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 ________.