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Classroom Lecture Notes - Kings

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Western University
Economics 2150A/B
Hugh Cassidy

Example from last week material. Consumer enjoys consuming X and Y but only if for every 3 units of X, they consume 7 units of Y. U = min{ax,by} U = min{7x,3y} Should switch the constants to make the totals equal. (7x, 3y) (7(3),3(7)) (21,21) 2 4 2 1/2 -1 -1/2 U = x y u1 = x y u2 = -xy u3 = -x y U1 does represent the same, square it U2 does not represent the same, by power of ½ and negative U3 does represent the same, because there are 2 negative components Chapter 4 Budget constraint Budget constraint (BC) – Is the set of affordable bundles given prices and income I ≥ PxX + PyY I = Income Px, Py = Prices If I= 12. Px = 2, Py =1. BC is 12 ≥ 2x + y Budget Line – set of all bundles the consumer can purchase, while spending all his income Graphing Budget Line, I = PxX + PyY Y= -Px/Py x X + I/Py Slope BL: -Px/Py Y –int = I/Py I = 12 Px = 2 Py = 1 Graph it Change in I, won’t change the slope, it shifts the budget line instead Change in Px or Py, will change the slope and 1 or both the intercepts Special Case: Px, Py change by the same percentage, then the slope doesn’t change. Leads to a shift in the budget line only. Optimal Choice Objective: to maximize utility Constraint: Budget constraint, which included income and prices - Consumer wishes to maximize his level of utility (highest indifference curve), while staying in his budget constraint - Maximize U (x,y) subject to a constraint. I = PxX + PyY - Move the BL up as much as possible to the edge of the Budget Constraint area. Should only have one point contacting the budget line, the line will be tangent to the BL. - (x,y) bundle that solves this problem is called the optimum. Interior Optimum: positive amounts of both goods, x*>0, y*>0 Corner Optimum: zero amount of one good: x* = 0, y* = 0 Solving for Optimum - Different procedures depending on the preferences (Perfect Compliments, Perfect Substitutes, Cobb-Douglas) c d Cobb – Douglas U = Ax y - Always interior optimum - Optimum occurs when slope BL = slope of indifference curve - MRS = Px/Py - At optimum, MRS = Px/Py o Tangency Condition - MRS = MUx/MUy o So MRS = Px/Py = MUx/MUy = Px/Py o MUx/Px = MUy/Py o Where MUx/Px is the marginal utility of X per dollar - If MRS > Px/Py  MUx/Px > MUy/Py o Can’t be optimal, should spend more on X and less on Y Solving for the bundle: two equations, I = PxX + PyY Budget Line MRS = Px/Py Tangency Condition Example I = 12, Px = 2, Py = 1 U = xy Solve for optimal bundle 1) BL 12 = 2x + y 2) Tangency Condition MRS = MUx/MUy set equal to Px/Py - MRS = c/d x y/x = 1/1 x y/x = y/x - MRS = Px/Py  y/x =2  y = 2x - Substituting: 12 = 2x + (2x) = 4x X= 3 - Y = 2x = 2(3), y = 6 - Optimum is (x,y) = (3,6) Ray: A line the origin of all possible optimal (x,y) bundles, derived from the tangency condition of MRS = Px/Py Previous Example now i
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