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Mathematical Economics Questions and Answers

31.

In CES production function :
\(Q = M\,[\alpha X_{1}^{-\rho} + (1 - \alpha) X_{2}^{-\rho}]^{-1/\rho}\)
Where Q is output and \(X_{1}\) and \(X_{2}\) are labour and capital. Elasticity of substitution is given by

Exam Year(s)

UGC NET - 2012,

Answer: C

For the CES production function \(Q = M[\alpha X_{1}^{-\rho} + (1 - \alpha)X_{2}^{-\rho}]^{-1/\rho}\), the elasticity of substitution is \(\sigma = \frac{1}{1 + \rho}\). It is constant at all points, which is why the function is called a constant elasticity of substitution function.

When \(\rho = 0\) the elasticity is one and the function becomes the Cobb-Douglas form; when \(\rho \to \infty\) it is zero (fixed proportions) and when \(\rho = -1\) it is infinite. The expressions \(-\rho\) and \(-1/\rho\) do not give these limits, and M is the efficiency parameter, so options A, B and D are wrong.

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32.

In Linear Programming problem involving two variables, multiple optimal solutions are obtained when one of the constraints is

Exam Year(s)

UGC NET - 2012,

Answer: A

In a linear programming problem with two variables, multiple optimal solutions arise when the objective function is parallel to a constraint that forms a boundary of the feasible region. The objective line then touches an entire edge of the feasible region at the optimum, and every point on that edge gives the same value.

A perpendicular objective function does not give this result, and two constraints parallel to each other only restrict the region without making the optimum non-unique, so options B and D are wrong; option C is wrong because option A holds.

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33.

In an open economy model of input output analysis, if A is the input coefficients matrix and \((I - A)\) is technology matrix, the following is the viability condition to be satisfied

Exam Year(s)

UGC NET - 2012,

Answer: D

For the open input-output model, the viability condition (Hawkins-Simon) is that all the principal minors of the matrix \((I - A)\) must be positive, so that the system gives non-negative output for any non-negative final demand. This is option D.

The determinant being non-zero (option A) is needed only for the inverse to exist, a positive determinant (option B) is necessary but not sufficient, since the other minors also have to be positive, and a negative determinant (option C) is not viable.

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34.

For the Cobb-Douglas Production function
\(Q = A L^{\alpha} K^{1 - \alpha}\)
The elasticity of substitution is

Exam Year(s)

UGC NET - 2012,

Answer: C

For the Cobb-Douglas production function \(Q = A L^{\alpha} K^{1 - \alpha}\), the elasticity of substitution is one at every point, as it is a special case of the CES function with \(\rho = 0\).

Zero is the elasticity for fixed proportions (the Leontief function) and infinity for perfect substitutes, so options A and B are wrong, and A (the efficiency parameter) is a constant that does not measure the elasticity.

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35.

The input coefficients/elements of which of the following matrices are interpreted as inputs required directly and indirectly per unit of final demand.

Exam Year(s)

UGC NET - 2012,

Answer: C

The elements of the matrix \((I - A)^{-1}\), the Leontief inverse, are interpreted as the inputs required directly and indirectly per unit of final demand. This is option C.

The matrix A gives only the direct input coefficients, \((I - A)\) is the technology matrix and not an inverse, so options A and B are wrong, and 'all of the above' is also wrong.

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36.

Which of the following is the condition(s) of economic viability and technological feasibility of Leontief's static model ?

Exam Year(s)

UGC NET - 2012,

Answer: D

The conditions of economic viability and technological feasibility of Leontief's static input-output model are the Hawkins-Simon conditions, which require that the principal minors of the matrix (I minus A) be positive so that a non-negative output vector exists for every final demand. The Perron-Frobenius root is a result on the eigenvalues of non-negative matrices, and the Kuhn-Tucker conditions are for non-linear programming. The equality of the primal and dual objective values belongs to linear programming duality. So the keyed answer is option D.

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37.

The production function \(Q = A\,[\beta K^{-a} + \alpha L^{-a}]^{-1/a}\) is homogeneous of degree

Exam Year(s)

UGC NET - 2012,

Answer: B

The function is the constant elasticity of substitution form, and it is homogeneous of degree one. If both inputs are multiplied by a factor \(\lambda\), the bracket becomes \(\lambda^{-a}\) times the original bracket, because each term contains one input raised to the power \(-a\). Raising to the power \(-1/a\) gives \(\lambda\) times the original output. So \[ Q(\lambda K, \lambda L) = \lambda Q(K, L) \] and the degree is one, option B. Zero, infinity and \(a\) do not follow, since the degree of homogeneity is fixed by the outer exponent matching the inner one.

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38.

Given the total cost function \(C = 7Q^{2} + 26Q + 5\), MC at Q = ₹ 5 is

Exam Year(s)

UGC NET - 2012,

Answer: D

Marginal cost is the derivative of the total cost function with respect to output. With \(C = 7Q^{2} + 26Q + 5\), \[ MC = \frac{dC}{dQ} = 14Q + 26 \] and at the output of 5 \[ MC = 14 \times 5 + 26 = 96. \] So the marginal cost is Rs 96, option D. The fixed cost of 5 drops out when differentiating. Rs 100, Rs 200 and Rs 85 do not come from the derivative; for comparison, the total cost at an output of 5 is Rs 310, and 14 times 5 alone is only 70.

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39.

Which one of the following is not an assumption of linear programming ?

Exam Year(s)

UGC NET - 2013,

Answer: C

Increasing returns to scale is not an assumption of linear programming. The technique assumes proportionality, so output grows in the same proportion as inputs (constant returns), together with additivity, divisibility and constant prices.

Constant output prices, constant input prices and technologically fixed factor proportions are all standard assumptions, since the objective function and the constraints must be linear. That is why the one allowing increasing returns is the answer to this 'which is not' question.

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40.

Input-Output Technique was invented by

Exam Year(s)

UGC NET - 2013,

Answer: B

The Input-Output technique was invented by Wassily Leontief, who won the Nobel Prize in 1973 for it. The method sets out the interdependence of the sectors of an economy in a table of flows.

Gunnar Myrdal worked on development and cumulative causation, Chenery on development patterns and Robert Solow on growth theory, so they are not the inventors. Leontief's table lets an analyst trace how a change in demand for one sector's output spreads through the other sectors that supply it with inputs.

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