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Basic Mathematics for Economists - Rosser - Chapter 11

11 Constrained optimization Solve constrained optimization problems by the substitution method. • Use the Lagrange method to set up and solve constrained maximization and constrained minimization problems. • Apply the Lagrange method to resource allocation problems in economics. | 11 Constrained optimization Learning objectives After completing this chapter students should be able to Solve constrained optimization problems by the substitution method. Use the Lagrange method to set up and solve constrained maximization and constrained minimization problems. Apply the Lagrange method to resource allocation problems in economics. 11.1 Constrained optimization and resource allocation Chapters 9 and 10 dealt with the optimization of functions without any constraints imposed. However in economics we often come across resource allocation problems that involve the optimization of some variable subject to certain limitations. For example a firm may try to maximize output subject to a budget constraint for expenditure on inputs or it may wish to minimize costs subject to a specified output being produced. We have already seen in Chapter 5 how constrained optimization problems with linear constraints and objective functions can be tackled using linear programming. This chapter now explains how problems involving the constrained optimization of non-linear functions can be tackled using partial differentiation. We shall consider two methods i constrained optimization by substitution and ii the Lagrange multiplier method. The Lagrange multiplier method can be used for most types of constrained optimization problems. The substitution method is mainly suitable for problems where a function with only two variables is maximized or minimized subject to one constraint. We shall consider this simpler substitution method first. 11.2 Constrained optimization by substitution Consider the example of a firm that wishes to maximize output Q f K L with a fixed budget M for purchasing inputs K and L at set prices PK and PL. This problem is illustrated in Figure 11.1. The firm needs to find the combination of K and L that will allow it to reach 1993 2003 Mike Rosser K Figure 11.1 the optimum point X which is on the highest possible isoquant within the budget constraint .