# Example

We want to analyze equilibrium price and quantity of an agricultural product. We have:

????_{????} = +200 − 2???? + 2???? ???????????????????????? ????????????????????????????????

????_{????} = +3???? + 2.5???? ???????????????????????? ????????????????????????????????

????_{????} = ????_{????} ???????????????????????????????????????????? ????????????????????????????????????

Where:

????_{????} = Quantity demanded

????_{????} = Quantity supplied

P = Price per unit

I = Consumer income

R = Rainfall in inches

In this model an increase in consumer income will cause the quantity demanded to increase. Moreover, an increase in rainfall will result in greater supply. The amount of rainfall is not under our control and for all practical and modeling purposes it is exogenous—determined outside our model in the skies! We know that consumer income affects quantity demanded but we are not

particularly interested in knowing how it is determined. We just want to know how it affects equilibrium P and Q. So we treat it as exogenous as well.

Constants: +200, -2, +2, +3, +2.5

Exogenous variable: I, R

Endogenous variables: P, ????_{????}, ????_{????}

Solve the model for the equilibrium values of P and Q by setting ????_{????} = ????_{????} and pretending I and R are known:

????^{∗} = 40 + 0.40 × ???? − 0.50????

????^{∗} = 120 + 1.20 × ???? + 1.00????

So both the equilibrium price ????^{∗} and equilibrium quantity ????^{∗} depend on income and rainfall. But look at the signs. An increase in consumer income will cause both equilibrium price and equilibrium quantity to increase. However, an increase in rainfall will cause equilibrium quantity to increase but equilibrium price to decrease. Do these results make sense? Assume I $100 and R = 20 inches. Then

????^{∗} = 40 + 0.40 × 100 − 0.50 × 20 = 70

????^{∗} = 120 + 1.20 × 100 + 1.00 × 20 = 260

## Graphs

To do graphs we measure each endogenous variable on one axis and graph the equations assuming the exogenous variables are known. This way, the exogenous variables will be in the intercepts. Although, occasionally, they might also be in the slopes. Always look for the exogenous variables in the intercepts and slopes!

## Comparative Statics

We first ask what happens to our solution if income increases to I = 150, ceteris paribus.

????^{∗} = 40 + 0.40 × 150 − 0.50 × 20 = 90

????^{∗} = 120 + 1.20 × 150 + 1.00 × 20 = 320

Therefore,

Δ????^{∗} = 90 − 70 = 20

Δ????^{∗} = 320 − 260 = 60

To figure this out graphically, you look at the intercepts and find I in there. Then you see what happens to the intercepts if income increases to I = 150.

Note that you could get this result directly from the solution itself. Go back to the solution

????^{∗} = 40 + 0.40 × ???? − 0.50????

????^{∗} = 120 + 1.20 × ???? + 1.00????

Then, change I by ΔI leaving other exogenous variables unchanged:

Δ????^{∗} = +0.40 × Δ???? = 0.40 × (150 − 100) = 20

Δ????^{∗} = +1.20 × (150 − 100) = 60

What if the amount of rainfall increases by 10 inches? Go back to the solution and change R by 10 and see the effect:

????????^{∗} = −0.50???????? = −0.50 × 10 = −5

????????^{∗} = +1.00???????? = +1.00 × 10 = +10

So if rainfall increases, supply will increase forcing a reduction in price.

## Disequilibrium or Transitional Dynamics

In terms of the following figure, comparative static analysis is figuring out that the increase in income causes the market to go from point A to point C. Transitional dynamic analysis asks the question, “How does the market travel from point A to point C. What path does it take?” To be able to answer this question we need to make additional assumptions. For now assume that as income increases, consumers jump to point B on their new demand function creating an excess demand for the good equal to AB. As a result price increases along the supply curve until we reach point C. The movement from point A to B to C is the transitional dynamics. This is not the only possible path followed by the market during the time of disequilibrium. It is also possible to move from point A to C slowly on the supply function. There might be other possible disequilibrium dynamic paths. It all depends on the model and the assumptions we are willing to make. You will see many examples of transitional dynamic in this class.

## Economic Interpretation

Finally we need to make sense out of all these algebra and graphs in a simple language. Why is it that ????^{∗} and ????^{∗} change and why do they change the way they do as income increases? The good news is the interpretation comes from the model equations themselves. You don’t need to Google anything. You basically look at the equations and infer the cause and effect relationships, and describe them. An important thing to remember (and occasionally people forget it) is that the interpretation should come from the model itself (okay, I repeated myself). You should not bring any extraneous information to the table. The model is your world, stick with it.

For example, an increase in income causes demand to increase (from demand function). As producers respond to the increase in demand (from supply function), price increases also.

## Solving the Model Above

The model above was just sued to describe different terms. We are not going to see it again. So we did not spend time to show you how to solve it. But if you insist, here is how to solve it. This is the model:

????_{????} = +200 − 2???? + 2???? ???????????????????????? ????????????????????????????????

????_{????} = +3???? + 2.5???? ???????????????????????? ????????????????????????????????

????_{????} = ????_{????} ???????????????????????????????????????????? ????????????????????????????????????

Plug the demand and supply quantities in the equilibrium condition:

+200 − 2???? + 2???? = 3???? + 2.5????

Solve this for P and call the solution ????^{∗}:

5???? = 200 + 2???? − 2.5????

????^{∗} = 40 + 0.40 × ???? − 0.50????

Plug this in either demand or supply function to get ????^{∗}. Supply function is easier:

????_{????} = +3???? + 2.5????

????_{????} = +3(40 + 0.40 × ???? − 0.50????) + 2.5????

????^{∗} = 120 + 1.20 × ???? + 1.00????