MRT as the Derivative of the Feasible Frontier Function g(t)

An individual's utility U is a function of their consumption c, given by the function U(c) = 40√c. The individual's consumption is, in turn, dependent on the number of hours they work h, according to the function c(h) = 100 - 2h. Using the rule for differentiating a composite function, find the marginal utility with respect to hours worked, dU/dh.

Analyzing Production Efficiency

Marginal Profit of Labor

A company's total revenue (R) is a function of the quantity of units sold (q), so R = f(q). The quantity of units sold is, in turn, a function of the price (p), so q = g(p). To find the rate at which revenue changes with respect to price (dR/dp), you must apply the rule for differentiating a function of a function. Arrange the following steps in the correct logical order to perform this calculation.

A firm's profit (Π) is a function of the quantity of goods it produces (q), and the quantity produced is a function of the hours of labor employed (L). To find the marginal profit with respect to labor (dΠ/dL), one must calculate the marginal profit with respect to quantity (dΠ/dq) and the marginal product of labor (dq/dL), and then add these two rates of change together.

A firm's profit (Π) is a function of the quantity of goods it sells (q), which in turn is a function of its advertising expenditure (A). To find the overall effect of advertising on profit, one must use the rule for differentiating a composite function. Match each economic concept to its correct mathematical representation.

A factory's total production cost (C) is determined by the number of units (q) it produces, according to the function C(q) = 1000 + 10q + 0.1q². The number of units produced is a function of the hours of labor (L) used, given by q(L) = 20L. When 5 hours of labor are used, the marginal cost with respect to labor (the rate at which cost changes for each additional hour of labor) is ____.

Strategic Decision-Making for Profit Maximization

A company's daily production output, Q, is determined by the amount of capital, K, it employs, according to the function Q(K) = 100√K. The company is expanding, and its capital stock grows over time, t (in months), as described by the function K(t) = 4 + 2t. What is the rate at which the company's daily output is changing with respect to time when t = 6 months?

A manufacturing firm's total production cost is a function of the number of units it produces, and the number of units produced is a function of the hours of labor employed. The rate of change of total cost with respect to labor hours can be found using the formula: dC/dL = (dC/dq) * (dq/dL). What is the correct economic interpretation of the dq/dL term in this formula?