Math Review and PPF

August 22, 2017

Graphs Review

Graphs help us visualize the relationship between two variables.
Two perpendicular lines are called axes and represent measurement in the two-dimensional plane.

Scatterplot - a graph where the relationship between two variables is shown as data points scattered around graph space. To draw it, we need data.

Continuous plot - a graph where the relationship between two variables is shown as a continuous curve. To draw it, we need either a formula showing dependence between variables, or data points and interpolation technique.

We say that variables have positive relationship if a higher value of one variable correspond to a higher value of another variable.
If all values lie on a straight line we say the relationship is linear.
If we control one variable and increment its value by the small fixed amounts, and the amounts of other variable changes in an increasing fashion, then the graph would become steeper. We call it convex graph.
If it changes in decreasing fashion, the graph will be less steep. We call it concave graph.

We say that variables have a negative relationship if a higher value of one variable correspond to a lower value of another variable.
Same situation is for linear, convex, concave graphs.

The degree of how a graph is "tilted" to axes is called slope.
When the graph is represented by a linear function, slope can be calculated by taking any two points on a graph and calculating following ratio:
\[ \frac{\Delta y}{\Delta x} \]
When the graph is non-linear, the slope is not constant and is calculated as a slope of the tangent line to a particular point.
The slope is closely related to the notion of derivative - sensitivity to change of the function value with respect to a change in its argument.
The value of derivative for convex graph is increasing, for concave graph - decreasing, for linear graph - constant.

Majority of economics problems stems from the problem of maximazation (minimazation).
If we can describe the problem in mathematical form, we can maximize the objective function using derivative. The value for derivative at maximum (minimum) point is 0.
If the optimization is constraint, that is, we maximize a function subject to some constraints, then the choice of maximum point should be done among feasible set of solutions.

We are given four points: point 1 \(x=1,\, y=5\), point 2 \(x=2,\, y=3\), point 3 \(x=3,\, y=2\), point 4 \(x=4,\, y=1\). Draw the points and continuous graph and identify: Is it increasing? Is it linear? Is it convex?
We are given three points: point 1 \(x=0,\, y=10\), point 2 \(x=0.5,\, y=5\), point 3 \(x=1,\, y=0\). Draw the points and continuous graph and identify: Is it increasing? Is it linear? If so, calculate the slope.

We are given four points: point 1 \(x=0,\, y=0\), point 2 \(x=2,\, y=1\), point 3 \(x=4,\, y=2\), point 4 \(x=6,\, y=3\). Draw the points and continuous graph and identify:
Is it increasing or decreasing? Is it linear and if so, calculate the slope? If not, identify if the graph is convex or concave?
Find maximal point of the function \(y=f(x)=-\frac{1}{2}x^2+5\) subject to \(0\le x\le 6\).

Definition: The production possibilities frontier (PPF) is the boundary between combinations of goods and services that can be produced and the combinations that cannot be produced, given the available factors of production - land, labor, capital, and entrepreneurship - and the state of technology.

PPF only holds given fixed available resources and technology. If any of these changes, the PPF changes as well!
The whole space represents all possible combinations of the two products we produce. Not all of them are attainable within the current technology and resources. The PPF shows the boundary between two continuums - attainable combinations and unattainable combinations.
If we can attain certain combination, it might not necessarily be the efficient one. By efficiency we suggest that we do not take the full advantage of all available resources and technology. Efficient combinations are part of the PPF curve.

If we are at efficient production and we would like to produce more of one of the product, we need to decrease the amount of other product in return. This shows the tradeoff that we should make in order to change structure of production.
If we however produce inefficiently, we can increase the production of either product without any cost. This refers us to a free lunch - situation in which we attain higher production without a tradeoff.

Suppose we have two goods: A and B. If we use all our resources to produce only one good, we can produce either 100 units of good A or 5 units of good B. Producing 1 unit of good B will allow up to 95 of good A. Producing 2 units of good B - 85 of good A. Producing 3 units of good B - 70 of good A. Finally, when 4 units of good B are produced, we can only produce up to 50 of unit A.
Draw the PPF based on data provided.
Can we produce 2.5 units of good B and 71 units of good A? Is pair \(A=71,\,B=2.5\) attainable?
Can we produce 4.5 units of good B and 25 units of good A? Is it efficient?

In reality, there is no easy transformation of resources and technology between production of two goods. We cannot simply forego the production of 1 integrated circuit to produce 10 more shirts as they use different kind of resources.
What we really want to know is how given aggregated resources (labor and physical capital), we can produce different amount of average good (or a basket of goods).
Most countries are endowed with certain resources (labor represented by working population and physical capital by technological development of a country or a company).
Knowing that, we can place a country or a company and use it to project how PPF will shift if amount of resources change (population may grow) or how two countries may benefit from trading.

You have two goods: C and D.
You have the following production possibilities:
\[ C=50, \, D=0 \,\,\,\,|\,\,\,\, C=0, \, D=50 \\ C=47, \, D=10\,\,\,\,|\,\,\,\, C=10, \, D=47\\ C=40, \, D=20\,\,\,\,|\,\,\,\, C=20, \, D=40\\ C=30, \, D=30 \]
Draw PPF
Draw points \(C=25, \, D=25\), \(C=45, \, D=20\), \(C=20, \, D=45\), \(C=20, \, D=20\) on the graph and identify whether they are attainable or unattainable. If they are attainable, identify if they are efficient or inefficient.

We reviewed the main types of graphs and their properties.
We learned definition of the production possibilities frontier and how to draw it from data.
We qualified the combinations of two goods in several categories: unattainable and attainable; efficient and inefficient. Concept of the tradeoff and free lunch was introduced.