My class was canceled due to the coronavirus and I am supposed to solve the following problem:
Assume that the consumer consumes only two goods, and his utility function is $u=x_{1}x_{2}$. Last year the prices of both goods were 10 and the consumer income was 100. This year the price of the first good is still 10, but the price of the second rose to 25. Calculate Paasche and Laspeyres price indices.
Paasche index=$\frac{\sum \left ( P_{i},t \right )\left ( Q_{i},t \right )}{\sum \left ( P_{i},0 \right )\left ( Q_{i},t \right )}$,
where $P_{i},0$ is the price of the individual item at the base period and $P_{i},t$ is the price of the individual item at the observation period, the same with quantity
Laspeyres Price Index=$\frac{\sum \left ( P_{i},t \right )\left ( Q_{i},0 \right )}{\sum \left ( P_{i},0 \right )\left ( Q_{i},0 \right )}$
I know how to calculate them using formulas if I have quantity as well as price of products.
But how should I calculate it, if I have income and utility function?
Thanks
It seems that you have to calculate the optimal quantity in the previous year and this year first. This can be done with the method of Lagrange multipliers. The maximizing functions are
$$\mathcal L_1=x_1x_2+\lambda\left( 100-10x_1-10x_2 \right)$$
$$\mathcal L_2=x_1x_2+\lambda\left( 100-10x_1-25x_2 \right)$$
The (optimal) quantities are $(x_1^*,x_2^*)=(5,5)$ for the previous year and $(x_1^*,x_2^*)=(5,2)$ for the current year. With this quantities and the given prices it is straightforward to calculate the Laspeyres price index and the Paasche index.