Problem Set 1

Note: You might find the "Geometry of Univariate Functions" computer simulation to be helpful in answering some of these questions. Usage details are listed for problem 2, but if you play around with the simulation, you can use it for many of the other problems as well.

MATHEMATICAL FUNCTIONS IN ECONOMICS: Match each of the following functions with the most likely application in economics selected from the list below. Briefly explain how you decided in each case.

Function list:		Application list:
(1) y = 200 - 4x	a.	Production function
		x = Labor; y = Output
(2) y = -200 + .25x	b.	Demand equation (axes inverted)
		x = Price; y = Quantity demanded
(3) y = 400 - x²	c.	Production possibilities curve
		x = Clothing; y = Food
(4) y = 25 + 28x - 9x² + x³	d.	Total cost function
		x = Output; y = Total cost
(5) y = x^1/2	e.	Savings function
		x = Disposable income; y = Savings

EXPLORING POLYNOMIAL FUNCTIONS: Consider the following polynomial functions:

(1) y = 10	(2) y = 10 + 3x
(3) y = 10 + 3x - 1.5x²	(4) y = 10 + 3x - 1.5x² + .25x³

Construct a single table in which you fill in the value of y for each function for whole-number values of x from 0 to 5.

Plot all 4 functions in a single diagram.

Briefly explain why the shape of the function changes the way it does as you add each successive term to it.

Note: This is a good problem to work on with the "Geometry of Univariate Functions" computer simulation.
(1)	Select the "Cubic" function for equation 1.
(2)	Start by leaving the first parameter value alone and setting each of the other parameter values = 0
(3)	Set the y-value range minimum = -15 and maximum = 25
(4)	Select 8 grid lines for the y axis
(5)	Click "Setup" and then when it's ready, click "Plot1"
(6)	To check your answers in part a, click the diagram at each of the whole-number x values, and the yellow output box will list the corresponding y value.
To plot the next function, just change the value of the new parameter and click "Plot1" again. These six steps are illustrated on the last page of this problem set.

3.	CONSUMPTION AND SAVINGS: Consider the following consumption and savings functions: C = 140 + .8DY S = -140 + .2DY where C = consumption, S = savings, and DY = disposable income.
	a.	In a single diagram, plot the consumption and savings functions. Put DY on the horizontal axis, and plot a DY range of approximately 0 to 1200.
	b.	Provide a brief economics-based interpretation of what the intercept and slope terms mean for each function.
	c.	What expression do you get when you add together the consumption and savings functions? (In other words, C+S=?.) Briefly interpret the economic meaning of your result.

CONSUMER DEMAND: Suppose you sell loaves of SuperBread. Right now you bake nine loaves a day and can sell them for $4 apiece (it is truly great bread). You hypothesize two alternative (inverse) demand equations for your bread, as indicated in the table below:

Quantity	Equation (1): P=13-Q		Equation (2): P=36Q^-1
baked	Price	Total revenue	Price	Total revenue
1
3
6
9	4	36	4	36
12

Complete the table.

Use a single set of vertically stacked diagrams to do the following:

(1)

In the top diagram, sketch the two demand equation, with Q on the horizontal axis and P on the vertical axis.

(2)

In the bottom diagram, sketch the two resulting total revenue (TR) equations, with Q on the horizontal axis and TR on the vertical axis. Keep in mind that TR = PQ.

Note: Use the same scale for Q in both diagrams so they line up with each other.

5.	FUNCTIONS AND SLOPES SLOPES: A function has the form: y = 2x^1/2.
	a.	What is the value of its y-intercept?
	b.	What is the name of this type of function?
	c.	Is the function concave, convex, neither or both?
	d.	Plot this function, incrementing x by one unit from 0 to 4.
	e.	Calculate the slope over each one-unit increment up to x=4.

INTERPRETING SLOPES IN ECONOMICS:

What information is conveyed in the slope of each of the following?

(1) A production function (Q = f(L))

(2) A total cost function (TC = f(Q))

(3) A production possibilities frontier (PPF: Qy = f(Qx))

Provide a brief explanation for the following three observations (in fact, the observations are related to each other, so try to provide an explanation which links them together):

(1)

Production functions eventually tend to become flatter as L rises.

(2)

Total cost functions eventually tend to become steeper as Q rises.

(3)

PPFs tend to become steeper as Qx rises.

7.	Staple your work--stapling always carries the weight of one problem on each problem set.