Question

1. The utility function for an individual is given by the equation: 𝑈 = 𝑋1 0.25𝑋2 0.75 and their budget constraint is 𝑃1𝑋1 + 𝑃2𝑋2 = 𝑀 (i) Using the Lagrange, and showing all work, Derive the demand functions for good 𝑋1 𝑎𝑛𝑑 𝑔𝑜𝑜𝑑 𝑋2 (ii) Given that the price for good 1, p1= k2.5 and the price for good 2, p2=k5, and the consumer’s income is k100. Calculate and graphically present the optimal choice for good 1 and good 2

Answer 1

i

The Lagrangian is set up as follows :-

$L = X_1^{0.25} X_2^{0.75} + λ(M - P_1X_1 + P_2X_2 )$

First order conditions :-

$\frac{\delta L}{\delta X_1}=0.25(\frac{X_2}{X_1})^{0.75}-\lambda P_1=0.................(1)$

$\frac{\delta L}{\delta X_2}=0.75(\frac{X_1}{X_2})^{0.25}-\lambda P_2=0.................(2)$

$\frac{\delta L}{\delta \lambda}=M-P_1X_1-P_2X_2=0.................(3)$

$\frac{(1)}{(2)}$ gives

$\frac{0.25(\frac{X_2}{X_1})^{0.75}}{0.75(\frac{X_1}{X_2})^{0.25}}=\frac{P_1}{P_2}$

simplifying

$X_2=\frac{3P_1X_1}{P_2}$

Now, substituting the value of X₂ in (3) gives -

$M-P_1X_1-P_2(\frac{3P_1X_1}{P_2})=0$

solving

$X_1=\frac{M}{4P_1}$

Substituting this value of X₁ in the previously found value of X₂, we get -

$X_2=\frac{3M}{4P_2}$

These are the required demand functions.

ii

(ii) Given:

P₁ = k 2.5

P₂ = k 5

M = k 100

So X ₁ and X₂  becomes -

$X_1=\frac{100}{4\times2.5}=10$

$X_2=\frac{3\times 100}{4\times 5}=15$

The budget constraint becomes -

2.5X₁ + 5X₂ = 100

Graphical representation :-

Here, the green line shows the budget line while the pink curve shows the indifference curve. The intercepts of the budget line have been obtained as -

$X1 = \frac{100}{2.5} = 40 \\ X2 = \frac{100}{5} = 20$

E is the optimal bundle of the individual.