# Aggregated demand of households given utility function

I have an utility function given,

$$\ u_j(q_{j1},q_{j2} )=q^{3/4}_{1j}*q^{1/4}_{2j}$$

$$\ s.t.: y=p_1*q_{1i} +p_2*q_{2i}$$

I do know that the for $$\ q_{1j}$$ the marginal prospensity to consume is 3/4 of y given prices $$\ p_1$$. But I dont acutally get how my teacher calculated the aggregated demand: $$\ q^*_j(p_1,p_2)=(\frac{1/4y}{p}, \frac{3/4y}{p})$$.

I set up the Lagrangean but then end up with $$\ p_1=\frac{1}{3}\frac{q_2}{q_1}*p_2$$, which does not seem too helpful for me.

Can someone give me a hint how to derive aggregate demand from a given utility function, please.

Thanks

• There is a constraint missing here? – user20105 May 6 '19 at 13:40
• Oh, yes thats true. I added that. – Pete May 6 '19 at 17:24

## 1 Answer

You just need to use the condition $$MRS_{q_{1j},q_{2j}} = \frac{p_{1}}{p_{2}}$$ to obtain $$3 \frac{q_{2j}}{q_{1j}} = \frac{p_{1}}{p_{2}} \;\;\;\; \text{(1)}$$ Then solving for $$p_{1}$$ and plugging this into the budget constraint you obtain: $$y = 3p_{2} \frac{q_{2j}}{q_{1j}}q_{1j} - p_{2}q_{2j}$$ $$\Rightarrow q_{2j} = \frac{y/4}{p_{2}}$$ Accordingly, solving for $$p_{2}$$ from condition (1) and plugging this into the budget constraint yields: $$q_{1j} = \frac{3y/4}{p_{1}}$$.

From this you can obtain your function for the aggregated demand for goods $$q_{1j},q_{2j}$$, as a function of the prices.