Distinguish between ordinal and cardinal utility approach in consumer behaviour. Explain the assumptions of cardinal utility.

Kazi ya Lagrangian inajumuisha kazi ya matumizi na kizuizi cha bajeti.

$$\mathcal{L}(x_1, x_2, \lambda) = x_1 x_2 - \lambda(P_1 x_1 + P_2 x_2 - M)$$

Tunachukua derivatives za sehemu ya $\mathcal{L}$ kwa heshima ya $x_1$, $x_2$, na $\lambda$ na kuziweka sawa na sifuri.

1. $\frac{\partial \mathcal{L}}{\partial x_1} = x_2 - \lambda P_1 = 0 \implies x_2 = \lambda P_1$ (Mlinganyo 1)
2. $\frac{\partial \mathcal{L}}{\partial x_2} = x_1 - \lambda P_2 = 0 \implies x_1 = \lambda P_2$ (Mlinganyo 2)
3. $\frac{\partial \mathcal{L}}{\partial \lambda} = -(P_1 x_1 + P_2 x_2 - M) = 0 \implies P_1 x_1 + P_2 x_2 = M$ (Mlinganyo 3)

Kutoka Mlinganyo 1 na 2, tunaweza kutoa $\lambda$:
Kutoka (1): $\lambda = \frac{x_2}{P_1}$
Kutoka (2): $\lambda = \frac{x_1}{P_2}$

Sasa tunalinganisha maneno haya mawili ya $\lambda$:

$$\frac{x_2}{P_1} = \frac{x_1}{P_2}$$ $$P_2 x_2 = P_1 x_1$$

Tunatenga $x_2$:

$$x_2 = \frac{P_1 x_1}{P_2}$$

Badilisha usemi wa $x_2$ tuliojipatia kwenye Mlinganyo 3:

$$P_1 x_1 + P_2 \left(\frac{P_1 x_1}{P_2}\right) = M$$ $$P_1 x_1 + P_1 x_1 = M$$ $$2 P_1 x_1 = M$$

Sasa tunatenga $x_1$ ili kupata kazi ya mahitaji ya $x_1$:

$$x_1 = \frac{M}{2 P_1}$$

Badilisha kazi ya mahitaji ya $x_1$ tuliyoipata kwenye usemi wa $x_2$:

$$x_2 = \frac{P_1}{P_2} \left(\frac{M}{2 P_1}\right)$$ $$x_2 = \frac{M}{2 P_2}$$

Kazi za mahitaji ni:

$$\boxed{x_1 = \frac{M}{2 P_1}}$$ $$\boxed{x_2 = \frac{M}{2 P_2}}$$