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Optimizing Customer Satisfaction

Optimizing Customer Satisfaction. By: Brian Murphy. Scenario. Consumer has an income of $200 and wants to buy two fixed goods: hats and guns. Price of hats is $20 and price of shirts is $30.

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Optimizing Customer Satisfaction

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  1. Optimizing Customer Satisfaction By: Brian Murphy

  2. Scenario • Consumer has an income of $200 and wants to buy two fixed goods: hats and guns. • Price of hats is $20 and price of shirts is $30. • Consumer wants to buy a certain number of hats and shirts so that he spends all or nearly all of his income while optimizing his satisfaction.

  3. Modeling • Key Variables: • Income (I) = $200 • Number of Hats Purchased (H) • Number of Shirts Purchased (S) • Price of Hat (PH) = $20 • Price of Shirt (PS) = $30 • Income Equation: HPH + SPS = 200

  4. Modeling • In Microeconomics, consumer satisfaction is mathematically represented by a utility function. • Utility function is usually generated from historical market trends. • Most common utility function is of the form U=aXαYβ • For this problem, the consumer’s utility function is U(H, S) = 2H1/2G1/2

  5. Finding the Optimal Bundle • What we need to find optimum: • Marginal Utility (MU) – the change in utility as a result of a small change in quantity of one good (calculated as the partial derivative of the utility function with respect to the good). • Marginal Rate of Substitution (MRS) – utility gain from a small change in one good while the other good is held fixed (Calculated as the ratio of the two marginal utilities). • Price ratio (PR) – ratio of the price of the two goods.

  6. Finding the Optimal Bundle • Calculated Variables: • MUH = H-1/2S1/2 • MUS = H1/2S-1/2 • MRSH,S = MUH/ MUS = S/H • PR = 20/30

  7. Problem Visualized • Notes: • Y-int = I/Py • X-int = I/Px • Slope = -Px/Py S Budget Line 20/3 Indifference Curve** Optimal Bundle 0 10 H **The Utility function projects outward in the third dimension in a bowl shape. The indifference curve is simply a cross section of the utility function.

  8. Solve • At Optimal Bundle: • Nearly all the money is spent • Slope of Indifference Curve = Slope of Budget Line • Slope of Indifference Curve = -MUH/ MUS = -MRSH, S Thus: S/H = 20/30, H = 1.5S Plug into Income Equation: 20(1.5s) + 30s = 200 S* = 3.33 H* = 1.5s* = 5

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