Econ 533 Econometrics and Quantitative Methods One Variable Calculus and Applications to Economics

FUNCTIONAL RELATIONSHIPS Q = f (P). Q is the number of units sold, and P is the Price. Equation is read as “The Number of units sold is a function of price.” Q is the dependent variable. P is the independent variable.

MARGINAL ANALYSIS Marginal Value is the change in the dependent variable associated with a one-unit change in a particular independent variable. Marginal Profit is the change in total profit associated with a one-unit change in output. Average Profit is the total profit divided by output.

MARGINAL ANALYSIS The central point about a marginal relationship is that the dependent variable is maximized when its marginal value changes from positive to negative. Thus, managers need not focus on averages, as they would not be maximizing the function.

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RELATIONSHIPS AMONG TOTAL, MARGINAL, AND AVERAGE VALUES. The average profit curve must be rising if it is below the marginal profit curve. The average profit curve must be falling if it is above the marginal profit curve. Hence, average profit must be a maximum where marginal profit equals average profit.

RELATIONSHIPS AMONG TOTAL, MARGINAL, AND AVERAGE VALUES. Managerial Economics, 8e W.W. & Company 2013

RELATIONSHIPS AMONG TOTAL, MARGINAL, AND AVERAGE VALUES. Managerial Economics, 8e W.W. & Company 2013

THE CONCEPT OF A DERIVATIVE Y = f(X) The derivative of Y with respect to X is defined as the limit of  Y/  X, as  X approaches zero.

LINEAR RELATIONSHIPS BETWEEN Y AND X Managerial Economics, 8e W.W. & Company 2013

HOW THE VALUE OF ΔY>Δ X VARIES DEPENDING ON THE STEEPNESS OR FLATNESS OF THE RELATIONSHIP BETWEEN Y AND X Managerial Economics, 8e W.W. & Company 2013

DERIVATIVE AS THE SLOPE OF THE CURVE Managerial Economics, 8e W.W. & Company 2013

HOW TO FIND A DERIVATIVE Constant Rule: If Y = a Then dY/dX = 0 Product Rule: If Y = a.X b Then dY/dX = b.a.X b-1 Sum Rule: If U= g(X) and W = h(X) and Y = U + W Then dY/dX = dU/dX + dW/dX

HOW TO FIND A DERIVATIVE (CONT’D) Difference Rule: If U= g(X) and W = h(X) and Y = U - W Then dY/dX = dU/dX - dW/dX Product Rule: If Y = U.W Then dY/dX = U.dW/dX + W.dU/dX Quotient Rule: If Y = U/W Then dY/dX = [W.(dU/dX) – U.(dW/dX)]/W 2

HOW TO FIND A DERIVATIVE (CONT’D) Chain Rule: If Y = f(W) and W = g(X) Then dY/dX = (dY/dW).(dW/dX)

HOW TO FIND A DERIVATIVE (CONT’D) Managerial Economics, 8e W.W. & Company 2013

HOW TO FIND A DERIVATIVE (CONT’D) Managerial Economics, 8e W.W. & Company 2013

HOW TO FIND A DERIVATIVE (CONT’D) Managerial Economics, 8e W.W. & Company 2013

USING DERIVATIVES TO SOLVE MAXIMIZATION AND MINIMIZATION PROBLEMS Maximum or minimum occurs only if the slope equals zero. Whether maximum or minimum depends on the sign of the second derivative. For maximum, dY/dX = 0, and d 2 Y/dX 2 <0. For minimum, dY/dX = 0 and d 2 Y/dX 2 >0.

USING DERIVATIVES TO SOLVE MAXIMIZATION AND MINIMIZATION PROBLEMS (CONT’D) Managerial Economics, 8e W.W. & Company 2013

USING DERIVATIVES TO SOLVE MAXIMIZATION AND MINIMIZATION PROBLEMS (CONT’D) Managerial Economics, 8e W.W. & Company 2013

MARGINAL COST EQUALS MARGINAL REVENUE AND THE CALCULUS OF OPTIMIZATION  = TR – TC, where  equals total profit, TR equals total revenue and TC equals total cost. For  to be a maximum, d  /dQ = 0 and d 2  /dQ 2 must be <0. Thus, dTR/dQ = dTC/dQ or Marginal Revenue = Marginal Cost.

MARGINAL REVENUE EQUALS MARGINAL COST RULE FOR PROFI T Managerial Economics, 8e W.W. & Company 2013

PARTIAL DIFFERENTIATION AND THE MAXIMIZATION OF MULTIVARIATE FUNCTIONS  = f(Q 1, Q 2 ) Set partial derivatives equal to zero  /  Q 1 = 0 and  /  Q 2 = 0 Results in two equations in two unknowns Solve simultaneously for the two unknowns

PARTIAL DIFFERENTIATION AND THE MAXIMIZATION OF MULTIVARIATE FUNCTIONS (CONT’D) Managerial Economics, 8e W.W. & Company 2013