Final answer:
The rental rate for capital, when the marginal rate of technical substitution is 7 and the wage rate for labor is $8, is calculated to be $56 per unit.
Step-by-step explanation:
The marginal rate of technical substitution (MRTS) indicates the rate at which a firm can replace capital with labor, keeping output constant. In the context of a cost-minimizing firm that uses only labor and capital, the absolute value of the MRTS is provided as 7. This means that 7 units of capital can be substituted for each unit of labor without changing the output level. Given the wage rate for labor is $8, we utilize the concept that in a cost-minimizing situation, the MRTS is also the ratio of the inputs' prices. So, the marginal rate of technical substitution of 7 equates the ratio of the price of capital (rental rate) to the wage rate, i.e., (rental rate of capital) / ($8) = 7.
To find the rental rate for capital, we simply multiply the wage rate by the MRTS: $8 * 7 = $56. Therefore, the rental rate for capital is $56 per unit.