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# Intermediate microeconomics 3

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### Intermediate microeconomics 3

1. 1. Chapter Three Intermediate Microeconomics Perloff Consumer Behavior Properties of Consumer Preferences 1—Completeness—A consumer can rank bundles of goods in terms of preferences. 2—Transitivity—If I prefer ‘X’ to ‘Y’ and if I prefer ‘Y’ to ‘Z’, by rule I prefer ‘X’ to ‘Z’. 3—More is Better. 4—Continuity—If ‘C’ resembles ‘A’ and I prefer ‘A’ to ‘B’, I must prefer ‘C’ to ‘B’. 5—Strict Convexity. Indifference Curves 1—Bundles of indifference curves farther from the origin are preferred to those of indifference curves closer to the origin. 2—There is a unique indifference curve for each possible bundle. 3—Indifference Curves cannot cross. 4—Indifference Curves slope downward. 5—Indifference Curves cannot be thick. Utility Function—Our measure of pleasure is ordinal rather than cardinal. Ordinal—we know that the relative utility of bundle ‘A’ is better than bundle ‘B’. However, we don’t know how much better! Cardinal-- \$100 is twice as valuable as \$50. Willingness to Substitute between Goods—This is based on Marginal Utility MUX = dU X ---dQ X = UX Consumers must choose a bundle of goods with limited resources (the consumers income is limited). Marginal Rate of Substitution (MRS)—The maximum amount of one good that a consumer will sacrifice (trade) to obtain one more unit of another good. MRS = - dU/dQ1 ----------- = dU/dQ2 - U1 -----U2
2. 2. Consider a Cobb-Douglas Utility Function In this example, we have Utility (U) and two items that contribute to our utility (Q1 and Q2, maybe bananas and apples). U = Q1a Q21-a The Marginal Utility of additional unit of Q1 is: U1 = dU/dQ1 = aQ1a-1 Q21-a This is positive since: 1>a>0 Q1a-1 >0 Q2a-1 >0 However, although it is a fraction, it result of the equation is still positive Consider an example: If a=0.5 The 0.5-1 is -0.5 or the square root. The result of the square root of Q1 is always positive. However, the negative sign puts the result in the denominator of the equation. (if Q1 represents bananas, we cannot have a negative amount of bananas) (if Q2 represents apples, we cannot have a negative amount of apples) The Marginal Utility of additional unit of Q2 is: U2 = dU/dQ2 = (1-a)Q1a Q2-a Once again—this is positive. As a result, the addition of more inputs increases our utility (“More is Better”).
3. 3. However, using the second derivative, we find that this increase in utility is increasing at a decreasing rate. U11 = dU1/dQ1 = (a-1)aQ1a-2 Q21-a This is negative since: 1>a>0 However, although it is a fraction, it result of the equation is still positive Consider an example: If a=0.5 The 0.5-1 is -0.5 or the square root. The result of the square root of Q1 is always positive. However, the negative sign puts the result in the denominator of the equation. However a-1 is Negative resulting in U11 being negative! Q1a-2 >0 Q2a-1 >0 (if Q1 represents bananas, we cannot have a negative amount of bananas) (if Q2 represents apples, we cannot have a negative amount of apples) The Change in Marginal Utility of additional unit of Q2 is: U22 = dU2/dQ2 = (-a)aQ1a Q2-a-1 Once again—this is negative. As a result, the addition of more inputs increases our utility (“More is Better”). Using these calculations, we can develop the Marginal Rate of Substitution: MRS = -U1/ U2 U1 = dU/dQ1 = aQ1a-1 Q21-a U2 = dU/dQ2 = aQ1a Q2-a MRS = -U1 = -dU/dQ1 = aQ1a-1 Q21-a ---- ------------------U2 = dU/dQ2 = (1-a)Q1a Q2-a MRS = -U1 = -aQ2 ----------U2 = (1-a)Q1
4. 4. Example U = Q10.6 Q20.4 (original template) U1 = dU/dQ1 = aQ1a-1Q21-a U1 = dU/dQ1 = 0.6Q1-0.4 Q20.4 U2 = dU/dQ2 = 0.4Q10.6 Q2-0.6 U1 MRS = --U2 - 0.6Q1-0.4 Q20.4 = ----------------0.4Q10.6 Q2-0.6 U1 - 1.5Q2 MRS = --- = ----------------U2 Q1 Consider the results if: Q1 = 12 bananas Q2 = 6 apples MRS = (-1.5)(6/12) = -9/12 = -0.75 This individual is willing to trade 9 units of good 1 (Bananas) to receive 12 units of good 2 (apples) or ¾ of a banana to receive 1 apple. Note—the MRS is not consistent. It is dependent on both the shape of the Utility Curve and the place the individual is at at a given point in time. Perfect Substitutes—A person is indifferent between goods Perfect Complements—A consumer consumes in fixed proportions
5. 5. Budget Constraint The reality of a limited budget limits our actual choice of bundles To find an equilibrium,we use a constrained optimization to solve the problem. Many problems in economics and public policy involve the choice of some optimum solution subject to one or more constraints on the feasible choices. For example, in the model of consumer choice an individual consumer tries to maximize utility subject to his/her budget constraint; the theory of cost assumes a firm minimizes the cost of producing a given level of output subject to the constraint imposed on input choices by the firm’s production function. We Maximize our Utility subject to a budget constraint Theory & Example Maximize: U = Q1a Q21-a Subject to : P1Q1 + P2Q2 = Y Set Up the Lagrangian L = Q1a Q21-a - (P1Q1 + P2Q2 - Y) ∂L/∂Q1 = aQ1a-1Q21-a - P1 = 0 ∂L/∂Q2 = (1-a)Q1aQ2-a - P2 = 0 aQ1a-1Q21-a P1 ------------- - ---- = 0 (1-a)Q1aQ2-a P2 aQ2 P1 ------------- = ---(1-a)Q1 P2 Q2 = (1-a) P1Q1 ----------aP2 ∂L/∂ = P1Q1 + P2Q2 - Y = 0 P1Q1 + P2(1-a) P1Q1 ----------aP2 =Y
6. 6. P1Q1 + (1-a) P1Q1 ----------a =Y P1Q1 + 1P1Q1 - P1Q1 -----a =Y P1Q1 -----a =Y Q1* = a Y ---P1 Since the equation is symmetric, the answer to Q2* is: Q2* = a Y ---P2 As a result, our consumption of Q is a function of the price and our income. Example Maximize: U = Q10.6 Q20.4 P1 = \$15 Subject to : P2 = \$20 P1Q1 + P2Q2 = Y Set Up the Lagrangian L = Q10.6 Q20.4 - (15Q1 + 20Q2 - 300) ∂L/∂Q1 = 0.6Q10.4Q20.4 -  = 0 ∂L/∂Q2 = 0.4Q1a0.6Q2-a0.6 - 20 = 0 0.6Q1-0.4Q20.4  ------------- ---- = 0 0.4Q10.6Q2-a0.6  3Q2 -----2Q1 15 = --- Y = \$300
7. 7. Q2 = 0.5Q1 ∂L/∂ = 15Q1 + 20Q2 - 300 = 0 15Q1 + 20Q2 = 300 Substitute Q2 = 0.5Q1 15Q1 + 20(0.5Q1 ) = 300 25Q1 = 300 Q1* = 12 Q2* = 6