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Unformatted text preview: 1 Introduction to Microeconomics Lectures 19 and 20 Constrained Optimisation Simon Cowan Outline & Many problems in economics have an agent choosing to maximize (or minimize) an objective (e.g. utility, or costs) subject to a constraint & Three methods of solving such problems: & Method 1: draw and think about the economics & Method 2: use the constraint to substitute and convert to an optimisation problem with a single variable & Method 3: Lagrange Multipliers (the best method) & Applications: & Consumer choice and demand functions & Cost functions, saving and borrowing, social welfare maximization The consumer’s optimisation problem 1 2 1 2 1 1 2 2 , 1 2 1 2 1 1 2 2 Max ( , ) subject to The objective function is ( , ) (utility) The choice variables are and . The constraint is . As long as utility is strictly increasing in both go x x u x x p x p x m u x x x x p x p x m + = + = ods, this consumer will want to spend all her income. The slope of the indifference curve & The indifference curve is defined by u ( x 1 , x 2 ) = k & Along the indifference curve x 2 is a function of x 1 , so u ( x 1 , x 2 ( x 1 )) = k & Differentiating with respect to x 1 gives: MU 1 + MU 2 ( dx 2 / dx 1 ) = 0 where MU 1 = ∂ u / ∂ x 1 etc. & Thus dx 2 / dx 1 =  MU 1 / MU 2 Method 1: draw and use economics & Suppose that the utility function is wellbehaved (increasing in both goods, and with strictly convex indifference curves) & Note that the budget constraint is linear & It follows that & the optimal bundle is on the budget line: p 1 x 1 + p 2 x 2 = m & At the optimum the marginal rate of substitution (  1 *slope of the indifference curve) equals the price ratio: MU 1 / MU 2 = p 1 / p 2 & These two equations can be solved to give the optimal x 1 and x 2 A CobbDouglas example (i) 1 2 1 2 1 2 2 1 1 2 1 2 2 1 Suppose , income is 12, 2 and 4. This has positive marginal utilities for all 0 and 0: , Slope of the indifference curve is / / . Indifference curves are stri u x x p p x x u u x x x x MU MU x x = = = > > ∂ ∂ = = ∂ ∂ =  1 2 2 1 2 2 1 1 2 2 3 2 1 1 ctly convex. Fix some level 0 of utility. Along the indifference curve so / . The slope is / / and the second derivative is / 2 / 0. k k x x x k x dx dx k x d x dx k x > = = =  = > 2 Example ii 1 2 2 1 1 2 2 1 1 2 2 2 2 1 The price ratio is / 1/ 2. 1 So MRS = 2 Therefore 2 . Substituting into the budget constraint, 12 2 4 , gives 12 4 4 so 1.5 and 3. Note the implicit assumption that goods ar p p x p x p x x x x x x x x = = = = = + = + = = e perfectly divisible (differentiation wouldn't be feasible otherwise). Example iii 5 10 15 20 25 2 4 6 8 10 x 1 x 2 What happens when the utility assumptions are relaxed? 2 2 1 2 1 2 1 1 Suppose that there is a satiation point, so marginal utilities aren't positive everywhere....
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 Spring '10
 Vines
 Economics, Utility, p2 x2

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