Concepts of Mathematical Economics Questions and Answers [With PDF]
Economics uses a little mathematics to say things precisely: how quantity demanded changes with price, how cost changes with output, or where a market settles. This chapter covers the basic tools: variables, constants, functions, graphs, slope and intercepts. Here are 20 important questions with clear answers and worked examples.
This is Chapter 3 of our Economics course. It follows Concepts of Economics and prepares you for Theory of Consumer Behavior.
Variables and Constants
Q1. What is a variable?
A variable is a quantity that can take different values and can be measured with numbers. Examples in economics: price, quantity demanded, income, profit, consumption, investment, exports and imports.
Q2. What are the features of a variable?
- Its value can change.
- It is usually written with letters such as x, y, P, Q or Y.
- It can take many real-number values, such as a price of 10, 12 or 15.
- Some variables are independent and others are dependent.
Q3. How are variables classified?
By dependence, into independent variables and dependent variables.
Q4. What is an independent variable?
A variable whose value is chosen or given from outside the relationship. It does not depend on the other variable. Example: in the demand relationship, price is the independent variable.
Q5. What is a dependent variable?
A variable whose value is determined by the value of the independent variable. Example: quantity demanded depends on price.
Q6. What is a constant?
A quantity whose value does not change within the problem. It is written as a number (such as 5 or 100) or as a fixed symbol (such as a, b or c).
Q7. What is the difference between a variable and a constant?
| Point | Variable | Constant |
|---|---|---|
| Value | Changes | Does not change |
| Symbols | Usually x, y, z, P, Q | Usually a, b, c or fixed numbers |
| Types | Independent and dependent | Coefficients and parameters |
| Example | Price P, quantity Q | The 100 and 2 in Q = 100 – 2P |
Q8. What is a coefficient?
A constant that multiplies a variable. In 2X, the variable is X and 2 is its coefficient. In Q = 100 – 2P, the coefficient of P is -2.
Q9. What is a parameter?
A constant that stays fixed within one problem but can take different values in different problems. In y = a + bx, the letters a and b are parameters. Changing them gives a different line.
Functions
Q10. What is a function?
A function is a rule that links a dependent variable to one or more independent variables, so that each value of the independent variable gives exactly one value of the dependent variable. It is written y = f(x). Example: Q = f(P), which says quantity depends on price.
Q11. What are the features of a function?
- It has at least one independent variable and one dependent variable.
- The relationship between them is written mathematically.
- Each input gives only one output.
- It has exactly one dependent variable.
Q12. How are functions classified?
| Basis | Types |
|---|---|
| Number of independent variables | Univariate and multivariate functions |
| Form of the equation | Constant, linear, quadratic, cubic, polynomial, rational and modulus functions |
Linear, quadratic and cubic functions are all special cases of polynomial functions.
Q13. What are univariate and multivariate functions?
- Univariate: one independent variable. Example: Q = 100 – 2P.
- Multivariate: two or more independent variables. Example: Q = 100 – 2P + 0.5Y, where quantity depends on both price and income.
Q14. What is a linear function?
A function of the first degree whose graph is a straight line. Its general form is y = a + bx, where b is the slope and a is the vertical (y) intercept.
Q15. What is a quadratic function?
A second-degree function whose graph is a curve (a parabola). Its general form is y = ax2 + bx + c, where a is not zero. Example: a firm’s profit that first rises and then falls as output grows.
Q16. What is a cubic function?
A third-degree function with the general form y = ax3 + bx2 + cx + d, where a is not zero. Total cost curves in economics are often cubic.
Graphs, Slope and Applications
Q17. What is a graph, and what are common economic functions?
A graph is a picture of a function on two axes. The independent variable is normally on the horizontal (x) axis and the dependent variable on the vertical (y) axis. In demand and supply diagrams, price is traditionally placed on the vertical axis, even though quantity depends on price.
| Function | Example | Meaning |
|---|---|---|
| Demand | Qd = 100 – 2P | If price rises by 1, quantity demanded falls by 2 |
| Supply | Qs = 20 + 3P | If price rises by 1, quantity supplied rises by 3 |
| Consumption | C = 200 + 0.8Y | Autonomous consumption 200, and 0.8 of each extra income is spent |
| Total cost | TC = 100 + 5Q | Fixed cost 100, and 5 for each extra unit |
Q18. What is slope?
Slope is the change in the dependent variable divided by the change in the independent variable: slope = change in y / change in x. It shows how fast y changes when x changes by one unit.
Example: In Qd = 100 – 2P, when price rises from 10 to 11, quantity falls from 80 to 78. The slope is -2 / 1 = -2.
For a curved function, the slope changes from point to point. The derivative gives the slope (the rate of change) of the function at a single point. In economics, derivatives give marginal values. For example, if TC = 100 + 5Q + Q2, then marginal cost MC = d(TC)/dQ = 5 + 2Q.
Q19. What is an intercept?
An intercept is the point where a line or curve crosses an axis. The x-intercept is where it crosses the x-axis (y = 0), and the y-intercept is where it crosses the y-axis (x = 0). In Qd = 100 – 2P, the quantity intercept is 100 (when P = 0) and the price intercept is 50 (when Qd = 0).
Q20. How do you find market equilibrium using functions?
Equilibrium is where the demand and supply curves intersect, meaning quantity demanded equals quantity supplied. Set Qd = Qs and solve.
| Step | Working |
|---|---|
| Demand | Qd = 100 – 2P |
| Supply | Qs = 20 + 3P |
| Set Qd = Qs | 100 – 2P = 20 + 3P |
| Solve for P | 80 = 5P, so P = 16 |
| Find Q | Q = 100 – 2(16) = 68 (check: 20 + 3(16) = 68) |
The equilibrium price is 16 and the equilibrium quantity is 68.
Quick Revision Points
- A variable changes. A constant does not. A parameter is a constant that can differ between problems.
- A function gives exactly one output for each input.
- Linear: y = a + bx. The slope is b, and a is the y-intercept.
- Slope = change in y / change in x. The derivative is the slope at a point.
- Equilibrium: set demand equal to supply and solve.
Test yourself with the chapter quiz below, then continue to Theory of Consumer Behavior.
