### Solving for equilibrium price and quantity with incomes and substitutes added in

This is a common economics supply and demand problem.  It is a little more advanced because it requires the use of math to explain the typical supply and demand graphs used in economics.    The question gives:

The Florida Fruit Growers Association is interested in analyzing the US domestic market for oranges. Empirical research reveals the supply and demand functions given below:

QS=1,000 + 10,000P-5,000Pl-500Pc

QD=45,000-12,500P + 4I + 2,500Ps

Where Q is quantity measured in 50 pound cases, P is the price per case in dollars, Pl is the price of unskilled labor in dollars (the wage rate=\$6), Pc is the price of capital as a percentage, I is family income and Ps is the price of California oranges.

a. Derive (solve for) the supply and demand curves for Florida growers assuming the following values for the independent variables:

PL=\$3, PC=12% (do not convert to a decimal), I=25,000, and PS=\$6

b. What is the equilibrium price and quantity for oranges?

c. Suppose the price of labor increases to \$8 (Ps=\$8) what happens in the market?

To solve for (a) we need to plug in all of the values given above, so:

QS = 1,000 + 10,000*(price of a case) – 5,000*(price of unskilled labor) – 500*(price of capital as a percentage)

This gives us:

QS = 1,000 + 10,000*P – 5,000*3 – 500*12
Or
QS = 1,000 + 10,000*P – 15,000 – 6,000 = - 20,000 + 10,000P

It helps to then calculate the inverse supply function to assist with graphing.  To do this, add 20,000 to both sides, and divide by 10,000 to get:

P = 2 + QS/10,000

So no oranges will be supplied until the price of a crate reaches a minimum of \$2.
The associated supply curve will look like the one in the graph below, the intercept on the X axis occurs at -20,000 (which isn’t shown) and we see that positive amount is supplied once the \$2 threshold is met.  Also, the curve slopes upwards with a slope of 1/10,000 (rise over run, or price over quantity).

Next for the demand curve.
QD = 45,000 – 12,500 (price of oranges) + 4 (family income) + 2,500 (price of California oranges).
Or
QD = 45,000 - 12,500P + 4*25,000 + 2,500*6

Which gives us:
QD = 45,000 – 12,500P + 100,000 + 15,000 = 160,000 – 12,500P

To solve for our inverse demand function we add 12,500P to both sides, subtract QD from both sides, and divide both sides by 12,500 to get:

P = 12.8 – QD/12,500

We can then use this information to plot a demand curve.  The intercept on the Y axis occurs at a price of 12.8, and slopes downward with a slope of – 1/12,500 (rise over run, or price over quantity).

We can solve for equilibrium price and quantity in the orange market by setting either QD equal to QS or the two prices equal to each other and then solving.  To be consistent with prior posts, I will set QD equal to QS, solve for P, then plug in the solved P value into our QD and QS equations to back out equilibrium quantity.

So first set QS equal to QD:
QS = QD = - 20,000 + 10,000P = 160,000 – 12,500P

We want to get P by itself so first we can add 12,500P to both sides, and add 20,000 to both sides to get:

22,500P = 180,000

Then divide both sides by 22,500 to get:

P = 8

So the equilibrium price of the orange market is \$8.  We then plug 8 into our QS and QD functions to get:

QS = -20,000 + 10,000*8 = 60,000
And
QD = 160,000 – 12,500*8 = 60,000

So the equilibrium quantity of the orange market is going to be 60,000.

Finally, what happens if the price of a substitute rises?  The problem asks what will occur if the price of California oranges rises from \$6 to \$8.  We already know that when the price of a substitute goes up, we will demand more of the good in question (a rightward shift in demand, or an increase in demand).  To show this mathematically we just plug in the new value into our demand equation:

QD = 45,000 – 12,500 (price of oranges) + 4 (family income) + 2,500 (price of California oranges).
Or
QD = 45,000 - 12,500P + 4*25,000 + 2,500*8

This gives us:
QD = 45,000 -12,500P + 100,000 + 20,000
Or
QD = 165,000 – 12,500P

The inverse demand function is now:
P = 13.2 – QD/12,500
You can see in the graph below that this results in the demand curve shifting right/up, which will result in a higher equilibrium price and equilibrium quantity.

We can figure out equilibrium price and quantity by again setting QS = QD.
This gives us:
QS = QD = - 20,000 + 10,000P = 165,000 – 12,500P

Following the same process as above we get:
P = 8.222

So our equilibrium price has gone up by \$0.22.  We then plug this into our supply and demand equations and solve for our new equilibrium quantity:

QS = -20,000 + 10,000*8.222 = 62,222
And
QD = 165,000 – 12,500*8.222 = 62,222

Again our quantities match, so we know that we did the math right!

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