lambda - Extra Handout #2: The Importance of Lambda This is...

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Extra Handout #2: The Importance of Lambda This is meant to clarify some of the confusion related to the interpretation of lambda (the Lagrange multiplier) in a constrained optimization problem. Constraints play an important role in Economics – without the budget constraint (or at least a credit card limit), consumers would be able to purchase anything they want. Without a production possibility frontier (PPF), firms could produce any level of output desired. Mathematically, these constraints are used in the formation of a ‘Lagrangian Equation’, an equation used to maximize some objective given constraints. There is considerable use of lambda in this course’s online detailed texts (for starters see Chapter 5, pg. 4 and Chapter 6, pg. 15). This handout will illustrate: i. The importance of lambda to environmental economics ii. How to calculate lambda iii. The intuitive and mathematical interpretation of lambda iv. How lambda varies when the constraint is modified v. How lambda can be used by regulators to determine an optimal tax level Definition of lambda : Lamba is the marginal value associated with relaxing a constraint. Since this value is not expressed or contracted upon in a market, it is often called the “shadow value” or “shadow price” of the constraint. Example (1) – A single consumer and a budget constraint Suppose that a single consumer wants to maximize utility given their budget constraint. Total Utility = U (X, Y) = 2XY Price of X = P x = $2 Price of Y = P y = $1 Income = I = $30 The consumer wants to maximize U(X,Y)=2XY subject to the income constraint. To solve the problem, we set up a Lagrangian Equation. ) 2 30 ( 2 Y X XY L - - + = l Taking the first order conditions yields the following: 0 2 30 / 0 2 / 0 2 2 / = - - = = - = = - = Y X L X Y L Y X L l l l Up to here, this probably looks familiar from previous courses. You have probably solved these before by setting 2Y=4X, solving for X and Y, and ignoring ?. This is fine if all you are concerned with is the optimal X and Y. However, sometimes knowing the value of ? is important. Solving this, we get X=7.5 and Y=15. Solving for the maximum utility gives U(7.5,15)=2*7.5*15=225.
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Using either of the first equations, we can solve for ? (the Lagrange multiplier). At the maximum utility, ?=15. This can be interpreted as the ‘shadow value’ of the constraint. This means that if income were increased by one dollar (from $30 to $31), then total utility could increase by 15 units.
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This note was uploaded on 08/01/2008 for the course ECON 101 taught by Professor Wood during the Spring '07 term at University of California, Berkeley.

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lambda - Extra Handout #2: The Importance of Lambda This is...

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