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## Set up a lagrangian equation

### Assignment Instructions

##### Assignment ID: FG133134352

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

(cA,) = log+ 9/10 log.

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 and UcB denote ∂U/∂cA and∂U/∂cB, respectively,andUcA,cA andUcB,cB denotes ∂^2U/∂c^2Aand ∂^2U/∂c^2B, 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) In the cA-cB 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, pA, changes from pA = 10 to pA = 11, while the price of good B, pB, 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).

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