MEconPower02-02-12-2014 - Managerial Economics Chapter 2...

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Managerial Economics Chapter 2 Introduction (2): Optimization
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1. Functional Relation - definition output = function(inputs) total revenue = function(product) total cost= function(product) profit = total revenue(product) – total cost(product)
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- Function forms (1) linear: Y = a + bX (2) quadratic: Y = a + bX + cX 2 TR = 45Q – 0.5Q 2 (3) cubic: Y = a + bX + cX 2 + dX 3 TC = 2 + 57Q – 8Q 2 + Q 3 Y = dependent variable X = independent variable a,b,c,d = coefficients
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2. Optimization = finding optimum product (Q) by maximizing profit (π) max. profit = total revenue – total cost max. π = (45Q – 0.5Q 2 ) – (2 + 57Q – 8Q 2 + Q 3 )
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A. Optimization by graph $ Q $ Q
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- slope of total revenue = slope of total cost marginal revenue = marginal cost - slope of profit = 0 marginal profit = none
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B. Optimization by Calculus (1) slope - average slope = ∆TR/ ∆Q - marginal slope = lim(∆TR/ ∆Q) = dTR/dQ dTR/dQ = derivative of TR with respect to Q differentiation
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(2) Derivative(differentiation) rule function derivative TR = a dTR/dQ = 0 TR = a + bQ n dTR/dQ = bnQ n-1
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(3) Optimization rule - first derivative= 0 - second derivative = - maximum + minimum
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∏ = -2 -12Q +7.5Q 2 – Q 3 d∏/dQ = -12 + 7.5(2)Q – 3Q 2 = 0 Q* = 4 d 2 ∏/dQ 2 = 15 – 6Q = -9 ∏* = 6
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