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Basic Mathematics for Economists - Rosser - Chapter 5

5 Linear equations Solve sets of simultaneous linear equations with two or more variables using the substitution and row operations methods. Relate mathematical solutions to simultaneous linear equations to economic analysis. Recognize when a linear equations system cannot be solved. | 5 Linear equations Learning objectives After completing this chapter students should be able to Solve sets of simultaneous linear equations with two or more variables using the substitution and row operations methods. Relate mathematical solutions to simultaneous linear equations to economic analysis. Recognize when a linear equations system cannot be solved. Derive the reduced-form equations for the equilibrium values of dependent variables in basic linear economic models and interpret their meaning. Derive the profit-maximizing solutions to price discrimination and multiplant monopoly problems involving linear functions Set up linear programming constrained maximization and minimization problems and solve them using the graphical method. 5.1 Simultaneous linear equation systems The way to solve single linear equations with one unknown was explained in Chapter 3. We now turn to sets of linear equations with more than one unknown. A simultaneous linear equation system exists when 1. there is more than one functional relationship between a set of specified variables and 2. all the functional relationships are in linear form. The solution to a set of simultaneous equations involves finding values for all the unknown variables. Where only two variables and equations are involved a simultaneous equation system can be related to familiar graphical solutions such as supply and demand analysis. For example assume that in a competitive market the demand schedule is p 420 - 0.2q 1 and the supply schedule is p 60 0.4q 2 If this market is in equilibrium then the equilibrium price and quantity will be where the demand and supply schedules intersect. As this will correspond to a point which is on both 1993 2003 Mike Rosser the demand schedule and the supply schedule then the equilibrium values of p and q will be such that both equations 1 and 2 hold. In other words when the market is in equilibrium 1 and 2 above form a set of simultaneous linear equations. Note that in most of