Consider a household whose utility is determined by the consumption of two goods, A and B. Let cA and cB denote the consumption of good A and good B, respectively. The utility of this household can be represented by a utility function

U(cA,cB) = logcA + 9 logcB.

The prices of goods A and B are given by pA = 10 and pB = 9, respectively, and this household is endowed with budget I = 532.

(a) Check if the utility function (1) satisfies i) UcA > 0, UcB > 0 and ii) UcA,cA < 0, UcB,cB < 0, where UcA andUcB denoteU/cAandU/cB,respectively,andUcA,cA andUcB,cB denotes2U/c2Aand 2U/c2B, 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.