Lecture # 19 - Optimization with Equality Constraints (Cont.)

Lecture # 19 - Optimization with Equality Constraints (cont.) More Examples: 1. Utility Maximisation In our last lecture we assume a consumer with has utility function U (x, y)=Axαy1 α • − that faces the budget constraint px x + py y = m. The Lagrangian was: Z (x, y, λ)= · · α 1 α Ax y + λ [m px x py y] , so the first order conditions were: − − · − · px x + py y = m · · ∂Z α 1 1 α = αAx − y − λpx =0 ∂x − ∂Z α α =(1 α) Ax y− λpy =0 ∂y − − We can express the last two equations as follows: αAxα 1y1 α (1 α) Axαy α λ = − − = − − (1) px py α 1 1 α ∂U Notice that αAx y = = MUx, that is the marginal utility of good x. Similarly • − − ∂x α α ∂U (1 α) Ax y = = MUy. So we can write: − − ∂y MU MU λ = x = y (2) px py So, in order to maximise utility, consumers must allocate their budgets so as to equalize the ratio of marginal-utility-to-price The condition in (2) can be restated as: • MU p x = x (3) MUy py — Thelefthandsiderepresentsthemarginal rate of substitution between the two goods, which in turn comes from the slope of the indifference curve. Recall that the indifference curve gives the combination of goods x and y that ∗ yields a constant utililty level U0. Then in a given indifference curve we must have dy MUx dU = MUx dx + MUy dy =0 = · · → dx −MUy — The right hand side is the negative of the slope of the budget constraint. 1 2. Cost Minimisation: Suppose a firm uses capital and output to produce good Y according to the production • function Y = F (K, L) What is the least-cost input combination for the production of a specified output level • Y0? The constrained optimisation can be written as: • min rK + wL subject to F (K, L)=Y0 K,L The Lagrangian is: • Z = rK + wL + λ [Y0 F (K, L)] − The first order conditions are: • F (K, L)=Y0 ∂Z = r λFK =0 ∂K − ∂Z = w λFL =0 ∂y − where FK and FL are respectively the marginal product of capital and labor. We can express the last two equations as follows: r w λ = = (4) FK FL which means that, at the point of optimal output combination, the ration of marginal- product-to-input-price must be the same for each input 2 Alternatively: • F w L = (5) FK r This has a similar interpretation than in the case of utility maximisation: — Thelefthandsiderepresentsthemarginal rate of technical substitution of labor to capital, which in turn is the negative of the slope of the isoquant associated to the production level Y0 indifference curve. The isoquant gives the combination of inputs x and y that yields production ∗ level Y0. Then we must have dK FL dY = FK dK + MUL dL =0 = · · → dL −FK — The right hand side is the negative of the slope of the isocost. An isocost is the locus of the input combinations that entails a given cost level ∗ C0, so that C0 = rK + wL But we also need to verify the second order conditions. For the bordered Hessian we • need five derivatives: — ZKK = λFKK − — ZLL = λFLL − — ZKL = λFKL − — gx = FK — gy = FL As a result, the bordered Hessian is: • 0 FK FL H = FK λFKK λFKL − − FL λFKL λFLL − − and we require that H < 0 ¯ ¯ ¯ ¯ 3 Interpretation of the Lagrangian multiplier Consider the Lagrangian function: • Z (x, y, λ)=f (x, y)+λ [c g (x, y)] − In general, the solutions to the Lagrangian, let’s call them x and y , are both functions of • ∗ ∗ c, as is λ. So in turn the associated value of f (x, y) is also a function of c : • f ∗ (c)=f (x∗ (c) ,y∗ (c)) (6) f (c) is called the (optimal) value function of the problem • ∗ Under certain regularity condition, we can prove that • df (c) ∗ = λ (c) dc So we can interpret the Lagrange multiplier as the rate at which the optimal value of the • objective function changes with respect to changes in the constraint constant c Proof: Take the differential of (6) : • df ∗ (c)=df (x∗,y∗)=fx dx∗ + fy dy∗ · · but from the first order conditions we have fx = λgx and fy = λgy, so df ∗ (c)=λ [gx dx∗ + gy dy∗] · · Now, if you take the differential of the constraint g (x∗ (c) ,y∗ (c)) = c : gx dx∗ + gy dy∗ = dc · · df ∗(c) Therefore df ∗ (c)=λdc, or better dc = λ (c) In economic applications, c often denotes the available stock of a particular resource, and • f (x, y) denotes utility or profit.. Then λdc measures the (approximate) change in utility or profit when there is a change in the availability of the resource by dc. As such, λ is called a shadow price of the resource. 4.

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