1 Supply and Demand, Summaries of Law

Download 1 Supply and Demand and more Summaries Law in PDF only on Docsity! These notes essentially correspond to chapter 2 of the text. 1 Supply and Demand The first model we will discuss is supply and demand. It is the most fundamen- tal 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 verified 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: 53.752.51.250 1.25 1 0.75 0.5 0.25 0 Quantity Price 1.1.1 A simplification of reality The demand curve is really a simplification of reality. There are many fac- tors that go into determining the demand for a specific product (how many consumers in a market, the prices of related goods, the amount of income con- sumers 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 say- ing 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, 1 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 difficult 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 find 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 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 = 14.3− 0.05QD. 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 ofQD = 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 difficult, 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 find 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 2 1.2.2 Changes in supply What happens when the value of a variable held at its constant level changes? Suppose that Presource increases from $1.5 to $1.75. Now we need to recalculate the simple supply curve. We do this by plugging in the new constant value for Presource. We get: Pown = −4.45 + 0.025QS + 1.5 (1.75) Pown = −4.45 + 0.025QS + 2.625 Pown = −1.825 + 0.025QS If we graph the new and old supply curves we will have: 250200150100500 5 3.75 2.5 1.25 0 Quantity Price In this case the new supply curve, after Presource changed from 1.5 to 1.75, is the one to the left. Note that the supply curve has decreased (shifts to the left are decreases). Again, this should coincide with what you were taught in principles — when the price of a resource increases, the supply of a good decreases. 2 Equilibrium Determination Alfred Marshall, whose principles of economics text was most likely read by every economics student from 1900 — 1950, compared supply and demand to the blades of a pair of scissors. In order for the pair of scissors to function properly, both blades are needed. The same is true with supply and demand — in order to properly understand how prices bring about equilibrium, we need to use both supply and demand. I have no doubt that you all are capable of finding the equilibrium price and quantity if given a graph. Simply find the coordinates of the point where supply and demand intersect and you have your equilibrium price and quantity. However, accurately graphing the supply and demand functions and determining their price and quantity at the intersection point from a graph is a daunting task. It is much easier (especially if you don’t have the graph given to you) to determine the equilibrium price and quantity by simply solving a system 5 of equations. Using our supply and demand functions from above, can we determine an equilibrium price and quantity? We have: Demand function: QD = 286− 20Pown Supply function: QS = 88 + 40Pown With only these 2 equations we CANNOT solve for a unique price and quantity pair. Notice that we have 3 variables: QD, QS , and Pown but only 2 equations. However, we do know that a 3rd equation holds at the equilibrium point: QD = QS , which must be true if a market is in equilibrium. We now have 3 equations and 3 unknowns (although this does not guarantee that a solution exists). 2.1 Steps to solve for equilibrium prices and quantities Begin with your 3 equations: QD = 286− 20Pown QS = 88 + 40Pown QD = QS You can use whatever method you want to solve for the unknowns. Given the current set-up, I would say: 1. Substitute QS in for QD in the demand function. 2. Next, the left-hand side of the supply and demand functions are now both equal to QS . Set the two functions equal to each other. 3. Solve for Pown. 4. Plug Pown back into the demand function to find QD. 5. Plug Pown back into the supply function to find 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, 6 QD = 286− 20 ∗ (3.3) = 220 Finally, QS = 88 + 40 ∗ (3.3) = 220 Since 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... 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 func- tions has changed. In those cases, you would need to plug 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. 3 Times when QD 6= QS There are some cases when determining the equilibrium price and quantity can- not 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 floor. 3.1 Price Floor example Recall that a price control is a government mandated price. A price floor 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 equilib- rium price is $3.30 and the equilibrium quantity is 220. Suppose the government imposes a price floor 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 floor. I say this because if the price floor is BELOW the equilibrium price, then the price floor does not bind because the market price is greater than the price floor. Thus, the equilibrium price and quantity would be the price and quantity traded in the market. So, if the government had 7