Aggregated demand of households given utility function

Question

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

Answer 1

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.