1 CHAPTER 1. FUNCTIONS AND CHANGE Applications of Functions to Economics Definition. The cost function gives the total cost of producing a quantity of some good. The standard notation is: q = quantity, C(q) = cost. In this section, unless we explicitly say otherwise, we assume that C(q) is a linear function, i.e. C(q) =mq + b. Then the fixed cost is defined as b and the variable cost is defined as m. The revenue function gives the total revenue received for a quantity of some good. Typical notation: q = quantity, R(q) = revenue. Note that R(q) = pq, where p is the price per item. The profit function gives the total profit for a quantity of some good. Typical notation (q) = profit. Note that (q) = revenue minus cost = R(q) C(q). A break-even point is a quantity that produces 0 profit, i.e. (q) = 0. Example 1. (Hughes-Hallett, 4e, 1.4#15(a)) Production costs for manufacturing running shoes consist of a fixed overhead of $650,000 plus variable costs of $20 per pair of shoes. Each pair of shoes sells for $70. Find the total cost, C(q), the total revenue, R(q), and the total profit, (q), as a function of the number of pairs of shoes produced, q, and the break even point. Solution. Let s start with the cost function. In this section we generally assume that C(q) is linear: C(q) =mq + b where b represents the fixed cost (the cost that does not change with q) and m represents the variable cost. We were given m and b directly in the problem: m = 20 ($/pair) b = ($) C(q) = q Now for the revenue function. Recall that in general, R = pq. We were given p directly in the problem, so we can plug this in: R(q) = 70q Now for profit. Recall that in general, (q) = R(q) C(q) (q) =R(q) C(q) = 70q ( q) = 50q To find the break even point we solve (q) = 0. You must be able to put 0 in the right place! It is not (0)! solve (q) = 0 50q = 0 50q = q = = 13000
2 CHAPTER 1. FUNCTIONS AND CHANGE 19 Definition. Let C(q), R(q) and (q) be the cost, revenue and profit functions. We define the functions MC(q), MR(q) and M (q) as follows: MC(q) (naive version) = change in cost resulting from increasing q by 1 MR(q) (naive version) = change in revenue resulting from increasing q by 1 M (q) (naive version) = change in profit resulting from increasing q by 1 We name these marginal cost, marginal revenue, and marginal profit respectively. Marginal here refers to the fact that q has changed by only a small amount. Naive version means we will later redefine these function using derivatives. If C is a linear function, then the marginal cost equals the slope; similar comments apply to R and. Example 2. (Hughes-Hallett, 4e, 1.4#15(b)) Find the marginal cost, marginal revenue and marginal profit for the shoe company (see Example 1). Solution. Marginal cost = slope of C(q) = 20. Marginal revenue = slope of R(q) = 70. Marginal profit = slope of (q) = 50.
3 CHAPTER 1. FUNCTIONS AND CHANGE 20 Definition. The supply curve is the graph relating the quantity q that manufacturers are willing to supply, to the price p for which the item can be sold. Note that price is viewed as the input variable (however economists usually put p on the vertical axis based on historical tradition). As price increases, the quantity q also increases. The demand curve is the graph relating the quantity q that consumers are willing to buy, to the price p. Note that price is viewed as the input variable (however economists usually put p on the vertical axis based on historical tradition). As price increases, the quantity q decreases. The equilibrium point is the intersection of the supply and demand curves. The price value of this intersection is called the equilibrium price and the quantity value is called the equilibrium quantity. Typical notations for these values are p and q respectively. It is assumed that markets tend to move towards, and settle in, equilibrium points. Example 3. (Hughes-Hallett, 4e,1.4#24) One of the tables below represents a supply curve; the other represents a demand curve. (a) Which table represents which curve? Why? (b) At a price of $155, approximately how many items would consumers purchase? (c) At a price of $155, approximately how many items would manufacturers supply? (d) Will the market push prices higher or lower than $155? (e) What would the price have to be if you wanted consumers to buy at least 20 items? (f) What would the price have to be if you wanted manufacturers to supply at least 20 items? I : p ($/unit) q (quantity) II : p ($/unit) q (quantity) Solution. (a) Table II is supply because p and q both increase together. In other words, as p " we have q ". Table I is demand because p and q increase/decrease opposite to each other. In other words, as p " we have q #. (b) Use Table II, look up p = 155. Then q 14. (c) Use Table I, look up p = 155. Then q 24. (d) The market will push prices lower. There are (at least) two ways to see this. Suppose the suppliers have put 24 items on the market, but people only want to buy 14. So there are 10 items left over. What do they do to get rid of the extra 10 items? Lower prices or have a sale.
4 CHAPTER 1. FUNCTIONS AND CHANGE 21 Graphically, we have something like the graph below p supply p =155 demand q where the line p = 155 intersects the two curves above the equilibrium point (the intersection) because that s where we have a smaller value for q on the demand curve than on the supply curve. Since the market will tend to push things to equilibrium, this means the price needs to come down towards the intersection point. (e) Use Table I, look up q = 20. Then p = $143. (d) Use Table II, look up q = 20. Then p = $110 Example 4. Below are some generic supply and demand graphs. economic meaning of the vertical and horizontal intercepts. Interpret the p price p price p 1 supply demand p 0 q quantity q 1 q quantity Solution. For p 0 we have q = 0. This is the price at which manufactures make q = 0 items. Simpler and better: at this price, the manufacturers are not willing to make any items. For p 1 we have q = 0. This is the price at which consumers will buy q =0items. Simpler and better: at this price, consumers won t buy any items. For q 1 we have p = 0. This is the quantity that consumers will buy when p = 0. Simpler and better: this is the number of items that could be given away for free. Example 5. (Hughes-Hallett, 4e, 1.4#25) A company produces and sells shirts. The fixed costs are $7000 and the variable costs are $5 per shirt. (a) Shirts are sold for $12 each. Find cost and revenue as functions of the quantity of shirts, q. (b) The company is considering changing the selling price of the shirts. Demand is q = p, wherep is price in dollars and q is the number of shirts. What quantity is sold at the current price of $12? What profit is realized at this price?
5 CHAPTER 1. FUNCTIONS AND CHANGE 22 (c) Use the demand equation to write cost and revenue as a function of the price, p. Then write profit as a function of price. (d) Graph profit against price. Find the price that maximizes profits. What is this profit? Solution. (a) C(q) = q, R(q) = 12q. (b) q(12) = = 1520, (1520) = R(1520) C(1520) = (c) C(p) =C( p) = ( p) = p Note that R(p) is not 12( p) because the 12 has changed and is now represented by the variable p. Thus R(p) =R( p) = p( p) = 2000p 40p 2 (p) =R(p) C(p) = 40p p (d) A graph is shown below:
6 CHAPTER 1. FUNCTIONS AND CHANGE 23 It appears that the maximum is at about $27.50, and the profit is about $13, 250.
7 CHAPTER 1. FUNCTIONS AND CHANGE 24 Definition. A tax adjusted supply or demand curve takes the original curve, and substitutes a di erent value of p into the formula for q. Let q = D(p) be the demand curve, written as a function of p. Here are two types of shifted demand curves: D(p + c) flat tax/rebate of c: consumer pays p + c on each item D(cp) percentage tax/rebate of c: consumer pays cp on each item Let q = S(p) be the demand curve, written as a function of p. Here are two types of shifted supply curves: S(p + c) flat tax/rebate of c: seller gets p + c on each item S(cp) percentage tax/rebate of c: seller gets cp on each item Comments. For instance, if a supplier side tax of $5 is placed on each item, then when a consumer pays $p, the manufacture only receives p 5. Thus, we substitute p 5 into the supply equation. If a sales tax of 0.08% is placed on each sale, then when a consumer sees a price of $p, they actually end up paying 1.08p to purchase the item. Thus, we substitute 1.08p into the demand equation.
8 CHAPTER 1. FUNCTIONS AND CHANGE 25 Example 6. (Hughes-Hallet, 4e, 1.4#38) In Example 8 (from the text), the demand and supply curves are given by q = 100 2p and q =3p 50, respectively; the equilibrium price is $30 and the equilibrium quantity is 40 units. A sales tax of 5% is imposed on the consumer. (a) Find the equation of the new demand and supply curves. (b) Find the new equilibrium price and quantity. (c) How much is paid on taxes on each unit? How much of this is paid by the consumer and how much by the producer? (d) How much tax does the government collect? Solution. (a) If the price the consumer sees is $p then they actually pay $1.05p after the sales tax. Thus, the number of units the consumers will buy is q = 100 q = p p) The supply curve is unchanged because the manufacturer still receives $p from each unit sold. (b) The equilibrium point is now found from intersecting the new demand curve with the unchanged supply curve: demand = supply p =3p = 5.1p p = $29.41 The equilibrium quantity is q = (29.41) = (c) Each unit sells for $ Of this amount, 5% is taxes, i.e. $1.47 is the total amount of taxes per unit. In some sense, the consumer pays all of this and the produce pays none of it. But the right way to interpret this question is by comparing the prices to the pre-tax equilibrium. Pre-tax, the supplier got $30 per unit, and now they get $29.41 per unit, costing the supplier $0.59 per unit. Pre-tax the consumer paid $30 per unit, and now they pay 1.05(29.41) = $30.88 per unit, costing the consumer $0.88 per unit. Thus, the total tax per unit of $1.47 is split into $ $0.88. (d) The government collects $1.47 in taxes per unit. There are a total of units being sold for a total tax collection of 38.23($1.47) = $56.2.