Uploaded by Frank Chilufya

LAGRANGE

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Given chileshe’s utility for ribs , R, and chicken C, is given by U=10R2C,
(a) Derive the Marshallian demand equations for R and C.
MARSHALLIAN DEMAND FUNCTION is derived by maximising utility, subject to the
budget constraint.
Our objective function is U= 10R2C
Our constraint is M= PR* R+ PC*C
We formulate our lagrangian function L as follows:
𝐿 = 10𝑅 2 𝐢 + πœ†(𝑀 − 𝑃𝑅 𝑅 − 𝑃𝐢 𝐢)
The first order condition requires we obtain partial derivatives of the function with respect to
QR, Qc and λ and equate to zero.
πœ•πΏ
πœ•π‘…
πœ•πΏ
πœ•πΆ
πœ•πΏ
πœ•πœ†
= 20𝑅𝐢 − πœ†π‘ƒπ‘… =0 ………………EQUATION 1
= 10𝑅 2 − πœ†π‘ƒπΆ =0 …………………..EQUATION 2
= 𝑀 − 𝑃𝑅 𝑅 − 𝑃𝐢 𝐢=0 ………...EQUATION 3
From equation 1 we solve for λ
20𝑅𝐢 − πœ†π‘ƒπ‘… =0
20𝑅𝐢 = πœ†π‘ƒπ‘…
20𝑅𝐢/𝑃𝑅 = πœ† ……….EQUATION 4
From equation 2 we solve for λ
10𝑅 2 − πœ†π‘ƒπΆ =0
10𝑅 2 = πœ†π‘ƒπΆ
10𝑅 2 /𝑃𝐢 = πœ† ……………. EQUATION 5
Now since equation 4 and 5 are both equal to λ, we can equate the two equations as follows:
20𝑅𝐢 10𝑅 2
=
𝑃𝑅
𝑃𝐢
We then solve for R and C
20RCPC=10R2PR
C=10R2PR/20RPC
C=R2PR/2RPC
C=RPR/2PC…………………………………. EQUATION 6
R2/R= 20CPC/10PR
R=2CPC/PR ……………………………………EQUATION 7
To find the demand equation for R, we replace equation 6 into the constraint
𝑀 = 𝑃𝑅 𝑅 + 𝑃𝐢 𝐢
𝑀 = 𝑃𝑅 𝑅 + 𝑃𝐢 (
R𝑃𝑅
)
2𝑃𝐢
The PC will cancel out, so that,
M= 𝑃𝑅 𝑅 +
M=
𝑃𝑅 𝑅
2
3𝑃𝑅 𝑅
2
2M=3𝑃𝑅 𝑅
2𝑀
R= 3𝑃
𝑅
…………………………………………. DEMAND EQUATION FOR R
To find the demand equation for C, we replace equation 7 into the constraint
𝑀 = 𝑃𝑅 (
2C𝑃𝐢
) + 𝑃𝐢 𝐢
PR
M= 2PCC+ 𝑃𝐢 𝐢
M= 3𝑃𝐢 𝐢
𝑀
C= 3𝑃 ………………………………………………. DEMAND EQUATION FOR C
𝐢
(b) If income is k90, price of ribs k10, and price of chicken wing is k15,
What quantities of x and y will maximise utility?
2𝑀
R= 3𝑃
𝑅
2∗90
…………………………………………. DEMAND EQUATION FOR R
R= 3∗10 =
180
30
= 6 …………………. UTILITY MAXIMISING UNITS OF RIBS
𝑀
C= 3𝑃 ………………………………………………. DEMAND EQUATION FOR C
𝐢
90
C= 3∗15 = 90/45=2…… UTILITY MAXIMIZING UNITS OF CHICKEN
WE CAN PROVE THAT TOTAL EXPENDITURE IS EQUAL TO INCOME AT THIS
POINT.
PCC+ PRR=M
15*2 + 10*6=30+ 60=90.
(C) what is the level of utility at this optimal level?
U=10R2C,
R=6, C=2
U= 10*62* 2=10*36*2= 720
(d) Plot the budget line and include an indifference curve on the same diagram to show the
optimal combination obtained above
To draw the budget line, we need to obtain the two intercepts so that we join then with a
straight line.
With chicken on the y axis, we know that the maximum amount of chicken wings chileshe
can buy will be when all income of 90 is spent on chicken wings of k15 each= M/PC 90/15=
6C
With ribs on the x axis, we know that the maximum amount of ribs chileshe can buy will be
when all income of 90 is spent on ribs of k10 each=M/PR= 90/10= 9R.
Chicken
6
2
U=720
6
9
Ribs
(e )Mathematically prove that at the above optimal combination of goods, the slope of the
budget line equals the slope of the indifference curve
The slope of the budget line can be obtained in two ways. Change in y/change in x=-6/9=
0.67
Or by using the formula –px/py= - Pr/Pc= -10/15= -0.67
Meanwhile, the slope of the indifference curve, MRSxy is given by MUx/MUy,
Since ribs is on the x axis, and chicken on the y axis, and since U=10R2C
Mux= MU ribs= dU/dR =20RC
At equilibrium, R= 6, c=2,
Such that MuR= 20*6*2= 240
MuY= MU chicken= dU/dC=10R2
At equilibrium, R= 6, c=2,
Such that MuR= 10*62= 10*36=360
MUx/MUy = 240/360= 0.67
HENCE PROVED THAT THE SLOPE OF THE BUDGET LINE IS EQUAL TO THE
SLOPE OF THE INDIFFERENCE CURVE AT THE OPTIMALPOINT
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