Review of Basic Geometry, Derivatives and Algebrafor Microeconomics
(Stephen Schmidt, UnionCollege)
Economic analysis of the type we’ll discuss is most useful when its results are numerical, rather than simply graphical. For example, if one wants to reduce cigarette consumption by 50%, it is important to know by how much the tax on cigarettes should be raised, not just that it should go up. For this reason, the models we’ll develop in this class will focus on variables (for example, p, q, xB1B, or xB2B) that take numerical values, and on equations (for example, 5*xB1 B+ 3*xB2B = 20) that we can use to solve for those values. There are a few rules of algebra that are very helpful in solving these equations, and there are also some important concepts from geometry and elementary calculus that will come up regularly in the class. This handout goes over those rules and explains how to use them. You don’t need to worry about memorizing these rules, but you may want to keep the handout in your notebook for use while doing problem sets. After a few weeks you will get to know these rules well.
Equations, Functions and Slopes
We will usually express relationships between economic variables by means of functions.A function allows us to calculate the dependent variable if we know the values of the independent variables. For instance, if we are interested in prices and quantities, we might have the function p(qB1B, qB2B) = 20 – 3*qB1B + qB2B. (We might also write this function as an equation in the form p = 20 – 3*qB1B + qB2B; for this course it won’t usually matter which way we chose to do it.) This function tells us that if, for example, qB1B=3 and qB2B=5, then p=16. If qB2B rises to 6, then p will rise to 17. From this function we can see which prices go with which quantities, and how a change in one of the independent variables will affect the dependent variable.
If the function has only one independent variable, then we can also graph it. For example, the graph of the functions y(x) = 2 + 0.5x and y(x) = xP2P look like this:
If the function has more than one independent variable then we cannot graph it this way (unfortunately, most interesting economic functions have more than one independent variable) but we might assign specific numerical values to all but one of the independent variables, and then graph the relation between the dependent variable and the last independent variable. For example, if p(qB1B, qB2B) = 20 – 3*qB1B + qB2B, then we might draw the following graphs:
The first graph sets qB2B=5, which lets us write the function as p = 20 – 3*qB1B + 5 or just
p = 25 – 3*qB1B. The second graph sets qB1B=3, which simplifies the function to p = 11 + qB1B.
We will often be interested in knowing how fast the dependent variable changes when one of the independent variables changes. For example, we’re often interested in knowing how much cost goes up as a output increases; we call this amount the marginal cost of producing more units. To learn this, we use the slope of the function, which is given by the following equation:
where (xB1,B yB1B) and (xB0, ByB0B) are two points on the function. For example, if the function is y=2 + 2x, and we pick the points (1,4) and (2,6), then the slope is equal to (6-4)/(2-1), which is equal to 2, as the following picture shows:
and you can see that, if Y represents costs and X represents output, then cost rises by 2 when output rises by 1, so that the slope and the marginal cost are the same thing. Note that, since the slope of the line is the same everywhere, it does not matter which two points you pick; you will always get the same slope. Rather than calculate the slope each time, it is worth remembering a simple rule:
Rule 1: If y = a + bx, where a and b are any numbers, then the graph of the function is a straight line whose slope is b.
A function of the type y=a+bx is called linear because its graph is a straight line. More complicated functions, such as y=xP2P, are nonlinear and their graphs are not straight lines. In such cases it will matter which two points we pick to calculate the slope of the function. Rather than worry about which points we should pick, we usually draw a line which has the same slope at the function at the point we’re interested in (which we call a tangent line) and calculate the slope of that line. For example:
In this graph, the function y=xP2P is shown, along with a line which is tangent to it at the point (2,4). The slope of the tangent line is 4 (because (8-4)/(3-2) = 4) and we will therefore say that the slope of the function y=xP2P at the point (2,4) is 4. You can see from the graph that the slope is increasing: it turns out that at the point (1,1) the slope of this function is 2, and at the point (3,9) the slope of this function is 6.
Derivatives
To use this idea of the slope of a nonlinear function, we need to be able to find the slope of the tangent line, which requires calculus. The idea behind the derivative is to measure the slope of the function itself using two points which are very close together: as the points get arbitrarily close to one another, the slope of the line between those points will become the same as the slope of the tangent line, and will therefore become the slope of the function. As the distance between the points gets very close to 0, x and y will get close to zero, and we usually call them dx and dy instead (where the small d indicates a very small difference.) We will therefore often refer to the derivative as dy/dx instead of y/x (although the idea is the same in both cases.) The following picture demonstrates this idea:
Rather than go through this process each time we need to know the slope of a function, however, we have a few simple rules for most of the derivatives we’ll need in this class:
Rule 2: If y(x) = k (a constant not depending on x) then dy/dx = 0. This is because if y doesn’t depend on x (for example, if y=6) then the graph is a flat line of slope 0. Or, if you prefer, if y = 6, then it doesn’t change when x changes, and so the rate of change is 0.
Rule 3: If y(x) = bx (a linear function), then dy/dx = b. This reflects the fact that the slope of that line is b.
Rule 4 (power rule): If y(x) = xPnP, then dy/dx = n*xPn-1P. For example, if y = xP2P, then dy/dx = 2*xP1P = 2x. You can see that this is true, in fact, from the example above. It works even if n is a fraction, or negative: for example, if y = xP1/2P, then dy/dx = (1/2)* xP-1/2P.
Rule 5 (sum rule): If y(x) = a(x)+b(x), then dy/dx = da/dx + db/dx. This just says that when functions contain two or more pieces added together, you can take derivatives piece by piece. For example, if y(x) = xP3P – 5x + 6, then dy/dx = 3xP2P– 5. If y(x) = a+bx, then dy/dx = 0+b = b.
Partial Derivatives
When a function has more than one independent variable, for instance p=25 – 3qB1 B+ qB2B,
then we may want to know how p would change if one of the independent variables changed while the others stayed constant. We call this the partial derivative of p with respect to one of the variables, and we find it by treating the other variable as if it was a constant and taking the derivative normally. In this example, we say p/qB1B = -3 and p/qB2B = 1; we write p/qB1B rather than dp/dqB1B to remind ourselves that this is only a partial derivative and we have assumed that qB2B isn’t changing when qB1B changes. (This material is not covered in Math 10 or Math 13, so you may not have seen partial derivatives before. If you haven’t, don’t worry; they are exactly the same as regular derivatives as long as you remember that the other independent variables are being treated as constants.)
Using Slope to Predict Changes in Variables
The equation for the slope can be used to show how one variable will change if the other variable does. We can take the equation slope = y/x and multiply by x on both sides to get y = slope*x, and since we know that the slope is just dy/dx, we can write this as y = dy/dx*x. This lets us show how y will change if x changes, or vice versa. In the picture on the left below, the slope of the line is 2: you can see that if x rises by 3 (x=3) then y rises by 6 (because y = slope*x = 2*3=6). You can do it in reverse too; if you want y to fall by 0.5, then you need x to fall by 0.25 (because 0.5 =2*0.25). Of course, if the function is a curve rather than a line, as in the picture on the right, then this is only approximately true, not literally true, because the slope of the function is changing; but as long as x is small, it’s a good approximation.
Solving Equations
Often in Economics 241, we will study relationships between variables that are not functions. For example, we might have the equation 2x + 4y = 12, which describes a relationship between x and y, but does not take the form y(x) (nor does it take the form x(y)). We will want to be able to solve this equation to write y as the dependent variable and x as the independent variable. The following rules are often helpful:
Rule 6: You can add (or subtract) the same thing to both sides of an equation. For instance, if we know y – x = 10, then we can add x to both sides of the equation to get
y = 10 + x, which gives y as a function of x.
Rule 7: You can multiply (or divide) both sides of an equation by the same thing. For instance, if we know 5y = 60 + 10x, then we can divide both sides by 5 to get y=12+2x. Or, if we know xy=1, then we can divide both sides by x to get y=1/x.
Rule 8: You can raise both sides of an equation to the same power. For example, if we know yP1/2P = 4, then we can raise both sides to the 2 power (that is, square both sides) to get y=16. This is a very useful trick in Economics 31; so much so that you should know some rules for handling equations with powers in them.
Rule 9: (XPaP)PbP = XPa*bP. For example, (XP2P)P3P = XP6P. This rule is most useful for canceling powers out. For example, if you have yP1/2P = xP3/2P, then you can raise both sides to the 2 power. This produces yP1P = xP3P, and since yP1P=y (that’s how powers of 1 work) this gives you y= xP3P, which is a proper solution for y. (It’s also worth remembering that yP0P=1 no matter what the value of y is.)
Rule 10: (X)P-aP = 1/(XPaP). This is most useful for getting rid of negative powers. If you have the expression y = xP-1/2P, you can rewrite it as y = 1/(xP1/2P), which is often helpful for simplifying expressions.
Rule 11: XPaP*XPbP = XPa+bP, and (XPaP)/(XPbP) = XPa-bP. This also helps simplify expressions. For example, xP4/3P * xP2/3P = xP4/3+2/3P = xP2P, or (xP3/2P) / (xP1/2P) = xP3/2-1/2P = xP1P = x. This is probably the most useful rule of all the power rules.
Solving two equations for two variables
Sometimes we’ll be interested in finding the values of two variables at the same time. In that case we’ll need to have two equations, and be able to solve both equations for the two variables. The solution will be given by the point where the graphs of the two equations intersect, as you recall from supply and demand:
The P and Q that solve both equations (that make supply equal to demand) are the P and Q where the supply curve and demand curve intersect. The problem is to find the numbers for P and Q that are that point. The rule for doing so is:
Rule 13: When solving two equations, use one equation to find a function that relates the two variables: then use that function to eliminate one variable from the other equation. Use that to solve for one variable. Last,use either equation to find the value for the other variable.
For example, suppose you have the equations x+y=7 and 2x-y=8, and you want to find the values of x and y that satisfy both equations. You can subtract x from both sides of the first equation to get y=7-x. Then you can plug that value into the second equation to get 2x –(7-x) = 8, which simplifies to 3x –6 = 9, or 3x=15, or x=5. From that, you can see from the first equation that y=2.
It doesn’t matter which equation you use, or which variable you solve for. For example, you could instead add y to both sides of the second equation, which would give you 2x = 8+y, then subtract 8 from both sides to get y=2x-8. Plugging into the first equation gives you x + (2x-8) = 7, or 3x=15, or x=5, and again y=2.