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

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 becom

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 ________.