Mathematical Economics Questions and Answers

Answer: B

Substituting demand into supply gives the dynamic equation of the cobweb.

The price path converges only if the absolute value of the coefficient of the lagged price is below 1, so |delta beta| < 1, which means |delta| < |1/beta|. The option |delta| = |1/beta| gives a steady oscillation, and the other two are not conditions on the product.

Answer: B

The optimal solution of a linear program with two decision variables is either an extreme point or lies on a line segment connecting two extreme points, which happens when the objective is parallel to a binding constraint.

A linear program does not have only one solution, because there can be alternative optima. Not all resources need to be used up, since some constraints may have slack. So the 'all of the above' option is wrong as well.

Answer: B

A shadow price is the change in the value of the objective function per unit change in any one of the constraints.

In linear programming the shadow price (dual value) of a constraint tells how much the optimal value of the objective, such as profit or cost, would change if the right-hand side of that constraint, the resource available, were changed by one unit.

The first option reverses the roles of the constraint and the objective. The third option mixes primal and dual solutions in a way that is not the definition, and 'all the above' fails since at least two of the other statements are not the definition.

Answer: A

The conditions postulated for a viable input-output system are the Hawkins-Simon conditions.

The Hawkins-Simon conditions require that the principal minors of the matrix \((I - A)\) be positive. They ensure that the system can produce a non-negative output for every final demand.

The Kuhn-Tucker conditions apply to non-linear programming, and the Bhagwati-Eckaus and Von Neumann conditions belong to growth and planning models, not the viability of the input-output system.

Answer: C

The correct code is iv i ii iii.

In the CES production function \(\gamma\) is the efficiency parameter (iv), \(\delta\) is the distribution parameter (i), \(\rho\) is the substitution parameter (ii) and \(\mu\) is the returns to scale parameter (iii).

The substitution parameter \(\rho\) gives the elasticity of substitution \(\sigma = \frac{1}{1+\rho}\). The parameter outside the bracket, \(\gamma\), multiplies output and measures efficiency. The other codes give these roles to the wrong symbols.

Answer: B

The Linear Programming Technique was developed for the first time by Dantzig.

George Dantzig devised the simplex method in 1947, which made linear programming a practical technique.

Koopmans worked on activity analysis and the allocation of resources, Leontief on input-output analysis, and Mahalanobis on statistics and planning. They did not develop the technique.

Answer: C

Average cost equals marginal cost at an output of 15. With \(AC = \frac{1}{3}x^{2} - 10x + 9\), total cost is:

Setting AC equal to MC:

The values 12, 18 and 21 do not satisfy this equation.

Answer: C

The consumer surplus is 62.5. Equating demand and supply gives the equilibrium:

Consumer surplus is the triangle between the demand curve and the price line, with the demand intercept at 30:

Surplus cannot be negative, which rules out the negative options, and 125 is double the triangle's area.

Answer: D

The key says the problem has no solution, meaning no optimal solution. Because \(x_2 \lt 0\) is a strict inequality, the feasible region is open at \(x_2 = 0\), and the objective is improved by letting \(x_2\) approach zero from below, so the maximum (a supremum of 118.75, with \(x_3 = 95/8\)) is approached but never attained.

A feasible solution does exist, and a reader could call the first option true, but the question asks about the solution of the maximisation problem. No attained optimum and no ties for multiple solutions exist, so 'no solutions' is the keyed answer.