Chapter 5: The Economic Approach Applied to Single Variable Optimization A Brief Review of the Economic Approach ¥ The Economic Approach can be applied to optimization problems or equilibrium systems. Economic models are abstract, simplified descriptions of the optimization problem or equilibrium system. ¥ As applied to individual decision-makers, the Economic Approach says that agents act as if they optimize. In other words, they do the best they can under given conditions. ¥ The choices agents make are interpreted as being efficient (or optimal or best) from the perspective of the agent. ¥ The often seemingly ridiculous assumption of optimization (or rationality) is used because (1) it organizes variables in a coherent fashion, (2) agents really do face choices and optimization is one way to make decisions, (3) it helps us understand whatÕs going on, and (4) it generates testable predictions. Did you know that lifeguards are very much aware of the problem they face? They are trained to neither jump right in the water nor to choose the path of least water!1 ¥ Economists do not claim that individuals actually use sophisticated mathematics, rather, economists use mathematics to interpret observed variables with the aid of the optimization framework. Lifeguards are not aware of the mathematics involved in finding the optimal distance to run, then swim. They are conscious of their goal and the fact that either too little or too much running will raise the time it takes to get to the victim. They are trained to use rules of thumb, instinct, experience, and common sense to solve the problem. ¥ Assuming that individuals act as if they optimize enables accurate predictions to be made about the choices observed under varying conditions. It is the economist who imposes the optimization framework in order to make predictions about the lifeguardÕs changes in behavior (in this case, the distance chosen to run on the sand) as an exogenous variable changes (say, their top running speed). ¥ Often, the optimization framework has provided good predictions of actual decisions. For many analysts, this is the acid test: to them, if a theory makes good predictions, then it doesnÕt matter to them how unrealistic are the assumptions of that theory. Having come up with a prediction, the economist would observe lifeguards in action in order to see if the predictions matched the empirical data. 1In fact, Wabash College swim coach Gail Pebworth tells us that lifeguards must Òknow their beach.Ó Lifeguards are trained to constantly recalculate the optimal distance as the time of day and season changes on any particular beach. C5Read.pdf 1 The Structure of Optimization Problems 1) Setting Up the Problem A) Objective function (e.g., utility or profit) ¥ Consumers are utility maximizers ¥ Firms are profit maximizers ¥ Students may be GPA maximizers ¥ The lifeguard is a time minimizer B) Endogenous variables are things the agent CAN choose; C) Exogenous variables are things the agent CANNOT control. ¥ Consumers choose goods; they cannot control prices ¥ Firms choose output or (maybe) price, they cannot control a competitorÕs moves ¥ Students choose the amount of time they want to study, not the content of the course ¥ The lifeguard chooses the amount of time to run on the sand, not the location of the drowning victim Objective Endogenous Exogenous Agent Function Variable Variable Prices, Type and Income, Consumer Max utility number of Tastes and goods Preferences Competitive Price, cost Max profit Output Firm conditions Market Monopoly Max profit Output, Price demand, cost conditions Ability, Time Spent course Student Max GPA Studying in content, each course professor's grading scale Top velocity on land and Distance to Lifeguard Min time in water, run on sand location of victim Sometimes agents have constraints on their choices. That is they face some restriction on the possible values of the endogenous variables. For example, available income restricts the range of options in the consumerÕs choice problem. We will talk about this kind of CONSTRAINED OPTIMIZATION problem later. C5Read.pdf 2 2) Finding the Optimal Solution An example of an optimization problem is the lifeguard problem you explored in C4Lab.xls. The optimal value of Distance_in_Sand (how far to run before jumping in and swimming) was found using ExcelÕs Solver. After you have Set Up the Problem, Solver does the work for you and determines the optimal value of the endogenous variable. In earlier work (C3Lab.xls explored the profit maximization problem), we saw how tables and graphs (using either the Direct Method or Method of Marginalism) could be used to find an optimal solution. Now we will see yet another way to solve optimization problems. ASIDES: ¥ Note how the steps for these various solution strategies are identical. Try to spot the consistent patterns across the different solution strategies. ¥ There is no one best strategy. The idea is that by exploring these different ways to do the Economic Approach, we learn and understand. Solving the Lifeguard Problem with the Method of Marginalism via Calculus Review of the Problem: You are a lifeguard who can run faster than you can swim. You spot a drowning victim. What path should you take to minimize the time necessary to reach the victim? The essential trade-off is that running is faster than swimming, but if you swim all the way, the distance is shortest. What is the path of The drowning victim is least time? here W A b T E R x is a choice variable for the lifeguard. He or she can set it at x=0 (which means jumping right in) or You are here SAND x=c (which means running down the c beach until he or she is directly across from the victim before jumping in). Notation: T = time to reach victim (in minutes) x = distance along sand the lifeguard runs (in yards) b = length of perpendicular connecting victim to beach (in yards c = distance of victim from lifeguard along sand (in yards) vs = velocity of lifeguard running on sand (in yards/minute) vw = velocity of lifeguard swimming in water (in yards/minute) C5Read.pdf 3 Step 1: Setting Up the Problem 1) What is the Objective Function (Goal)? minimize the time it takes to reach the victim (T) ¥ recall that dividing distance (in say, yards) by velocity (yd/min) gets you time (min) ¥ total time to victim is the sum of running time and swimming time ¥ once you choose how far to run on the sand, the swimming distance is also determined because the lifeguard makes a bee-line (along the hypotenuse) for the victim 2) What are the Endogenous Variables? distance to run along the sand (x) before jumping into the water 3) What are the Exogenous Variables? velocity on sand (vs), velocity in water (vw), position of victim (that is, b and c) Step 2: Finding the Optimal Solution The problem can be stated mathematically in this way: (c - x)2 +b2 x minT= + x vs vw ¥ The variable x, under the Òmin,Ó identifies the endogenous variables in the problem. It is clear that there is only one choice variable in this problem. ¥ The first term on the right-hand side (x/vs) is the time it takes to run on the sand; the second is the time it takes in the water. ¥ We divide distance (in yards or miles or feet) by velocity (in yards per minute or miles per hour or feet per second) to obtain travel time. C5Read.pdf 4 Before we continue, letÕs quickly review the other methods of solving this problem: A Review of the Direct Method (using Totals): Two alternative ways of applying it: (1) TABLE: For any set of values of the exogenous variables, have Excel calculate the value of the function for different values of x. Choose the value of x that gives the lowest value for the function, total time to victim. Solving the Lifeguard Problem Using the Direct Method Vs 300 Vw 100 c 100 b 100 Distance On Total Time Sand Time on Sand Time in Water to Victim (yards) (minutes) (minutes) (minutes) 0 0.0000 1.4142 1.4142 5 0.0167 1.3793 1.3960 10 0.0333 1.3454 1.3787 15 0.0500 1.3124 1.3624 20 0.0667 1.2806 1.3473 25 0.0833 1.2500 1.3333 30 0.1000 1.2207 1.3207 35 0.1167 1.1927 1.3094 40 0.1333 1.1662 1.2995 45 0.1500 1.1413 1.2913 50 0.1667 1.1180 1.2847 55 0.1833 1.0966 1.2799 60 0.2000 1.0770 1.2770 65 0.2167 1.0595 1.2761 70 0.2333 1.0440 1.2774 75 0.2500 1.0308 1.2808 80 0.2667 1.0198 1.2865 85 0.2833 1.0112 1.2945 90 0.3000 1.0050 1.3050 95 0.3167 1.0012 1.3179 100 0.3333 1.0000 1.3333 C5Read.pdf 5 (2) GRAPH: Have Excel graph the value of the function for different levels of x, and choose the x that corresponds to the lowest value of the function, total time. A Totals Graph of the Lifeguard Problem 1.60 1.40 1.20 min 1.00 Time on Sand (minutes) 0.80 Time in Water (minutes) 0.60 Time to Victim (minutes) 0.40 0.20 0.00 0 102030405060708090100 Distance on Sand (yards) Although the optimal distance on sand is about 65 yards, that U-shaped curve (or bowl) is pretty flat. That means that mistakes to the right or left (too much or too little running on sand) aren't that big a deal.