Question 1 Consider the budget constrained utility maximisation problem between two goods, Good, and Goody. Let x be the quantity of Good, whose price is $13 per unit and y be the quantity of Good, whose price is $15 per unit The total budget is $560. (Therefore, the budget constraint is 13x + 15y = 560). The Utility function is given by U(x, y) = x(y + 1) (a) Solve the problem to maximise the utility, constrained by the budget using Lagrange Multipliers. (b) Show that the solution found does actually maximise the utility function (amongst feasible x and y choices). (c) Calculate the optimal values for x*,y*, λ* and U* (d) Consider now the dual optimisation problem of minimising the cost (Budget, B(x, y)) to achieve the utility U* as calculated in (c). Relate the optimal values of this problem to those found in (c) and also B*.
Question 1 Consider the budget constrained utility maximisation problem between two goods, Good, and Goody. Let x be the quantity of Good, whose price is $13 per unit and y be the quantity of Good, whose price is $15 per unit The total budget is $560. (Therefore, the budget constraint is 13x + 15y = 560). The Utility function is given by U(x, y) = x(y + 1) (a) Solve the problem to maximise the utility, constrained by the budget using Lagrange Multipliers. (b) Show that the solution found does actually maximise the utility function (amongst feasible x and y choices). (c) Calculate the optimal values for x*,y*, λ* and U* (d) Consider now the dual optimisation problem of minimising the cost (Budget, B(x, y)) to achieve the utility U* as calculated in (c). Relate the optimal values of this problem to those found in (c) and also B*.
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Transcribed Image Text:Question 1
Consider the budget constrained utility maximisation problem between two goods,
Good, and Goody.
Let x be the quantity of Good, whose price is $13 per unit and
y be the quantity of Good, whose price is $15 per unit
The total budget is $560. (Therefore, the budget constraint is 13x + 15y
The Utility function is given by U(x, y)
= x(y + 1)
=
560).
(a) Solve the problem to maximise the utility, constrained by the budget using Lagrange
Multipliers.
(b) Show that the solution found does actually maximise the utility function (amongst
feasible x and y choices).
(c) Calculate the optimal values for x*, y*, λ* and U*
(d) Consider now the dual optimisation problem of minimising the cost (Budget, B(x, y))
to achieve the utility U* as calculated in (c). Relate the optimal values of this problem
to those found in (c) and also B*.
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Transcribed Image Text:(a) Solve the problem to maximise the utility, the constraint by the budget using Lagrange multipliers.
Step 1: Initialize the Lagrange function:
L(x, y, ) = (13x + 15y - 560) x(y + 1)
Step 2: Take the partial derivatives and multiply them by zero:
ÔL/ôx = y + 1 - 13λ = 0 (1)
ÔL/ôy = x - 15λ = 0 (2)
13x + 15y - 560 = 0 (3)
Step 3: Concurrently solve the systems of equations (1), (2), and (3):
x = 15λ in equation (2).
When we substitute x in equation (3), we get:
13(15)) + 15y - 560 = 0 195A + 15y - 560 = 0 15y = 560 - 195X (4)
Substituting x = 15 and equation (4) into equation (1)
yields: y + 1 - 13 = 0 y = 13 - 1 (5)
Substituting equation (5) for equation (4):
15(13λ - 1) = 560 - 195X
195A - 15 = 560 - 195A
390A = 575 = 575/390
A = 1.474358974 ≈1.474
Substituting 1.474 into equation (5) yields:
y = 13(1.474) - 1
y = 18.16666667≈ 18.167
Substituting x = 15A into equation (2) yields:
x = 15(1.474)
x = 22.11538462 ≈ 22.11
As a result, the best values are x = 22.11 & y = 18.167.
Transcribed Image Text:(b) Show that the solution found does actually maximise the utility function (amongst feasible x and y choices).
To demonstrate that the solution identified maximises the utility function, we must first determine whether it meets the second-
order requirements for a maximum. This necessitates computing and evaluating the second-order partial derivatives of the Lagrange
function at optimum values.
O²L/ox² = 0 O²L/oy² = 0 8²L/əxây = 1
We may deduce that the solution maximises the utility function since the second-order partial derivatives are constant and do not
rely on x or y.
(c) Calculate the optimal values for x*, y*, λ*, and U*
Determine the best values for x, y, and U: We get x = 22.11538462 22.11 y = 18.16666667 ≈ 18.167. A = 1474358974 ≈ 1.474
U( x, y) = x (y+1 ) = 22.11( 18.167 +1) = 423.88
(d) Consider now the dual optimisation problem of minimising the cost (Budget, B(x, y)) to achieve the utility U* as
calculated in (c). Relate the optimal values of this problem to those found in (c) and also B*.
The goal of the dual optimisation problem is to minimise the cost (budget) B(x, y) in order to maximise the utility U as estimated in
(c).
Because we now know the best values for x and y, we can plug them into the budget constraint equation:B 13(22.11538462) +
15(18.16666667) = B 287.43 + 272.43 = B
B≈ 560
The best solution for the dual optimisation issue is B = 560, which equals the entire budget.
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