced
Number of widgets produced = Hourly productivity * Worker's shift
= 3 widgets/hour * 8 hours
= 24 widgets
Step 2: Calculate the salary earned
Salary earned = Number of widgets produced * Price rate per widget
= 24 widgets * $5/widget
= $120
Now, let's use the utility function to find the hours of shirking and the income.
Utility(s, m) = s^(1/3) * m^(2/3)
s^(1/3) * m^(2/3) = Utility
(s^(1/3)) = (Utility / m^(2/3))
s = (Utility^3 / m^2)
Given Utility(s, m), we need to find the value of s that maximizes the utility function. However, we don't have a specific utility value to work with. Could you please provide the desired utility value so we can proceed with the calculation?
To find the hours of shirking and income, we need to maximize the utility function U(s, m) while taking into account the worker's effort and the price rate of the widgets produced.
Let's start by calculating the total number of widgets the worker can produce in an 8 hours shift:
Number of widgets produced = Hours of effort * Widgets produced per hour
Number of widgets produced = 8 hours * 3 widgets/hour
Number of widgets produced = 24 widgets
Now, let's calculate the income the worker will receive from selling these widgets:
Income = Number of widgets produced * Price rate per widget
Income = 24 widgets * $5/widget
Income = $120
Next, we can use the utility function U(s, m) to determine the amount of hours the worker will shirk and the corresponding salary. To maximize the utility function, we need to find the values of s and m that maximize the function's value.
However, the utility function given – U(s, m) = s^(1/3) * m^(2/3) – is not directly dependent on the hours of shirking (s). This means that no matter how many hours the worker shirks, it won't affect the utility value. As a result, we cannot determine the specific hours of shirking from the given utility function.
All we can determine from the information provided is that the worker will receive an income of $120 for an 8 hours shift.
To find the hours of shirking and income, we need to maximize the utility function by taking the derivative of the utility function with respect to each variable (s and m) separately and setting them equal to zero.
Let's start with the utility function:
Utility(s, m) = s^(1/3) * m^(2/3)
To find the optimal hours of shirking (s), we differentiate the utility function with respect to s and set it equal to zero:
dUtility/ds = (1/3) * s^(-2/3) * m^(2/3) = 0
Simplifying, we get:
1/3 * m^(2/3) / s^(2/3) = 0
Since m is a constant, we can ignore it. Now, solving for s:
1/s^(2/3) = 0
This equation has no real solutions. Therefore, there is no optimal hour of shirking that maximizes the utility function.
Moving on to income (m). We differentiate the utility function with respect to m and set it equal to zero:
dUtility/dm = (2/3) * s^(1/3) * m^(-1/3) = 0
Simplifying, we get:
2s^(1/3) / (3m^(1/3)) = 0
Since s is a constant, we can ignore it. Now, solving for m:
2/3m^(1/3) = 0
This equation also has no real solutions. Therefore, there is no optimal income that maximizes the utility function.
In conclusion, based on the given utility function and information, there are no specific hours of shirking (s) or income (m) that can be determined to maximize the utility function.