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Part I: The Nature of Managerial Economics satisfied, while taking the partial derivatives of the Lagrangian with respect to x and z and setting them equal to zero optimize your objective function.
Discovering the secret code: The Lagrangian Multiplier One of the neat things about managerial economics is that it has a lot of useful shortcuts — if you know the secret. One of those shortcuts is the λ used in the Lagrangian function. In the Lagrangian function, the constraints are multiplied by the variable λ, which is called the Lagrangian multiplier. This variable is important because λ measures the change that occurs in the variable being optimized given a one-unit change in the constraint. So, if you’re trying to minimize the cost of producing a given quantity of output, λ tells you how much total cost changes if you decide to produce one more unit of output. This shortcut enables you to quickly assess the relationships between constraints and the variable being optimized. Suppose that your firm has a contract that requires it to produce 1,000 units of a good daily. The firm uses both labor and capital to produce the good. The quantity of labor employed, L, is measured in hours, and the wage is $10 per hour. The quantity of capital employed, K, is measured in machine-hours, and the price per machine hour is $40. Given this information, your firm’s total cost, TC, equals
The firm’s production function describes the relationship between the amounts of labor and capital used and the quantity of the good produced
By contract, q must equal 1,000. You must determine the amount of labor and capital to use in order to minimize the cost of producing the 1,000 units of the good. And remember, at this point, you can use calculus to dazzle everyone! The steps you take in order to dazzle everyone are the following:
1. Create a Lagrangian function. Recognize that the variable you’re trying to optimize is total cost — specifically, you’re trying to minimize total cost. So, your objective function is 10L + 40K. Second, your constraint is that 1,000 units of the good have to be produced from the production function. So your constraint is
1,000 – 20L0.5K0.5 = 0.
