Economists often model economic relationships as linear functions. These may be represented as lines on graphs, or more compactly and usefully as mathematical equations. Economic research typically involves specifying some hypothesized relationship (as implied by theory) as a mathematical function, and then testing this hypothesis against real-world data. Print out this exercise and try it!

1. We begin with a very basic demand hypothesis: consumption of some market good Q declines as the price of Q rises, and increases as the price falls. So we collect some market data to test this hypothesis:

   

Quantity	Price
15	$1.40
21	$1.25
35	$1.00
24	$1.10
22	$1.30
19	$1.20
11	$1.40
16	$1.45
26	$1.05
29	$0.90
38	$0.90
28	$1.05


Plot these data points on the graph. Notice that these data are "noisy" like all market data.  Nevertheless, the scatter does show an overall downward trend, which is consistent with our hypothesis of an inverse relationship between quantity demanded and price. Use a straightedge to draw a line through these points that best represents their overall trend. Extend this line straight to the vertical axis.

2. You can characterize this (inverse) demand or "willingness-to-pay" relationship as a linear function of the form P = a₀ + a₁Q where a₀ is the vertical intercept and a₁ is the slope ("rise" over "run," which will negative for a downward-sloping function.) Derive the intercept and slope coefficients for this function: P = _____ + ______Q_D
   Now you have a predictive model of demand for Q. Suppose the market supplies 10 more units of Q. How much will the market price have to fall to get consumers to buy these extra units?

3. Actually economists generally represent demand functions the other way around, in the form Q = b₀ + b₁P. Determine the values of b₀ and b₁ for the line you have drawn through the data points. b₀ is the horizontal axis intercept and b₁ is the "run" divided by the "rise" (i.e., 1/ a₁ from the inverse demand function you got in the previous question). Derive the intercept and slope coefficients of this function: Q_D = _____ + _____P

4. Now plot these supply data by hand on a second graph, with Q on the horizontal axis and P on the vertical axis:
   
   

Quantity	Price
36	$1.55
44	$1.95
25	$1.30
32	$1.29
13	$0.85
19	$1.25
22	$1.20
29	$1.35
41	$1.68
34	$1.41


Use a straightedge to draw the line through these points that best represents their overall trend. Extend this line straight to the axis. 

5. Determine the formula for this (inverse) supply or "willingness-to-sell" function: P = _____ + _____Q_S.
   By how much does the price of Q have to increase in order to get producers to supply 5 more units?
   Determine the formula for the conventional supply function: Q_S = _____ + B_____P.
   If P falls by 40 cents, how much less will producers be willing to sell?

6. The market equilibrium occurs where supply and demand schedules intersect. You can determine this point graphically. Superimpose your supply and demand schedules on one graph and visually determine the equilibrium price and quantity: P_EQ = $_____; Q_EQ = $_____
   Now solve for P_EQ and Q_EQ mathematically (by substitution).
   Calculate the point elasticities of supply and demand at the market equilibrium point. 
   

Enough basic micro theory review! In the next exercise we'll let the computer do the hard work.