Slope of the Isoprofit Curve in Terms of Wage and Employment
The slope of an isoprofit curve () can be expressed in a useful form that depends only on the wage () and employment level (). This is achieved by taking the derivative of the isoprofit equation and then substituting the original profit formula, , into the result. This process yields the formula for the slope: .
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Introduction to Microeconomics Course
The Economy 2.0 Microeconomics @ CORE Econ
Ch.6 The firm and its employees - The Economy 2.0 Microeconomics @ CORE Econ
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Slope of the Isoprofit Curve in Terms of Wage and Employment
Using the Second Derivative to Prove Isoprofit Curve Concavity
A student wants to mathematically verify the shape of an isoprofit curve for a firm. The firm's profit (Π) is given by the equation Π = P * Q(N) - w * N, where P is a constant price, Q(N) is the quantity produced using N units of labor, and w is the wage. To find the slope of the curve in the (N, w) plane, the student first rearranges the equation to isolate the wage: w = (P * Q(N) - Π) / N. The student then attempts to find the derivative of w with respect to N and gets the following result: dw/dN = P * Q'(N). What is the primary conceptual error in the student's derivation?
Interpreting the Isoprofit Curve's Slope
Derivation of the Isoprofit Curve's Slope
Consider a standard isoprofit curve for a firm where profit is held constant. A positive value for the derivative of wage with respect to employment (dw/dN > 0) signifies that the firm must decrease wages as it increases employment to maintain that same profit level.
To mathematically verify that an isoprofit curve is upward-sloping, one must differentiate the wage (w) with respect to employment (N), starting from the profit equation Π = PQ(N) - wN, where Π and P are constants. Arrange the following steps into the correct logical sequence to perform this verification.
A firm's profit (Π) is held constant along an isoprofit curve, defined by the relationship Π = PQ(N) - wN, where P is price, Q(N) is output as a function of labor N, and w is the wage. The mathematical expression for the slope of this curve in the (N, w) plane is given by: dw/dN = [PQ'(N) - w] / N. An analyst examines this expression and concludes, "Since hiring more workers leads to diminishing marginal returns (Q'(N) decreases as N increases), the numerator PQ'(N) - w will eventually become negative. Therefore, isoprofit curves must eventually slope downwards." Why is this analyst's reasoning for the shape of the curve flawed?
Evaluating an Isoprofit Claim
Mathematical Verification and Interpretation of Isoprofit Curve Shape
A firm's isoprofit curve is defined by a constant profit level (Π) where Π = PQ(N) - wN. The slope of this curve in the (N, w) plane is given by the derivative dw/dN = [P*Q'(N) - w] / N. Match each mathematical component of this derivative expression to its correct economic interpretation.
A firm's isoprofit curve is defined by the equation w = (PQ(N) - Π) / N, where w is wage, N is employment, P is a constant price, Q(N) is the production function, and Π is a constant profit level. To mathematically verify that this curve is upward-sloping, one must show that its derivative, dw/dN, is positive. For the derivative to be positive, the marginal revenue product of labor, PQ'(N), must be greater than the ________.
Learn After
How Wage and Employment Levels Determine the Isoprofit Curve's Slope
Profit Maximization as Tangency Between an Isoprofit Curve and the No-Shirking Wage Curve
The historical Danish monopoly on trade with the Faroe Islands is often cited as an extreme example of a single-seller market. Which of the following features of this arrangement most critically demonstrates the high barrier to entry that sustained the monopoly?
A firm's profit is held constant along a curve that shows various combinations of the wage it pays (w) and the number of people it employs (N). The revenue generated per employee (y) is a fixed amount. The slope of this curve at any point, representing the trade-off between wage and employment, is given by the formula: (y - w) / N. If the firm moves from a point with low employment to a point on the same curve with high employment, how does the slope of the curve change?
Analyzing an Isoprofit Curve's Slope
Consulting Firm's Wage-Employment Trade-off
A firm's isoprofit curve shows combinations of wage (w) and employment (N) that yield a constant level of profit. The slope of this curve, representing the trade-off between wage and employment, is given by the formula (y - w) / N, where y is the constant revenue per employee. Match each scenario describing a point on the curve with the correct mathematical description of the slope at that point.
True or False: Consider a firm where the combinations of wage and employment that yield a constant profit are represented by a curve. According to the mathematical relationship governing this curve's slope, the wage reduction required to offset the cost of hiring one additional employee is greater when the firm's current profit-per-employee is high compared to when it is low, assuming the number of employees is the same in both scenarios.
A manufacturing company generates $90,000 in revenue per employee. It currently employs 100 workers at an average wage of $60,000. To hire one additional worker while maintaining the same total profit, the company must decrease the average wage for all employees by $____.
Strategic Implications of the Isoprofit Curve's Slope
Two firms, Firm A and Firm B, are operating at points on their respective isoprofit curves, which represent combinations of wage and employment that yield a constant level of total profit. At these specific points, both firms employ the same number of workers. However, Firm A generates a significantly higher profit-per-employee than Firm B. Given that the slope of an isoprofit curve is calculated as (profit-per-employee) / (number of employees), which of the following statements accurately compares the slopes at their current operating points?
A firm's profit (Π) is determined by its revenue per employee (y), the wage it pays (w), and the number of employees (N), according to the formula Π = (y-w)N. To find the slope of the curve where profit is held constant (the isoprofit curve), one must mathematically derive the expression for the trade-off between wage and employment (dw/dN). Arrange the following mathematical steps in the correct logical order to perform this derivation.
Consulting Firm's Wage-Employment Trade-off