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The gross wage paid by an employer, $$W^{\text{gross}}$$, is linked to the worker's take-home wage, $$W$$, through the direct tax rate on labor, $$t_d$$. The formula expressing this relationship is: $$W^{\text{gross}} = W(1 + t_d)$$.

Formula Relating Gross and Take-Home Wages

An employee receives a monthly take-home wage of $4,000. The direct tax rate on labor, which accounts for the difference between the employer's cost and the employee's net pay, is 20%. Using the standard formula that connects these three values, calculate the total gross monthly wage paid by the employer.

Calculating Gross Wage from Take-Home Pay

An employer's total cost for an employee's labor is the gross wage, $W^{\text{gross}}$. The employee receives a take-home wage, $W$, after direct labor taxes at a rate of $t_d$ are accounted for. Given the relationship $W^{\text{gross}} = W(1 + t_d)$, rearrange this formula to express the take-home wage ($W$) in terms of the gross wage ($W^{\text{gross}}$) and the tax rate ($t_d$).

Deriving Take-Home Wage from Gross Wage

True or False: To maintain a constant take-home wage for an employee, a doubling of the direct labor tax rate requires the employer to double the gross wage they pay.

The variable $$t_d$$ represents the percentage rate of direct taxation on labor. This rate encompasses all direct taxes applied to wages, such as income taxes and social security contributions, which create a difference between the employer's wage cost and the employee's take-home pay.

Definition of Direct Tax Rate on Labor ($$t_d$$)

An employee's take-home wage is $2,000. The direct labor tax rate is 25%. If the government increases the direct labor tax rate to 30%, what must happen to the gross wage paid by the employer for the employee's take-home wage to remain unchanged at $2,000?

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