Consider a household whose utility is determined by the consumption of two goods, A and

$B.$ Let $c_A$ and $c_B$ denote the consumption of good A and good $B,$ respectively. The utility of this

household can be represented by a utility function

$U(c_A,c_B)=log{c}_A+\frac{9}{10}log{c}_B.$

The prices of goods A and $B$ are given by $p_A=10$ and $p_B=9,$ respectively, and this household

is endowed with budget $I=532.$

$($a$)$ Check if the utility function $(1)$ satisfies i$)U_{c_A}>0,U_{c_B}>0$ and ii where

$U_{c_A}$ and $U_{c_B}$ denote del$\frac{U}{d}el{c}_A$ and del$\frac{U}{d}el{c}_B,$ respectively, and $U_{c_A,c_A}$ and $U_{c_B,c_B}$ denotes $de{l}^2\frac{U}{d}el{c}_A^2$ and

$de{l}^2\frac{U}{d}el{c}_B^2,$ respectively. Describe the economic meaning of these conditions.

$($b$)$ Write down this household's static optimization problem over the two goods.

$($c$)$ Set up a Lagrangian equation and derive the optimal conditions.

$($d$)$ Express the marginal rate of substitution $($MRS hereafter$)$ between the two goods A and $B$ in

an analytical form and describe the relationship between the MRS and the relative price $($between

the two goods A and $B)$ at the optimum. Explain the economic reason why this relationship has

to hold at the optimum.

$($e$)$ Solve the optimization problem.

$($f$)$ On the $c_{A^{-}}{c}_B$ plot, illustrate the optimum, the budget constraint, the indifference curve passing

the optimum. Describe how the indifference curve and the budget constraint meet with each other

at the optimum. Explain mathematically why they meet in that way.

$($g$)$ Assume that the price of good $A,p_A,$ changes from $p_A=10$ to $p_A=11,$ while the price of

good $B,p_B,$ remains the same. Find the new optimum. $($i$.$e$.,$ solve the optimization problem again

under the new prices.$)$

$($h$)$ Explain the wealth effect and the substitution effect caused by the price change described in

question $($g$).$