Lectures19and20ConstrainedOptimisation

# Lectures19and20ConstrainedOptimisation - Introduction to...

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Unformatted text preview: 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 well-behaved (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 Cobb-Douglas 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 = = = - = 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?...
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Lectures19and20ConstrainedOptimisation - Introduction to...

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