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MGMT 722
Greg Matlashewski

BUDGET CONTRAINTS: - Suppose that the consumer (household) consumes only two goods (X and Y). Given the Prices of the two goods (P X P Y and the consumer’s income (m), the possible quantities purchasable by the consumer (X ,oY )oare constrained by P X o P YY≤ o e.g., Suppose that a student has a budget for coffee and milk of $100/month. If the price of Coffee (X say) is $1/cup and the price of Milk (Y) is $2/litre, the budget constraint is $1*Q +C$2*Q ≤ $M00 Note: We can define one of the goods (Y say) as a composite commodity, representing all goods other than X. Budget Line: Definition: The maximum combination of two commodities purchasable by a consumer given the prices of the two commodities and the consumer’s money income Rearranging the budget equation for the assumption that all income is spent gives the Budget line Yo = m/P –YP /PX*XY 0 e.g. The budget line for Coffee and Milk with Milk as the Y commodity is →Q =M$100/$2 – $1/$2*Q = 50 C 0.5Q C Since the Opportunity Cost of X = -dY/dX (Recall: Opportunity Cost = the best foregone option) →Opportunity Cost of X = -dY/dX = P /P (= -slope of the budget line) X Y (=> dY = -P /X *YX) e.g. The opportunity cost of a unit of Coffee = -(-0.5) = 0.5 Milk i.e., one unit of Coffee costs ½ litre of Milk - 1 - Q Y Budget Line Constraint m/P Y slope = -P /P X1 Y slope = -P /P Xo Y m/P m/P ' Q X Xo X1 - The diagram above shows the budget line for m,Xo ,YP , and the budget line that ensues given a fall in the price of XX1o P . A change in one of the prices causes a rotation of the budget line around the intercept of the commodity whose price is unchanged Note: A change in Income causes a parallel shift in the budget line: an increase in income shifts it out and an increase in income shifts it in. PREFERENCES AND INDIFFERENCES CURVES For any two consumption bundles (X , Y ) and (X , Y ), define an individual’s preference 0 0 1 1 ranking of each of the two bundles by 1. (X1, 1 ) > (0 , 0 ) means the individual prefer1 (X1, Y )0to 0X , Y ) 2. (X1, 1 ) ~ (0 ,0Y ) means the individual is indifferent betw1en 1X , Y )0and0(X , Y ) Preference Assumptions - 2 - 1. Completeness For every pair of consumption bundles (X , Y ) and (1 , Y1), (X , Y 0 > (0 , Y )1or 1X , Y ) 0 0 0 0 > (X ,1Y ) 1r (X , Y1) ~ 1X , Y ) 0 0 i.e. The consumer prefers one or other bundle or is indifferent between them. In short the consumer does not make contradictory choices. 2. Transitivity (X 1 Y )1≥ (X , Y 0 and0(X , Y ) ≥ 0X ,0Y ) => (X2, Y2) ≥ (X , 1 ) 1 2 2 (where ≥ means either preferred to or indifferent to) 3. Non-Satiation (More is always preferred to Less) If both X or1Y are 1t least equal to X or Y res0ectivel0 and at least one of X or Y is a 1 1 greater amount than X or Y r1specti1ely, then (X 1 Y )1≥ (X , Y 0 0 This assumption is sometimes called the monotonicity of preferences 4. Convex The less one has of a good, the more one requires of the other good in exchange to remain indifferent. [Assumption 1 follows from the assumption that economic agents are rational Assumption 2 is actually an assumption since transitivity need not necessarily hold. Indeed, transitivity for groups of individuals often does not hold. Assumption 3 and 4 are actually assumptions about well-behaved indifference curves. Assumption 3 may also not hold particularly beyond certain amounts and Assumption 4 is likely but not obvious.] - 3 - Indifference Curves Definition: Combinations of two commodities between which the consumer is indifferent (or that give the same level of utility) Q Indifference Curves Y QX Marginal Rate of Substitution (MRS) of Y for X Definition: The amount of Y that the individual will give up for an increase in X while remaining indifferent to the combinations of X and Y → -ΔY/ΔX (or –dY/dX in calculus notation) where Y and X are quantities → the negative of the slope of an indifference curve (for Y on the vertical axis) (Since substitution implies giving up one of the commodities, MRS is a positive number) The Preference Assumptions => 1. Indifference Curves are negatively sloped (Non-satiation) i.e., an increase in one commodity => a decrease in the other for possible indifference => dQ YdQ X 0 or the MRS > 0 -> Indifference curves are not positively sloped (i.e., they don’t curl back) - 4 - 2. Indifference Curves do not intersect
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