Chapter 2 lecture outline AAEC 3315, Benson 1 Pages 21-45 in Nicholson and Snyder 1. Mathematics for economics 1.1. Much of the math used in microeconomics is focused on finding the optimal value of some variable 1.1.1. “Optimal” being the value that maximizes (or minimizes) some objective 1.1.2. Maximization/minimization of functions 1.2. The portions of this chapter that we skip involve 1.2.1. Identification of certain properties of functions that are useful in building economic models 1.2.2. Techniques to maximize a stream of objectives over time 1.2.3. Mathematical statistics, which is used when an economic model includes uncertainty regarding one or more variables 1.3. This isn’t a math class, but, since our economic models are built using mathematics, it is necessary to understand it (the same way that knowledge of the English language is necessary for understanding of the models I describe) 2. Maximization of a function of a single variable 2.1. Example: suppose that a manager wants to maximize the profits received from selling a particular good 2.1.1. Profits received (π) depend on the quantity of the good sold (q) (1) 2.1.2. If f(q) is known, we could graph it and find its maximum and optimal q* π Quantity q1 q2 q* q3 π3 π2 π1 π*Chapter 2 lecture outline AAEC 3315, Benson 2 2.1.3. If f(q) isn’t known, the manager could try changing the value of q to see what happens to profit 2.1.3.1. Say the manager starts with q = q1 and π = π1; next the manager tries increasing output to q2, and observes that profit increases to π2 2.1.3.2. Profits have increased in response to an increase in q, or (2) 2.1.3.3. As long as ⁄ , profits increase as the manager increases output 2.1.3.4. This occurs until q > q*, in which case increasing output decreases profit, or ⁄ 2.2. The limit of ⁄ as q goes to zero is called the derivative of the function π = f(q), and is denoted dπ/dq, df/dq or f '(q) 2.2.1. The formal definition of the derivative of f at point q1 is (3) 2.2.2. We can denote the value of a derivative at a specific point like this: (4) 2.2.2.1. So, for the example above, we could say that and that 2.2.2.2. What is the value of ? 2.3. First-order condition for a maximum: for a function of 1 variable to be maximized at some point, the value of the derivative of that function at that point must be zero (5)Chapter 2 lecture outline AAEC 3315, Benson 3 2.3.1. Why is this true? What would it mean if ⁄ ? ⁄ ? 2.4. Second-order conditions 2.4.1. It’s possible for the derivative of a function to be zero at a certain point, but not be maximized at that point 2.4.1.1. What is the value of the derivative at the minimum of a function? 2.4.1.2. What is the value of the derivative at x = 14 below?Chapter 2 lecture outline AAEC 3315, Benson 4 2.4.1.3. A zero value of a derivative is a necessary condition for maximization, but is not sufficient; that is, in order for a point to be a maximum, the derivative must be zero, but a derivative can be zero at places that aren’t maxima 2.4.2. The sufficient condition for our manager is that the profit made with a little more than q* and a little less than q* must be less than the profit made with q* 2.4.2.1. Or, ⁄ must be greater than 0 for q < q* and ⁄ must be less than zero for q > q*; this means that ⁄ must be decreasing at q* 2.4.2.2. The derivative of ⁄ must be negative 2.4.3. The derivative of a derivative is called a second derivative and is denoted 2.4.4. So the additional condition that guarantees that q* is a maximum is therefore (6) 2.5. So, equation (5) and equation (6) are together necessary and sufficient conditions for a maximum. 2.6. Example: suppose the relationship between profits (π) and quantity produced (q) is given by (7) 2.6.1. The value of q that maximizes profits can be found by differentiating π(q) (8) setting ⁄ , and solving for q (9) 2.6.2. Is q*=100 a “global maximum?”Chapter 2 lecture outline AAEC 3315, Benson 5 2.6.2.1. It is possible for a function to be maximized only within a certain range 2.6.2.2. Describe the 2nd derivative of the above function 2.6.2.3. What is the 2nd derivative of equation (7)? 2.6.2.3.1. 2.6.2.4. Since the 2nd derivative is always negative, the rate of increase in profits is always decreasing, and, after a certain point, the rate of increase turns negative and will never recover 2.7. What if output (q) is determined by labor (l), according to √ , the firm sells output at $50/unit, and hires labor at $10/unit 2.7.1. How much labor will maximize profits? 2.7.2. , and , so √ 2.7.3. Profit is maximized where ⁄ 2.7.4. What happens if l = 30, or l = 20? 3. Functions of several variablesChapter 2 lecture outline AAEC 3315, Benson 6 3.1. most interesting economic problems involve choosing or identifying the optimal value of multiple variables, not just one 3.1.1. You can’t identify the opportunity cost of a trade-off unless multiple variables are being considered 3.1.2. Consumers choose among multiple goods when maximizing utility, producers choose the quantity of multiple inputs when maximizing profits 3.1.3. We denote the dependence of one variable (y) on a series of other variables (x1, x2, x3, …, xn) as 3.2. We’re still interested in finding the point where y is maximized and understanding the trade-offs that are made to reach that point 3.2.1. Consider an agent changing the x values to find the maximum 3.2.2. With many variables, there is not a single derivative that we can use 3.2.2.1. The steepness of a slope of a mountain depends on the direction you travel 3.2.2.2. The change in the value of y of a multi-variable function depends on how much is being changed in each of the x variables 3.2.2.3. We consider derivatives, therefore, that are defined as the change in y when only one of the x variables is increased, and all the others are held constant 3.2.2.4. The partial derivative of y with respect to (in the direction of) x1 is written 3.2.3. Calculating partial derivatives is done exactly as derivatives of
View Full Document