1 Supply and Demand, Exams of Law

Download 1 Supply and Demand and more Exams Law in PDF only on Docsity! These notes essentially correspond to chapter 2 of the text. 1 Supply and Demand The …rst model we will discuss is supply and demand. It is the most fundamental model used in economics, and is generally used to predict how equilibrium prices and quantities will change given a change in the underlying determinants of supply and demand. 1.1 Demand Recall the Law of Demand from your principles of economics courses: Law of Demand: There exists an inverse relationship between the price of a good and the quantity demanded of the same good This law has been veri…ed using "real-world" data for many goods. If the law holds, then we can draw a demand curve if we place price on the y-axis and quantity on the x-axis. Demand curves for which the law of demand holds will be downward-sloping. They may be linear or non-linear, although we will generally work with linear demand curves for simplicity. Graphed below are portions of a linear and non-linear demand curve: 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Quantity Price 1.1.1 A simpli…cation of reality The demand curve is really a simpli…cation of reality. There are many factors that go into determining the demand for a speci…c product (how many consumers in a market, the prices of related goods, the amount of income consumers have, etc.), but when we graph the demand curve we only consider the price of the good and the quantity demanded. In a sense, we are saying that the quantity demanded of a good is only a function of the price of the good (or the own-price of the good as I have called it). Mathematically, we say that QD = f (Pown). For the more complex case we could write, QD = f (Pown;Psub; Pcomp; Income;# of consumers). However, to show this in a picture we would need more and more dimensions (2 dimensions for QD and Pown, 3 dimensions for QD, Pown and the price of one substitute, 4 dimensions for QD, Pown, the price of one substitute and the price of one complement, etc.). Since it is di¢ cult to draw and picture such higher dimension objects we only consider the graph of QD and Pown. 1.1.2 Demand functions and inverse demand functions As you can see above, we will be working with demand equations in the course. When QD is isolated, so that QD = f (Pown), this is called a demand function. If Pown is isolated, so that Pown = f (QD), then this is called the inverse demand function. (Note: If you are going to graph a demand curve you need to use the inverse demand function, since price is on the y-axis and quantity is on the x-axis and we typically think of graphing equations of the form y = f (x).) To …nd the inverse demand function when given the demand function you simply have to solve for Pown. Suppose that you have the linear demand function QD = 12 6Pown. Then the inverse demand function 1 would be: Pown = 2 1 6QD. In general, our linear demand functions will take the form of QD = a bPown. (Note: Be careful here. The inverse demand functions may also be generally written as Pown = a bQD. However, the a’s and b’s will not be the same. If a demand function is written as QD = a bPown, then the inverse demand function is actually Pown = a b 1 bQD. If an inverse demand function is written as Pown = a bQD, then the demand function is actually QD = a b 1 bPown. The main point: KNOW WHICH FUNCTION YOU ARE WORKING WITH!!!) Examples A simple demand function example is one where QD is only a function of Pown. Thus, QD = 286 20Pown is a simple demand function. If we rewrite this as the inverse demand function we get: Pown = 14:30:05QD. We can now graph the inverse demand function on our plane using routine methods. The number 14.3 is the price (or y) intercept. The slope of the line is (0:05). Note that the slope of the demand curve will always be negative if the law of demand holds. A more complex demand function takes the form of QD = f (Pown;Psub; Pcomp; Y ). You should note that Y is income. Writing this out we get: QD = 171 20Pown + 20Psub#1 + 3Psub#2 + 2Y . The inverse demand function would be: Pown = 8:55 0:05QD + 1Psub#1 + 0:15Psub#2 + 0:1Y . Attempting to graph this would be di¢ cult, so we hold the values of the variables other than Pown and QD at their constant (or average or ceteris paribus) levels. Suppose Psub#1 = 4, Psub#2 = 10 3 , and Y = 12:5. We then plug these constant values in to the complex demand (or inverse demand) function to …nd the simple demand (or inverse demand) function. Plugging them in gives: Pown = 8:55 0:05QD + 1 (4) + 0:15 10 3
+ 0:1 (12:5) Simplifying gives: Pown = 8:55 0:05QD + 4 + 0:5 + 1:25 Or Pown = 14:3 0:05QD Thus a simple demand function assumes the values of other variables are held at their constant level. 1.1.3 Changes in demand What happens when one of the values of a variable held at its constant level changes? Suppose that Psub#1 increases from $4 to $4.5. Now we need to recalculate the simple demand curve. We do this by plugging in the new constant value for Psub#1. We get: Pown = 8:55 0:05QD + 1 (4:5) + 0:15 10 3
+ 0:1 (12:5) Pown = 8:55 0:05QD + 4:5 + 0:5 + 1:25 Pown = 14:8 0:05QD If we graph the new and old demand curves we will have: 0 5 10 15 13 14 15 16 17 Quantity Price The new demand curve (after Psub#1 increases) is the demand curve to the right with the higher intercept (the unfortunate part of using the actual functions is that I cannot place labels inside the box when I graph them). Recall that a shift to the right is an increase in demand. This should make sense because as the price of a substitute good increases, the demand for our good also increases. Thus the factors other than Pown determine where the demand curve is placed on the graph. 2 3. Solve for Pown. 4. Plug Pown back into the demand function to …nd QD. 5. Plug Pown back into the supply function to …nd QS . You should make sure that QD = QS . (The purpose of this 5th step is to check your algebra.) Now, to do the work: QS = 286 20Pown QS = 88 + 40Pown Next, 286 20Pown = 88 + 40Pown Next, Pown = 198 60 = 3:3 Next, QD = 286 20  (3:3) = 220 Finally, QS = 88 + 40  (3:3) = 220 Because QD = QS at Pown, we have solved for the equilibrium price and quantity. Either that or we made so many mistakes along the way that things just worked out. I’ll assume it’s done correctly... Putting the supply and demand functions on the same graph (demand is in black; supply is in red) reveals that our algebra is indeed correct. 0 1 2 3 4 5 0 100 200 300 Quantity Price You should also be able to recalculate the equilibrium price and quantity given that one or more of the underlying factors of the supply or demand functions has changed. In those cases, you would need to substitute the new constant value into either the new supply or demand function, recalculate the simple supply or demand function, and then work through the steps to solve for the equilibrium price and quantities. 5 2.2 Changes in Supply and Demand While understanding how supply and demand interact to determine equilibrium price is important, it is perhaps more important to determine how equilibrium price and quantity will change if either supply or demand (or both) changes. For most cases it is fairly straightforward – if supply increases (again, this increase would be caused by an underlying change in a resource price or some other factor that a¤ects supply) there will be an increase quantity and a decrease in price, while if there is a supply decrease there will be a decrease in quantity and increase in price. For a demand increase we would see an increase in both price and quantity, while for a demand decrease we would see a decrease in both. If both supply and demand change, then the change in equilibrium price and quantity will depend on the relative shifts of the two curves. While we more formally discuss estimating supply and demand curves in a few weeks, you can still think about the general direction of the equilibrium price changes by just using information you can pick up from daily observation. If you observe that bad weather has been a¤ecting crops, it is pretty straightforward to reason that there will be a supply decrease and that prices of the a¤ected crops (as well as prices in which the crops are a resource) would increase. Alternatively, consider what would happen if a government announced a plan (perhaps a tax or subsidy) to increase the amount of corn-based ethanol in production. While supply of corn might increase overall, the supply of corn that consumers purchase to eat could decrease as more corn is shifted to ethanol production. When in business, it’s important to recognize which factors will a¤ect your business, and what the general e¤ects might be (this way even if you do not have good data with which to make precise estimates you still know the general e¤ect). 3 Times when QD 6= QS There are some cases when determining the equilibrium price and quantity cannot be done as described above. Typically, these cases involve some restriction imposed on either price or quantity in the market. We will work through an example of a price ‡oor. 3.1 Price Floor example Recall that a price control is a government mandated price. A price ‡oor is a price set by the government which the market price cannot fall below. A price ceiling is a price set by the government which the market price cannot rise above. Suppose we have the following supply and demand functions: QD = 286 20Pown QS = 88 + 40Pown These are the same supply and demand functions from above, so the equilibrium price is $3:30 and the equilibrium quantity is 220. Suppose the government imposes a price ‡oor of $4. We now know that the price cannot fall below $4. How would we go about solving for the price and quantity traded (to be honest, it is really NOT an equilibrium quantity because QD 6= QS which is why I call it the quantity traded) in the market? I propose the following steps: 1. First, calculate the equilibrium price and quantity without imposing the price ‡oor. I say this because if the price ‡oor is BELOW the equilibrium price, then the price ‡oor does not bind because the market price is greater than the price ‡oor. Thus, the equilibrium price and quantity would be the price and quantity traded in the market. So, if the government had decided to set a price ‡oor of $3 in this market, the outcome would just be the equilibrium price and quantity because $3 < $3:30. (It is possible that a price ‡oor set below the true equilibrium price may end up being a focal point in the market, but the “conventional wisdom”is that non-binding price controls have no e¤ect on the market.) 2. If the price control does bind then you need to calculate QD and QS by substituting the value of the price control ($4 in the example) into the demand and supply functions. We …nd that QD = 206 when Pown = $4 and that QS = 248. Note that this is NOT an equilibrium solution because QD 6= QS . 6 3. Finally, choose the quantity level that is lower: in this case, QD < QS . The reason that we choose the lesser amount is that even if you have 248 units for sale, if people only want to buy 206 units at the price you are charging then only 206 units will be traded. 7