Edit: Updated August 2018 with more examples and links to relevant topics.

Summary:  To solve for equilibrium price and quantity you should perform the following steps:

1) Solve for the demand function and the supply function in terms of Q (quantity).

2) Set Qs (quantity supplied) equal to Qd (quantity demanded). The equations will be in terms of price (P)

3) Solve for P, this is going to be your equilibrium Price for the problem.

4) Plug your equilibrium price into either your demand or supply function (or both--but most times it will be easier to plug into supply) and solve for Q, which will give you equilibrium quantity.

The present value is the value in today's dollars of an asset, benefit, or cost that will occur in the future. Perhaps the easiest way to think of a present value is to ask yourself how much you would be willing to accept to sell your rights to a future asset.

For example, is you were to plant an orange tree today you make expect to get $50 worth of oranges in the future. Since you have to wait for the oranges (and their $50 value), the value in today's dollars will be less. If someone is patient and the time they have to wait for the oranges is small, then their willingness to accept amount (the present value) may be $45 or $40. If the person is impatient or has to wait a long period of time, then the willingness to accept amount (the present value) may be much lower approaching $5 or so.

The present value of a future payoff should never be larger than the face value of the payoff. This is a result of the idea of "discounting" which means that people tend to prefer having resources now rather than in the future. For example, imagine you were given the choice between:

Sometimes you will be asked to define a price function. When you receive such a question, it is probably in regards to a supply and demand graph, and you will be asked to graph said price function. The three components of a price function include: price (as you probably expected), an intercept, and quantity. It is also possible for a price function to include many other variables such as preferences, prices of related goods, income, number of buyers, etc. but this is for more complicated price functions (and upper division style classes).

A price function generally looks like:

This post goes over the math required to solve for the profit maximizing price and quantity of a price discriminating monopoly operating in two markets. Consider the following problem:


A cable company sells subscriptions in San Francisco and Boston. The demand function for each of the two groups, which are separate and do not have the ability to re-sell to members of the other community, are Psf = 480 - 4Qsf and Pb = 400 -2 Qb. The cost of providing the cable service for the firm is TC = 500 + 4Q, where Q = Qsf + Qb. If the company can price discriminate between the two markets, what are the profit maximizing prices and quantities for the San Francisco and Boston markets?

To solve this problem, we need to review the steps for finding the profit maximizing price and quantity for a monopoly. We find that we need to find the price and quantity where marginal revenue (MR) is equal to marginal cost (MC).

This post goes over an example of solving for equilibrium price and quantity using the method detailed in the prior equilibrium solving method post. In this example we are given a Qs equation as shown:

Qs= -7909.88 + 79.0988P

Note that this gives us a positive sloping supply curve and that price has to be at least 100 in order for the supplier to produce anything at all (we can figure this out by dividing the intercept 7909.88 by the coefficient on the price 79.0988).

The next step is exploring the demand equation. In this example we are given a demand function as follows:

Qd= 38650 - 40P

Here we have a downward sloping demand curve and the quantity demanded at a price of 0 will be 38650. Once the price reaches 966.25 we will see a quantity demanded of 0 (found by dividing 38650 by 40).

This post highlights the differences between expected value and expected utility and demonstrates how the difference between the two is in how they are calculated. Expected value shows us the value that is to be expected from engaging in a lottery (or risky situation) where there are 2 or more possible outcomes. Likewise, Expected utility shows us the utility that is expected out of a lottery with two or more possibilities. Remember that utility shows the satisfaction or happiness derived from a good/service/money while value simply shows us the monetary value. That is why the two terms are measured differently and show us different things. The rest of this post will describe how to calculate expected value and expected utility and has solved examples demonstrating the importance of the difference between them.

A good rule of thumb is to read the problem, and identify all of the key information. You need to know 4 things:
1) what is the probability of outcome 1?
2) what is the monetary value of outcome 1?, these go into the first term of your equations.
3) what is the probability of outcome 2? This can either be stated explicitly in the problem, or calculated from the probability given for outcome 1.
4) what is the monetary value of outcome 2?, these go into the second term of your equations.

Equations:

This post goes over the math required to show the difference between surplus and equilibrium in a perfectly competitive and monopolistically competitive market. Note that a monopolistically competitive market's math and graph will be the same for a monopoly or an oligopoly.  Here are the equations to work with:

P = 40 - 8Q
MC = 8

This intensive economics question goes over calculating equilibrium price and quantity, then using those numbers to get consumer and producer surplus, and finally implementing a tax to see how that will change the previous results:

1. The inverse demand curve (or average revenue curve) for the product of a perfectly competitive industry is give by p=80-0.5Q where p is the price and Q is the quantity. The short-run industry marginal cost function is MC=50+0.25Q

a) Calculate the equilibrium price and quantity assuming perfect competition and profit maximization and hence calculate the consumer and producers' surplus.

This post is going over the economics of a dominant firm problem using algebra and math.  For more information on the economic theory of a dominant firm or price leader model go here.  The actual question being solved is:

In the model of a dominant firm, assume that the fringe supply curve is given by Q= -1+0.2P, where P is market price and Q is output. Demand is given by Q=11 –P. What will price and output be if there is no dominant firm? Now assume that there is a dominant firm, whose marginal cost is constant at $6. Derive the residual demand curve that it faces and calculate its profit-maximizing output and price.

In order to solve this you need to follow these steps:

This economics post will go over the profit maximization behavoir of a perfectly competitive firm.

For a related numerical example look here, for a graphical example look here, and finally for a word problem based example look here.

Remember that when calculating the profit maximizaing point for any firm, it is imperative that we set marginal revenue equal to marginal cost (MR=MC).  If we are at any other point, then there are potential gains to be made.  Imagine if MR<MC, at this point we are losing money on the margin.  We are selling too many goods or services, and need to scale back.  If MR>MC, then we are selling too few, and we could sell more goods or services because our marginal gain is greater than our marginal cost.  Below is a graphic showing this relationship for a perfectly competitive firm: