ECON 222: MATHEMATICS FOR BUSINESS
TUTORIAL TEST 4
PRACTICE QUESTIONS
Question 1
Given the production function
calculate both the marginal rate of technical substitution and the slope of the isoquant
at the input combination .
Question 2
Solve the following problem:
using the step-by-step procedure.
Question 3
This question refers to the following optimization problem:
Assume that the domain of both the objective function and the constraining function is the non-negative quadrant, that is,
The graphical representation of the constraint can be seen in Figure 1. The equation of the tangent line at the point is
Figure 1
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ECON 222: MATHEMATICS FOR BUSINESS
TEST 4
ANSWERS TO PRACTICE QUESTIONS
Question 1
Given the production function
calculate both the marginal rate of technical substitution and the slope of the isoquant
at the input combination .
Answe
The first partials of the production function are
Therefore,
Marginal Rate of Technical Substitution at is
Slope of the isoquant at is
Note
The equation of the isoquant is
Note that the input combination satisfies this equation.
Question 2
Solve the following problem
using the step-by-step procedure.
Answe
Step 1: First-order conditions
Step 2: Solve for x and y
Only one solution:
Step 3: Compute the Hessian determinant
Step 4: Evaluation of at the solution of
Step 5: Locating maxima/minima
From Step 4, we know that
Evaluation of at shows that
Step 6: Optimum value
Evaluating the objective function at
We find
Question 3
The question refers to the following optimization problem:
Assume that the domain of both the objective function and the constraining function is the non-negative quadrant, that is,
The graphical representation of the constraint can be seen in Figure 1. The equation of the tangent line at the point is
The optimization problem has been given to an economist and a mathematician. The economist claims that the problem has no solution and the mathematician asserts that there is a unique solution to the problem.
(a) Solve the optimization problem using the step-by-step procedure.
(b) Who is right, the economist or the mathematician? To gain a positive mark you must justify your answer.
Figure 1
Answe
(a) Solve the optimization problem using the step-by-step procedure
Step 0: Lagrangian function
Step 1: First-order conditions
Step 2: Solve for x, y, and
Eliminating the Lagrange multiplier from the first two equations of , yields
Substituting into the last equation of , we find
,
so that
Finally, from the first equation we find
To sum up, the admissible solution of the is
Step 3: Bordered Hessian determinant
, ,
Therefore,
Step 4: Expand
Step 5: Locating maxima/minima
Step 6: Optimum value
The objective function is
(b) Who is right, the economist or the mathematician? To gain a positive mark you must justify your answer.
The mathematician is right.
If you move along the constraint toward the , you will keep on increasing the value of the objective function
but you cannot go beyond .
The constrained-maximum problem has a solution, which is
This point is feasible because belongs to the non-negative quadrant:
Moreover,
satisfies the constraint of the problem:
In fact,
The value of the constrained maximum is
The Lagrange multiplier technique cannot find the solution because
is a corner solution.
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