Marginal Analysis for optimal decisions

Optimization Techniques In Economics different optimization techniques as a solution to decision making problemsOptimization implies either a variable is maximized or minimized whichever is required for efficiency purposes, subject to different constraints imposed on other variablesE.g. Profit Maximization, Cost Minimization, Revenue Maximization, Output MaximizationA problem of maxima & minima requires the help of differential calculus

Profit Maximization

Profit Maximization

Profit MaximizationTotal Profit Approach for Maximization=TR-TC=> The difference to be maximized in order to Max. Profit

Marginal Analysis to profit maximizationMarginal Analysis requirement for profit Maximization, Marginal Revenue = Marginal Cost (MR) (MC)Marginal Value represents slope of Total value curves,Thus slopes of TR &TC should be equal

Two output level showing same slope, i.e. MR=MC

TRTCQOTR, TCABQ2Q1

Interpretation of the previous diagram MR=MC is a necessary condition for Maximization, not a sufficient one as this condition also hold for loss maximizationSufficient condition requires that reaching a point of maximization, profit should start declining with any further rise in output, i.e. Slope of TC should rise & Slope of TR must fall after reaching the point of Maximization, Change in MC>Change in MR*Case Study to be discussed: An alleged blunder in the stealth bombers design

Using derivatives to solve max and min problemsOptimization With CalculusTo optimize Y = f (X):First Order Condition: Find X such that dY/dX = 0Second Order Condition:A. If d2Y/dX2 > 0, then Y is a minimum.ORB. If d2Y/dX2 < 0, then Y is a maximum.

CENTRAL POINTThe dependent variable is maximized when its marginal value shifts from positive to negative, and vice versa

The Profit-maximizing ruleProfit(p) = TR TCAt maximum profit dp/dQ = dTR/dQ - dTC/dQ = 0So,dTR/dQ = dTC/dQ (1st.O.C.)==> MR = MCd2TR/ dQ2 = d2TC/dQ2 (2nd O.C.) ==> dMR/dQ < dMC/dQ

This meansslope of MC is greater than slope of MR function

Constrained OptimizationTo optimize a function given a single constraint, imbed the constraint in the function and optimize as previously defined

OPTIMAL ADVERSTING EXPENDITURESOptimal solution: Buy 2 TV ads and 4 radio adsOptimal choice between number of TV and Radio AdsObjective Function Maximize benefits (measured in sales)Budget constraint of $2000, given PTV = $400 &PR=$300Optimal condition: MBTV/PTV=MBR/PR

Number of AdsMBTVMBTV/PTVMBRMBR/PR14001.03601.223000.752700.932800.72400.842600.652250.7552400.61500.562000.51200.4

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