All Terrain Thinking

A Compendium of things I think are Important

Algebra: Calculating why it's not in your wallet

What will happen to the price of a commodity if there is an increase in supply? What will happen to the level on national income if the government increases defense spending? How can we explain the slowdown in the growth in output in the US? Why is tuition increasing so fast in US universities? What will be the rate of interest on government securities next month?

These are just some questions which you may have come across in your economics courses. But how do we approach them given that the answer to each requires a rather involved logical chain. For example, let's look at the first question and follow through the logic that you discussed in your introductory course. If there is an increase in supply, there will be a surplus in the market. The supplier in the market will respond to this surplus by lowering prices which will raise demand and lower supply. Eventually the fall in supply and the rise in demand will eliminate the surplus and the market will reestablish equilibrium at a lower price.

How can we capture this logic? In most courses we do it some graph similar to what we have below. The increase in supply (S1q to S2) means that at the original equilibrium price (P1) there is now a surplus (S>D). Equilibrium is reestablished at P2.

In many instances, however, economists are interested in using algebra to express the relationships conveyed in the graph above. They would do is create a model, which is nothing more than a set of equations. There a number of ways to classify the equations. One would be by the mathematical structure of the equation. For us the most important distinction would be between linear and nonlinear.

Nonlinear

Polynomial Y = a + bX + cX2

Logarithmic Y = 4Xb lnY = ln4 + b lnX

Linear

Single variable Y = a + bX

Multi variable Y = a + bX + cZ

[Note: One of the important differences between these equations is the interpretation of the coefficients. In the linear equation we would interpret b as a rate of change (b = DY/ DX): b tells us what the change in y would be for a one unit increase in x. In the log linear equation we would interpret the coefficient as an elasticity ( b = %DY/ %DX ): b tells us the percentage change in y that would accompany a 1 percent change in x].

Now let's return to the supply-demand model and build a mathematical model. We need to make some decisions here as to what kind of model we want to build. To make our life easy we will assume that the model is linear. It will consist of three equations: (1) the demand curve, (2) the supply curve, and (3) the equilibrium condition. The first two of these would be called behavioral equations because they describe behavior, while the last would be called an equilibrium equation. In very general terms we could develop a qualitative model and specify the model as:

Before you panic because you have never seen anything like this, remember that it is a model that you have seen before and now we can try to make sense of it. The equations are clearly linear which means that they would appear as straight lines on a graph. Given that they are linear, the coefficients of price (P) are:

The coefficient b tells us how quantity demand changes when price changes, and from our economics we have come to expect that they would be negatively related-an increase in price will reduce demand. For supply, meanwhile, price and quantity are expected to be positively related.

To answer the opening question with this model, we need to solve it for the unknown variables (Qs, Qd, and P). Given that we are looking to solve for P, I suggest that we substitute equations (1) and (2) into equation (3) to obtain the following equation:

Collecting terms and bringing P to the left side of the =, we get an equation specifying the solution-the equilibrium value of P.

What do we know about the right side? We know that b<0 and d>0 so (d-b)>0. We are almost there, but first we need to know how to show an increase in supply. If you return to the supply equation, c is the shift parameter: an increase in c would represent an increase in supply-an outward shift in supply. Returning to equation (5) we can conceptually rewrite it as:

What we have is a linear equation specifying the relationship between P and c. The red term represents the intercept and the green term represents the slope. Now we are home. We know that the slope = DP/Dc and we also know that -[1/(d-b)] is a negative number since (d-b)>0. Putting them together we have that an increase in c will result in a decrease in P, but that is what we already found with our graphs and our narrative.

Before we leave this, let's redo things with actual numbers and build a quantitative model. Our model will be:

Following the procedure above, we substitute equations (1') and (2') into equation (3') to obtain the following equation:

Collecting terms and bringing P to the left side of the =, we get an equation specifying the solution-the equilibrium value of P equal to 40.

Now let us increase supply by 30 so that our supply equation becomes Qs = 40 + 1P. Following the same procedure we end up with equation (4''):

Collecting terms and bringing P to the left side of the =, we get an equation specifying the solution-the equilibrium value of P equal to 40. As we expected, the equilibrium price has fallen.

Not as bad as it looked? To see if you really understand it, please try the following exercises.

• What will happen to price if there is an increase in demand?
• How will the price change be affected by the price elasticity of demand? [Hint: a more elastic demand would be represented by a larger value for the coefficient (absolute value) of the demand curve]

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