Microeconomic Theory
12th Edition
ISBN: 9781337517942
Author: NICHOLSON
Publisher: Cengage
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(A) We want to minimize the cost function C(x, y) = 9x + 12y subject to the constraint U(x, y) =x (y + 3) = 128.9 and non-negativity constraints × ≥ 0 and y ≥ 0
We set up the Lagrangian:
L(x, y, λ) = C(x, y) - λ(U(x, y) - 128.9) Now, we calculate the partial derivatives with respect to x, y, and 1:
∂L/∂x = 9 - λ(y + 3)
∂L/∂y = 12 - λX
∂L/λ = 128.9 - U(x, y)
Setting these partial derivatives equal to zero, we have:
9 - λ(y + 3) = 0
2 12 - λx = 0
128.9 - x(y + 3) = 0
From equation 2, we get λ= 12/x. Substituting A into equations 1 and 3:
9 - (12/x)(y + 3) = 0
2. 128.9 - xY - 3x = 0
Now, we can solve this system of equations simultaneously. Let's start by solving equation 2 for X:
128.9 - xy - 3x = 0
xy = 128.9 - 3x
Now, substitute this into equation 1:
9 - (12/x)(y + 3) = 0
9-(12/x)(128.9 - 3x + 3) = 0
9- (12/x)(131.9 -3x) = 0
9 -1,582.8/x + 36x = 0
Multiply through by x to get rid of the fraction:
9x - 1.582.8 ÷…
A decrease in people's disposable income OA. increases investment demand. B. increases consumption. OC. decreases saving. O D. increases saving and decrease consumption. OE. increases saving.
Note:-
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(a) Dividing MUX and MUY by their respective prices compute the weighted
marginal utility or marginal utility of money expenditure and draw up Table 2 showing diminishing returns for the consumption of the 6 units?
(b) Briefly state what you can infer on this equation; MUX/PX = MUY/PY as far as equi marginal principle and managerial economics is concerned?
C)Given that MUX/PX and MUY/PY are equal to 6 when 5 units of X and 3 units of Y are purchased. By purchasing these combinations of X and Y, calculate
the amount the consumer will spend and derive the maximum satisfaction from combination of these units.
(d) Applying the above principle illustrate in terms of a figure, and draw marginal
utility curves for goods X and Y. You need to use marginal utility and price
i.e. Marginal utility per Kwacha spent on good X = MUX/PX, and that of Y =
MUY/PY. The MUX/PX curve should be shown in figure (a) while the
MUY/PY curve should be shown in Fig (b). Please do not draw negative…
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