Mathematical Economics Questions and Answers

Answer: C

X is a Proper subset of Y.

If every element of X is in Y, X is a subset of Y. If, in addition, Y has some element not in X, the two sets are not equal, and X is a proper subset, written \(X \subset Y\).

A union set combines two sets; a power set is the set of all subsets of a set; the empty set has no elements at all. None of these describes the stated relationship.

Answer: C

The inverse function is x = f^-1(y).

If \(y = f(x)\), the inverse function reverses the mapping: it takes a value of \(y\) and returns the \(x\) that produced it. So \(x = f^{-1}(y)\). For example, if \(y = 2x + 3\), then \(x = \frac{y - 3}{2}\).

\(y = f(x)\) is the original function. \(y = f^{-1}(x)\) only relabels the variables, and \(x = f(y)\) applies the original function to \(y\), which is not the inverse.

Answer: A

The differential coefficient is \(-\frac{1}{2}\).

Differentiate the demand function with respect to price:

\[ \begin{aligned} q &= 9 - \tfrac{1}{2}p \\ q'(p) &= -\tfrac{1}{2} \end{aligned} \]

The constant 9 drops out, and the slope is negative, as the law of demand requires: quantity falls as price rises. \(\frac{1}{2}\) loses the sign, and 9 is the intercept, not the slope.

Answer: A

The product is the matrix A itself.

The identity matrix \(I\) has 1s on the diagonal and 0s elsewhere. It is the multiplicative identity for matrices: \(AI = IA = A\) for any square matrix \(A\) of the same order, just as multiplying a number by 1 leaves it unchanged.

The identity itself would result only if \(A\) were the identity. The zero matrix comes from multiplying by the zero matrix, and the last option is the transpose of \(A\), which multiplication by \(I\) does not produce.

Answer: B

The isoquant is \(K = 25L^{-1}\).

Set output at 100 and solve for \(K\):

\[ \begin{aligned} 20K^{0.5}L^{0.5} &= 100 \\ (KL)^{0.5} &= 5 \\ KL &= 25 \\ K &= \frac{25}{L} = 25L^{-1} \end{aligned} \]

This rectangular hyperbola is the Q = 100 isoquant. The option \(25K^{-1}\) gives \(K\) in terms of itself, and \(25L\) would make the isoquant slope upward, which an isoquant cannot do when both inputs are productive.

Answer: D

The determinant is 95.

Expand along the first row:

\[ \begin{aligned} |A| &= 4(5 \times 4 - 2 \times 0) \\ &\quad - 6(2 \times 4 - 2 \times 9) \\ &\quad + 1(2 \times 0 - 5 \times 9) \\ &= 80 + 60 - 45 = 95 \end{aligned} \]

The middle term is \(-6 \times (8 - 18) = +60\). Taking it as \(-60\) gives \(80 - 60 - 45 = -25\), and stopping after the first term gives 80, one of the distractors.

Answer: A

The marginal revenue function is 4000 − 20q.

First express price in terms of quantity, then find total revenue and differentiate:

\[ \begin{aligned} q &= 400 - 0.1p \Rightarrow p = 4000 - 10q \\ TR &= pq = 4000q - 10q^{2} \\ MR &= \frac{d(TR)}{dq} = 4000 - 20q \end{aligned} \]

For a straight-line demand curve, MR has the same intercept and twice the slope. Marginal revenue is a function of quantity, so the options in terms of \(p\) are wrong, and \(+20q\) would make MR rise with output.

Answer: C

The point elasticity is − 0.25.

At \(P = 12\): \(12 = 60 - 3q\), so \(q = 16\). From \(P = 60 - 3q\), \(q = 20 - \frac{P}{3}\) and \(\frac{dq}{dP} = -\frac{1}{3}\).

\[ \begin{aligned} e &= \frac{dq}{dP} \cdot \frac{P}{q} \\ &= -\frac{1}{3} \times \frac{12}{16} = -0.25 \end{aligned} \]

Demand is inelastic at this price. The option 0.25 is the same figure without the minus sign that a downward-sloping demand curve gives.

Answer: D

The statements not associated with Leontief's static model are I (final demand is a stochastic vector), II (the reciprocal of the Perron-Frobenius root is the maximum growth rate) and IV (the expression with the matrix RB, which belongs to a dynamic model with capital coefficients). So the code is I, II, IV, option D.

Statement III, that input coefficients are fixed technological parameters, is the basic assumption of the static model, so options A, B and C, which include III, are wrong. In the static model the final demand is a given (exogenous) vector, not a random one.

Answer: C

Two commodities are substitutes if a rise in the price of one raises the demand for the other, so that both cross effects are positive: \(\delta X_{1}/\delta P_{2} \gt 0\) and \(\delta X_{2}/\delta P_{1} \gt 0\). This is option C.

If both cross derivatives are negative (option D) the goods are complements. Options A and B give opposite signs for the two effects, which cannot hold for a pair of substitutes, since the relation of substitution is symmetric.