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Course: Microeconomics Text: Varian’s Intermediate Microeconomics 1 The principal behavioral postulate is that a decision-maker chooses its most preferred alternative from those available to it. The available choices constitute the budget set. How is the most preferred bundle in the budget set located? 2 x2 x1 3 Utility x2 x1 4 Utility Affordable, but not the most preferred affordable bundle. x2 x1 5 Utility The most preferred of the affordable bundles. Affordable, but not the most preferred affordable bundle. x2 x1 6 x2 One will choose the bundle on the outermost indifference curve. Therefore, it is the one that just touch the budget constraint. Affordable bundles x1 7 x2 (x1*,x2*) is the most preferred affordable bundle. x2* x1* x1 8 The most preferred affordable bundle is called the consumer’s ORDINARY DEMAND at the given prices and budget. Ordinary demands will be denoted by x1*(p1,p2,m) and x2*(p1,p2,m). 9 When x1* > 0 and x2* > 0 the demanded bundle is INTERIOR. If buying (x1*,x2*) costs $m then the budget is exhausted. 10 Assuming monotonicity and smooth indifference curve, for an interior solution (x1*,x2*), it satisfies two conditions: (a) the budget is exhausted; p1x1* + p2x2* = m (b) the slope of the budget constraint, -p1/p2, and the slope of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*). 11 Condition (b) can be written as: dx2 dx1 MRS MU 1 MU 2 p1 p2 So, at the optimal choice, the marginal willingness to pay for an extra unit of good 1 in terms of good 2 is the same as the price you actually need to pay. If the price is lower than your willingness to pay, you would buy more, otherwise buy less. Adjust until they are equal. 12 The consumer problem can be formulated mathematically as: One way to solve is to use Lagrangian method: 13 Assuming the utility function is differentiable, and there exists interior solution The first order conditions are: 14 So, we have the same condition as before that MRS equals price ratio. 15 So, for interior solution, we can solve for the optimal bundle using the two conditions: 16 Suppose that the consumer has CobbDouglas preferences. a b U( x1 , x 2 ) x1 x 2 Then MU1 MU2 U x1 U x2 a1 b x2 ax1 a b 1 bx1 x 2 17 So the MRS is MRS U / x1 U / x2 a 1 b 1 2 a b 1 1 2 ax x bx x ax2 . bx1 At (x1*,x2*), MRS = p1/p2 so * 2 * 1 ax bx p1 p2 x * 2 bp1 ap2 * 1 x . 18 So now we know that bp1 * * x2 x1 ap 2 * * p1x1 p 2x 2 m. (A) (B) 19 So now we know that Substitute bp1 * * x2 x1 ap 2 * * p1x1 p 2x 2 m. (A) (B) and get bp1 * * p1x1 p 2 x1 m. ap 2 This simplifies to …. 20 * x1 am ( a b )p1 . 21 * x1 am ( a b )p1 . Substituting for x1* in * * p1x1 p 2x 2 m then gives * x2 bm ( a b )p 2 . 22 So we have discovered that the most preferred affordable bundle for a consumer with Cobb-Douglas preferences a b U( x1 , x 2 ) x1 x 2 is * * ( x1 , x 2 ) ( am , bm ( a b )p1 ( a b )p 2 ) . 23 x2 a b U( x1 , x 2 ) x1 x 2 * x2 bm ( a b )p 2 * x1 am ( a b )p1 x1 24 With the optimal bundle: * * ( x1 , x 2 ) ( am , bm ( a b )p1 ( a b )p 2 ) . Note the expenditures on each good: a b ( p1 x , p2 x ) ( a b) m, (a b) m . * 1 * 2 It is a property of Cobb-Douglas Utility function that expenditure share on a particular good is a constant. 25 26 1. Tangency condition is only necessary but not sufficient. 2. There can be more than one optimum. 3. Strict Convexity implies unique solution. 27 But what if x1* = 0? Or if x2* = 0? If either x1* = 0 or x2* = 0 then the ordinary demand (x1*,x2*) is at a corner solution to the problem of maximizing utility subject to a budget constraint. E.g. it happens when the indifference curves are too flat or steep relative to the budget line. 28 The indifference curves are too steep in this case to touch the budget line. The willingness to pay for good 2 is still lower than the price of good 2 even when x2 is zero. 29 The mathematical treatment for corner solution is more complicated. We have to consider also the inequality constraints. (x1 ≥0 and x2 ≥0) We skip the details here. 30 x2 MRS = 1 x1 31 x2 MRS = 1 Slope = -p1/p2 with p1 > p2. x1 32 x2 MRS = 1 Slope = -p1/p2 with p1 > p2. x1 33 x2 MRS = 1 x2 * m p2 Slope = -p1/p2 with p1 > p2. * x1 0 x1 34 x2 MRS = 1 Slope = -p1/p2 with p1 < p2. * x2 0 x * 1 m p1 x1 35 So when U(x1,x2) = x1 + x2, the most preferred affordable bundle is (x1*,x2*) where m (x , x ) p ,0 1 * 1 * 2 if p1 < p2 and m (x , x ) 0 , p 2 * 1 * 2 if p1 > p2. 36 x2 m p2 MRS = 1 Slope = -p1/p2 with p1 = p2. x1 m p1 37 x2 m p2 All the bundles in the constraint are equally the most preferred affordable when p1 = p2. x1 m p1 38 x2 x1 39 x2 x1 40 x2 Which is the most preferred affordable bundle? x1 41 x2 The most preferred affordable bundle x1 42 x2 Notice that the “tangency solution” is not the most preferred affordable bundle. The most preferred affordable bundle x1 43 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 x1 44 x2 U(x1,x2) = min{ax1,x2} MRS = - x2 = ax1 MRS = 0 x1 45 x2 U(x1,x2) = min{ax1,x2} MRS = - MRS is undefined x2 = ax1 MRS = 0 x1 46 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 x1 47 x2 U(x1,x2) = min{ax1,x2} Which is the most preferred affordable bundle? x2 = ax1 x1 48 x2 U(x1,x2) = min{ax1,x2} The most preferred affordable bundle x2 = ax1 x1 49 x2 U(x1,x2) = min{ax1,x2} (a) p1x1* + p2x2* = m (b) x2* = ax1* x2 = ax1 x2* x1* x1 50 (a) p1x1* + p2x2* = m; (b) x2* = ax1*. -Intuitively, it is clear to have (b) because we don’t want to spend money on something that cannot raise my utility. -Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = m which gives * x1 m p1 ap2 ; * x2 am p1 ap2 . 51 x2 U(x1,x2) = min{ax1,x2} * x2 x2 = ax1 am p1 ap 2 * x1 m x1 p1 ap 2 52 Recall MRS p1 p2 As price is the same for all individuals, every individual has the same MRS at the optimal consumption bundle, even if they have very different preferences. Thus, it can be interpreted as a social willingness to pay for good 1 in terms of good 2. 53 In this chapter we put together budget constraint and preference together to analyze consumption decision. Typically, if we have interior solution, then the optimal condition is MRS=p1 / p2. It can also have corner solution if one good is not consumed at all. 54 Next, we will look at comparative statics. In particular, how will prices and income changes affect the optimal consumption bundle? 55