What WeÕve Done So Far: Analyzing Single Variable Unconstrained Optimization Problems Chapters 3, 4, 5, 6, and 7

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1 Chapter 8: Two- (and n-) Variable Unconstrained Optimization via CALCULUS What WeÕve Done So Far: Analyzing Single Variable Unconstrained Optimization Problems Chapters 3, 4, 5, 6, and 7 DEFINITION: Single Variable Unconstrained Optimization means that the agent controls ONE variable (like quantity produced, distance to run on sand, or size of shipment). INITIAL SOLUTION: Remember the five steps for solving unconstrained optimization problems with a single variable. STEP 1) Write the problem in a mathematical expression in the following form: max (or min) endogenous variable objective function In this step you specify what you want to do to the objective function, identify the endogenous variable, and write the objective function as a function of the endogenous and exogenous variables. This step requires that you have already SET UP THE PROBLEM (and know the goal, endogenous, and exogenous variables). STEP ) Find the derivative of the objective function with respect to the endogenous variable. In this step you explore how the objective function varies as the endogenous variable changes. By plugging in values for the exogenous variables, we can determine the value of the derivative for any value of the endogenous variable. Only one value of the endogenous variable, however, is the optimal or best value. C8Read.pdf 1

2 STEP 3) Set the derivative equal to zero. In other words, find the value of the endogenous variable for which the change in the objective function is zero. This is the one position of the endogenous variable that is the optimal value. Setting the derivative equal to zero works because the derivative is simply the slope of the objective function. The value of the endogenous variable which makes the slope zero corresponds to the Òtop of the hilló (for a maximization problem) or the Òbottom of the bowló (for a minimization problem). 1 STEP 4) Solve for the optimal value of the endogenous variable as a function of the exogenous variables, called a "reduced-form expression." This boils down to using algebra to write the solution in a nice form. STEP 5) Find the optimal value of the objective function. Plug in the optimal value of the endogenous variable into the objective function to find out the optimal value of the objective function, called the "maximum (minimum) value function." COMPARATIVE STATICS: Explore how the optimal value of the endogenous variable and the optimal value of the objective function vary as an exogenous variable changes, ceteris paribus. This can be done by graphing the reduced forms (endo*=ä(exo other exo var) and objfn*=ä(exo other exo var)). Taking the derivative of a reduced form with respect to an exogenous variable is a quick way to make a qualitative prediction and to determine if the relationship is linear or non-linear. NOTE: An alternative to calculus (for finding the optimal solution) and the Method of the Reduced Form (for comparative statics) is to use ExcelÕs Solver and the Comparative Statics Wizard. By repeatedly solving the problem for a variety of values of a single exogenous variable, you can track optimal values as functions of exogenous values. This numerical comparison of different concrete solutions is called the Method of Actual Comparison. 1 The second derivative is required to determine whether you are actually at a minimum or a maximum. In order to keep things simple, all of the problems in this book have nice, global optimum solutions, so all you need to do is make sure the first-order condition is satisfied. C8Read.pdf

3 A Schematic Representation of Where WeÕve Been and Where WeÕre Headed... Survey the map below and make sure you understand how the things we've done so far tie together: Economic Approach Optimization Equilibrium Single Variable Unconstrained Steps 1 and : Setup and Solving Constrained Multivariable We are here! Chaps 8+9 Totals: Tables Graphs Marginals: Table Graph Chap 3 Calculus Chap 5 Excel's Solver Chap 4 Chap 7 Step 3: Comparative Statics Method of Actual Comparison Method of Reduced Form Comparative Statics Wizard Chap 6 Paper and Pencil Recalculation Calculus Chap 5 C8Read.pdf 3

4 Analyzing (and n-) Variable Unconstrained Optimization Problems THE GENERAL RECIPE We now want to look at the optimization procedure when there is more than one endogenous variable. The steps are basically the same, but they are slightly more complicated because of the fact that we have more than one choice variable to worry about. We are in search of the top of a 3D hill, not a D curve. DEFINITION: Two (or n-) Variable Unconstrained Optimization means that the agent controls TWO (or more) variables. The agent can set these variables at any values. INITIAL SOLUTION: Steps for Solving Multi-Variable Unconstrained Optimization Problems STEP 1) Write the problem in a mathematical expression in the following form: max (or min) endogenous variables objective function Note how there are now two (or more) endogenous variables. STEP ) Find the partial derivatives of the objective function with respect to the endogenous variables. The partial derivative of a function with respect to a variable is obtained by asking how the function changes as that variable alone is changed, holding all other variables (including other endogenous variables) constant. One can think of a partial derivative as a marginal objective function with respect to a particular endogenous variable. Notation: f(x,y) x is the partial derivative of f (which is a function of both x and y) with respect to x. The Ò Ó symbol (like a cursive "" with the tail cut off) is common notation for the partial derivative operator (analogous to the ÒdÓ operator weõre used to). In fact, the same derivative rules apply. Just remember to treat other endogenous variables as constants when you are taking a partial derivative. Example: fx,y =x y 3. What are the two partial derivatives of f(x,y)? f( x, y ) = xy 3 x f( x, y ) y = 3 x y C8Read.pdf 4

5 STEP 3) Set the partial derivatives equal to zero. You should have as many equations as there are endogenous variables. The endogenous variables should all carry asterisks (*) to denote that they are being placed at their optimal values. These equations may be UNRELATED (and therefore easy to solve) or RELATED (and therefore harder to solve). The solutions you obtain for the endogenous variables (in step 4) are the optimal values. The equations may be so intertwined (with messy powers and complicated expressions) that NO CLOSED FORM solution is possible. In other words, they cannot be solved analytically or cleanly in terms of the optimal value of the endogenous variable as a function of exogenous variables alone. We will show examples of all three of these cases below. STEP 4) Solve for the optimal values of the endogenous variables. That is, solve the set of n simultaneous equations (called first-order conditions) for the n optimal values of the endogenous variables, with one equation for each endogenous variable. When we use the first-order conditions to solve for the various endogenous variables, we are essentially rewriting this set of simultaneous equations as endogenous variable 1 *=f 1 (exogenous variables) endogenous variable *=f (exogenous variables)... endogenous variable n *=f n (exogenous variables) STEP 5) The final step is to evaluate the objective function at the optimal values of the endogenous variables: objective function* = Ä(endogenous variables* exogenous variables) COMPARATIVE STATICS: Explore how the optimal values of the endogenous variables and the optimal value of the objective function vary as an exogenous variable changes, ceteris paribus. This can be done by graphing the reduced forms (endo 1 *=Ä(exo other exo var), endo *=Ä(exo other exo var), and so on for the endo* and objfn*=ä(exo other exo var)). Taking the derivative of a reduced form with respect to an exogenous variable is a quick way to make a qualitative prediction and to determine if the relationship is linear or non-linear. NOTE: An alternative to calculus (for finding the optimal solution) and the Method of the Reduced Form (for comparative statics) is to use ExcelÕs Solver and the Comparative Statics Wizard. By repeatedly solving the problem for a variety of values of a single exogenous variable, you can track optimal values as functions of exogenous values. This numerical comparison of different concrete solutions is called the Method of Actual Comparison. C8Read.pdf 5

6 EXAMPLE OF A TWO-VARIABLE PROFIT MAXIMIZATION PROBLEM The Story: Suppose a firm is trying to decide, not only how much to produce (Q), but also how much advertising to do (A). It is, as usual, trying to maximize profits, defined as total revenues minus total costs. Total Revenues: As usual, total revenues equal price times quantity. However, two factors influence the firm's price: (1) the price of a close substitute and () advertising by the firm. If the firm did not advertise at all in Case 1 below or did very little advertising in Case below, the firm's price would be the same price as that of the close substitute product. Advertising allows the firm to sell its product at a higher price than that of the close substitute product and so generate more revenues. Total Costs: Total costs are the sum of costs of production and costs of advertising. More on the Variables: Q, the number of units to produce, is measured in physical units of stuff produced (e.g., barrels of oil, bushels of wheat, reams of paper.) A, the number of units of advertising, is also measured in actual units of advertising, like number of commercials per week or pages of print ads. The given price of the close substitute product is P. This may be confusing for P does not represent the price of the firm's product, P=price of close substitute's product. The Functions: Revenues: The general expression of the revenue function is price times quantity. Advertising increases the firm's price to P times the advertising factor. We will explore two concrete possibilities of the advertising factor (or, how total revenue is affected by advertising): Case 1) Advertising Factor = (1 + A/Q) so that the firm's price = P (1+A/Q) Total Revenue = P (1+A/Q) Q = PQ + PA C8Read.pdf 6

7 Case ) Advertising Factor = A so that the firm's price = PA Total Revenue = P A Q = PAQ Advertising is much more powerful here than in Case 1. Costs: Production Costs are, as before, modeled simply by Q. This says that the cost of producing any given amount is the square of that amount. Advertising Costs will have two variants: The simple case will simply be that advertising costs equal A 3. A more complicated version says that advertising is costly in and of itself and it raises production costs (because, for example, higher quality must be delivered when it is touted in an ad) so that advertising costs are A 3 + AQ. Thus, Total Costs can be of two types: Case 1) Q + A 3 Case ) Q + A 3 + AQ By analyzing three different profit functions (based on these alternative revenue and cost functions), we will see examples of the three types (unrelated, related, and no closed form) of Two Variable Unconstrained Optimization problems. In the next few pages we explore the following types: Type 1: Unrelated Partial DerivativesÑCase 1 Revenues with Case 1 Costs Type : Related Partial DerivativesÑCase Revenues with Case 1 Costs Type 3: No Closed FormÑCase Revenues with Case Costs TYPE 1: UNRELATED Partial Derivatives LetÕs suppose the firm has a revenue function based on the Revenues Case 1 Advertising Factor, (1 + Q/A), so that the revenue function is PQ + PA and the total cost function that is Q + A 3. C8Read.pdf 7

8 We can work our way through the five steps in solving the problem: STEP 1) Write the problem in a mathematical expression: max π = PQ + PA - (Q + A 3 ) Q, A STEP ) Find the partial derivatives of the objective function with respect to the endogenous variables. π Q = P Q π A = P 3 A STEP 3) Set the partial derivatives equal to zero. You should have as many equations as there are endogenous variables. The endogenous variables should all carry asterisks, *, to denote that they are being placed at their optimal values. P Q = 0 P 3 A = 0 These two equations are said to be UNRELATED (and therefore easy to solve). STEP 4) Solve for the optimal values of the endogenous variables. That is, solve the set of n simultaneous equations (called first-order conditions) for the n optimal values of the endogenous variables, one equation for each endogenous variable, Obviously, Q*=P/ and A*=(P/3) 1/. If P=$10/unit, then Q*=5 and A*Å1.8. STEP 5) The final step is to evaluate the objective function at the optimal values of the endogenous variables: π = P( P + P 3 ) P P 3 3 If P=$10/unit, then π*å$37.. Read, "If the price is ten dollars per unit, then the maximum profit possible is thirty-seven dollars and twenty cents." C8Read.pdf 8

9 COMPARATIVE STATICS: Having found reduced-form expressions, we can explore comparative statics via the Method of the Reduced Form. It is especially easy in this case, since we have only one interesting exogenous variable, the price of the close substitute product (P). Thus, comparative statics amounts to finding and evaluating dq*/dp, da*/dp, and dπ*/dp at given values of the exogenous variables. dq*/dp = 1/ is positive which means that as you increase P, Q* increases. In addition, we say that Q* is linear in P because any increase in P always results in the same 1/ unit increase in Q*. Clearly, a presentation graph of Q*=Ä(P) is a straight line. da*/dp and dπ*/dp are also positive (for positive values of P), but A* and π* are non-linear in P. This means that the change in A* or π* for a given change in P depends on the initial position of P. A presentation graph of A*=Ä(P) or π*=ä(p) is a curve. So, our qualitative predictions from this problem are that increases in price of the substitute product (P) will lead to increases in optimal, observed quantity produced, advertising, and profits. Perhaps the most interesting result of the comparative statics analysis is that, as the price of the close substitute product increases, a profit-maximizing firm will increase its advertising. TYPE : RELATED Partial Derivatives LetÕs suppose the firm has a revenue function based on Revenues Case : Advertising Factor, A, so that the revenue function is PAQ and the Case 1 Total Costs so that the total cost function that is simply Q + A 3. We can work our way through the five steps in solving the problem: STEP 1) Write the problem in a mathematical expression: max π = PAQ - (Q + A 3 ) Q, A STEP ) Find the partial derivatives of the objective function with respect to the endogenous variables. π Q = PA Q π A = PQ 3A STEP 3) Set the partial derivatives equal to zero. You should have as many equations as there are endogenous variables. The endogenous variables should all carry asterisks (*) to denote that they are being placed at their optimal values. PA Q = 0 PQ 3A = 0 These two equations are RELATED since they both contain Q* and A*. C8Read.pdf 9

10 STEP 4) Solve for the optimal values of the endogenous variables. That is, solve the set of n simultaneous equations (called first-order conditions), for the n optimal values of the endogenous variables, one equation for each endogenous variable, Although there are many strategies for solving systems of equations, substitution and dividing one equation by the other are often easy ways to isolate variables on one side. As an example of substitution, look at the solution strategy below: PA Q = 0 > Q = PA then, substituting Q into the second equation P PA 3A and solving for A A = P 6 = 0 which means that by substituting back into Q Q = P P 6 = P 3 1 NOTE: Q*=PA*/ is NOT a reduced form because it has A* (an endogenous variable) on the righthand side. A true reduced form is endo*=ä(exogenous variables alone). A = Q = P 6 is a reduced form. P 3 1 is a reduced form. If P=$10/unit, then Q*=83.33 and A*= STEP 5) The final step is to evaluate the objective function at the optimal values of the endogenous variables: π = P P 3 1 P 6 P 3 1 P 6 3 If P=$10/unit, then π*=$315. Read, "If the price is ten dollars per unit, then the maximum profit possible is two thousand three hundred and fifteen dollars." C8Read.pdf 10

11 COMPARATIVE STATICS: As before, armed with the reduced forms, we can explore dq*/dp, da*/dp, and dπ*/dp. They are all positive and non-linearñeach of the three presentation graphs with respect to P is a curve. This means that increases in P lead to increases in Q*, A*, and π*, but the actual amount of the increase for a unit increase in P depends upon the initial value of P. For example, an increase in P from 10 to 11 yields an increase in A* of about 3.4 units (from to 0.17); while an increase in P from 0 to 1 yields an increase in A* of about 6.83 (from to 73.5). TYPE 3: NO CLOSED FORM LetÕs suppose now the firm has Case Revenues (Advertising Factor is A) so that the revenue function is PAQ, and a Case total cost function, Q + A 3 + AQ. We will work our way through the five steps in solving the problem: STEP 1) Write the problem in a mathematical expression: max π = PAQ - (Q + A 3 + AQ ) Q, A STEP ) Find the partial derivatives of the objective function with respect to the endogenous variables. π Q = PA Q AQ π A = PQ 3A Q STEP 3) Set the partial derivatives equal to zero. You should have as many equations as there are endogenous variables. The endogenous variables should all carry asterisks (*) to denote that they are being placed at their optimal values. PA Q A Q = 0 PQ 3A Q = 0 These two equations are RELATED since they both contain Q* and A*. Look at the two first-order conditions carefully. Can you think of a way to solve for A* and Q* as a function of P the way did in the previous example? C8Read.pdf 11

12 STEP 4) Solve for the optimal values of the endogenous variables. That is, solve the set of n simultaneous equations (called first-order conditions), for the n optimal values of the endogenous variables, one equation for each endogenous variable, In Type : Related Partial Derivatives above, we said, ÒAlthough there are many strategies for solving systems of equations, substitution and dividing one equation by the other are often easy ways to isolate variables on one side.ó ThatÕs not going to be enough, in this case. In fact, itõs going to take a lot of work to reduce these two equations to the ÒcleanÓ form of Q*=Ä(P) and A*=Ä(P). Running into a NO CLOSED FORM barrier is not the end of the world. The solution actually exists and is captured by the step 3 partial derivative equations that are set equal to zero. What is needed are numerical methods to approximate the solution for given values of P. HereÕs where Microsoft Excel can come to the rescue. We can use ExcelÕs Solver to find a concrete, numerical solution to the problem for a given value of P. If P=$10/unit, then Q*=3.68 and A*=.78. MORE ON NO CLOSED FORM: Let's make sure you understand what's going on here. The problem is NOT that we have a "letter" for P instead of a number. Even if you had, say, P=$5/unit, you still would find the two equations with two unknowns (Q* and A*) extremely difficult to solve with algebra in terms of Q*=some number and A*=some number because of the powers ( and 3) in the equations. But just because you can't get a "clean" answer via algebra doesn't mean that an answer is nonexistent. It's there alrightñyou just can't express it as a pretty, reduced form and you need other methods (here's where Excel's Solver helps) to get the answer. STEP 5) The final step is to evaluate the objective function at the optimal values of the endogenous variables: It actually can be doneñif you are really persistent and find a book that shows the biquadratic (or quartic) roots. But the point is that itõs tedious and time consuming and that there really are problems that have absolutely no closed form, analytical, ÒcleanÓ solutions. We'll pretend that this is one of those... C8Read.pdf 1

13 Since we donõt have a reduced form for the optimal values of the endogenous variables (Q*=Ä(P) and A*=Ä(P)), we cannot find a maximum value function, π*=ä(p). We are left, once again, with numerical solutions. If P=$10/unit, then π*=$9.6. As before, the solution is there, we just can't get at it through the maximum value function. COMPARATIVE STATICS: We can NOT explore dq*/dp, da*/dp, and dπ*/dp since we donõt have reduced forms. Our only alternative is to patiently use the Method of Actual Comparison to build a presentation table and graph based on numerical solutions. We want something like the table below to be able to make comparative statics predictions and draw presentation graphs. Price of the Close Substitute Q* A* π* Before, the only way to do this was to crunch through repeatedly resolving the same optimization problem as a single exogenous variable (in this case, P) changed. With ExcelÕs Solver and the Comparative Statics Wizard, the tediousness of this is alleviated. We will do this in the next chapterõs lab. C8Read.pdf 13

14 Summary: This reading reviewed what we've done so far with single variable unconstrained optimization problems and introduced two variable unconstrained optimization problems. Two (and n-) variable unconstrained optimization problems can be solved via calculus, just like single variable problems, by following the same five steps. The big difference between single and multi-variable unconstrained optimization problems is that the former has only one first-order condition while the multi-variable case has as many firstorder equations as there are endogenous variables. This forces us to use partial derivatives to find the first-order conditions. You take the partial derivative of the objective function with respect to each endogenous variable. This gives a set of equationsñone equation for each partial derivative. The fact that we have a simultaneous system of equations (instead of a single first-order condition like in the single variable unconstrained optimization case) can cause troubleñof two kinds: The equations can be difficult to solve. You need to be good at algebra, be careful, and be patient. It's tough to be all three! If the equations are unrelated, then it's usually pretty easyñit's essentially the same as the single variable situation. If they are related, it starts to get tougher. You might have to substitute one equation into another or divide one by another to cancel out a variable. The secret is to look for ways to reduce the system from n equations and n unknowns, to n-1 equations and n-1 unknowns. The equations can be impossible to solve. There is no closed form. The fact that an analytical solution could not be found used to present a formidable hurdle. Nobody wants to crank though repeated resolving of an optimization problem with pencil and paper. The chances of mistake are high and it's pretty boring. Microsoft Excel's Solver and the Comparative Statics Wizard really pay off here. Unlike the single variable case where it may have seemed that using Excel was not really worth it, it is clear that the computer offers a tremendous savings in time and effort. We leave it to you to judge exactly how powerful Excel can be when you tackle two variable unconstrained optimization problems in the next lab, C9Lab.xls. C8Read.pdf 14