econ 533 econometrics and quantitative methods one variable calculus and applications to economics
TRANSCRIPT
Econ 533Econometrics 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.
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RELATIONSHIPS AMONG TOTAL, MARGINAL, AND AVERAGE VALUES.
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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
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HOW THE VALUE OF ΔY>Δ X VARIES DEPENDING ON THE STEEPNESS
OR FLATNESS OF THE RELATIONSHIP BETWEEN Y AND X
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DERIVATIVE AS THE SLOPE OF THE CURVE
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HOW TO FIND A DERIVATIVE
• Constant Rule:• If Y = a
• Then dY/dX = 0
• Product Rule:• If Y = a.Xb
• Then dY/dX = b.a.Xb-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)]/W2
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)
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HOW TO FIND A DERIVATIVE (CONT’D)
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HOW TO FIND A DERIVATIVE (CONT’D)
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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 d2Y/dX2 <0.• For minimum, dY/dX = 0 and d2Y/dX2>0.
USING DERIVATIVES TO SOLVE MAXIMIZATION AND MINIMIZATION PROBLEMS (CONT’D)
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USING DERIVATIVES TO SOLVE MAXIMIZATION AND MINIMIZATION PROBLEMS (CONT’D)
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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 d2/dQ2 must be <0.
• Thus, dTR/dQ = dTC/dQ or Marginal Revenue = Marginal Cost.
MARGINAL REVENUE EQUALS MARGINAL COST RULE FOR PROFI T
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PARTIAL DIFFERENTIATION AND THE MAXIMIZATION OF MULTIVARIATE FUNCTIONS
• = f(Q1, Q2)
• Set partial derivatives equal to zero
• /Q1 = 0 and /Q2 = 0
• Results in two equations in two unknowns• Solve simultaneously for the two unknowns
PARTIAL DIFFERENTIATION AND THE MAXIMIZATION OF MULTIVARIATE FUNCTIONS (CONT’D)
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