Math for Economists Trivia Quiz

Answer: First Order Condition

The first order condition uses the first derivative to find a value for the variable in question that is at a local extreme on the original function.
When the derivative is equal to 0, this means the line tangent to the function at that point is perfectly horizontal, ensuring that the function is at a local maximum or minimum. The first order condition, or the first derivative test, cannot determine whether the point is at a maximum or minimum alone. This requires the use of the second order condition. Together, they form the basic concept of optimization.

Answer: 4x - 6

To find the marginal cost, one must differentiate the cost function with respect to the variable in question, in this case x.
By the exponent rule, the derivative of 2x^2 with respect to x yields 4x, and the derivative of -6x with respect to x yields -6.
Using the summation rule of derivatives, the first derivative of the cost function is 4x - 6.
5 is the fixed cost of firm, and is not included in the marginal cost. It does not change as x increases, and is therefore irrelevant in terms of marginal thinking.

Answer: Integration

The marginal cost is the derivative of the total cost, and therefore in order to find the total cost given the marginal cost, one must compute the anti-derivative, or indefinite integral. As per the fundamental theorem of calculus, integration can be reversed by differentiation.
Using indefinite integrals to calculate the total cost of a firm given its marginal cost yields an infinite amount of positive fixed costs, as the constant of integration must be added to the total cost.
Therefore, if the only given information is the marginal cost, it is impossible to compute the fixed cost of a firm.

Answer: U= X^a,Y^b

The Cobb-Douglas function (named for Charles Cobb and Paul Douglas) is convex to the origin and never meets with the X-axis or the Y-axis, ensuring that corner solutions are impossible, and the consumer cannot specialize in one good. An example would be consumption and leisure.
U= min(X,Y) is a Leontief function (named for Wassily Leontief), reserved for a consumer with perfectly complimentary preferences. The curves will be L shaped, and the properties of the Leontief function ensure that the consumer must purchase both goods in a fixed proportion. An example would be peanut butter and jelly.
U= 2x + Y and U= X + 2Y are both linear utility functions, reserved for a consumer with preferences that are perfectly interchangeable, or perfect substitutes. That is, given that their prices are the same, the consumer will purchase either of the two goods and receive the same level of utility. An example would be Pepsi and Coke.

All three types of utility functions mentioned above can also be used as production functions to solve a firm's problem.

Answer: Lagrange Multiplier

The Lagrange multiplier (named after Joseph Louis Lagrange) introduces a new variable into the function, represented by the Greek letter "Lambda". This new variable is then multiplied into the constraint when set to zero, and added to the function to be optimized.
The new function L, now including all variables to be optimized, in addition to Lambda, multiplied by the constraint set to zero, is differentiated and set equal to zero (First order condition):
- once with respect to each variable in the original function to be optimized
- once with respect to "Lambda"
By solving the resulting system of equations, which will subsequently eliminate the added variable "Lambda", one can find the required optimum with respect to each variable.

Answer: Marginal Rate of Transformation

The Production Possibility Frontier is a graphical representation of the production rate of two selected goods. When the competitive equilibrium lies on the PPF, no improvement to either the consumer or the firm can be achieved without hurting the other party.

This equilibrium is referred to as Pareto efficiency, named after Vilfredo Pareto, an Italian Economist. The Marginal Rate of Transformation, the slope of the PPF, is negatively sloped and extends below the x-axis due to Governmental purchases. The Laffer Curve is a graphical representation of the relationship between taxation rate and tax revenue, whereas the Isoquant is another name for the Production Function.

Answer: False

The leader in Stackelberg competition, or any other form of sequential competition, carries a "first-mover's advantage". To solve a sequential game, one must solve the follower's problem first, and derive a perfect "reaction function." This is called backwards induction, or retrograde analysis. A reaction function is the best response to any choice made by the leader. Knowing fully well the follower's reaction function, the leader chooses the quantity that maximizes his utility or profit. The follower then utilizes his reaction function to choose the quantity that optimizes his utility or profit.
Therefore, the follower's reaction function in a Stackelberg oligopoly can be represented as:
Y2 = f(Y1) where Y2 is the follower's optimal quantity and Y1 is the leader's optimal quantity, given the follower's reaction function.

Answer: Price = Marginal Cost

As the number of firms increase in the market, so does the firm's elasticity of demand. In a perfectly competitive market, the elasticity of demand becomes perfectly elastic, and the firms are price-takers. That is, they will charge the lowest possible price they can afford. If not, they risk losing all their customers to other firms (products are assumed to be identical).
The lowest possible price that a firm can charge is at its marginal cost. Charging above that will result in zero market share, and charging below that will result in economic losses.
There are virtually no exit and entry barriers in perfectly competitive markets, ensuring zero economic profit in the long-run as P=MC.

Answer: C'/(1+r)

The present value of any future parameter can be calculated by dividing that parameter by (1 + the interest rate). Assuming one consumes 11 apples in the future period, and the interest rate is determined to be 10% over two periods, then the present value of future consumption would be: 11/(1+0.1) = 10 apples.

The value of 11 apples in the future period is equivalent to the value of 10 apples today, as the interest rate distorts future value. In order to make proper decisions in the lifetime budget constraint, one must discount the effect of the interest rate, and this is achieved by finding the present value of future goods.

Answer: Slutsky Equation

Ukrainian mathematician and economist Eugen Slutsky derived the Slutsky equation to determine how a change in prices would change the demand of a certain good. First, the equation analyzes the substitution effect, or the factor of exchange between two goods.

When the price of one good decreases, for example, the opportunity cost of purchasing the other good increases due to substitution effect. Secondly, the equation analyzes income effect, or the effect of change in the consumer's purchasing power. Both these effects together combine to create the total effect, but can do so in different ways.

The behavior of the SE and IE will determine whether the good is a normal good, inferior good, or Giffen Good.