Question: Income vs Substitution Effects.

Assume that preferences are represented by the utility function \(u\left(x_{1}, x_{2}\right)=\sqrt{x_{1}}+x_{2}\)

1. Calculate the consumer's optimal choices, and his corresponding utility, at \(p_{1}=1\), \(p_{2}=2, m=1\)
2. Now suppose the price of good 1 increases to \(p_{1}=2\), while there is no change in \(p_{2}\).

1. Calculate the compensating variation: how much would \(m\) need to increase, in order for the consumer to be unaffected by the price increase? (i.e., to obtain the same utility level as in part (a))
2. Calculate his optimal bundle, if you increase his wealth by the amount calculated above in (i)

(c) Finally, calculate his optimal bundle at \(p_{1}=2, p_{2}=2, m=1\) (new prices, original wealth). Compared to part (a), how much does his consumption of good 1 change due to the substitution effect, and how much due to the income effect? (Hint: you can use your answer in part (b) (ii) to calculate the change due to substitution effect).

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