Question

5. Consider the representative household in the static two-good consumption model whose preferences are represented by u(ci, c2)= / 2, with each good priced at P and P2. The household receives an exogenous amount of income, Y. (a) Using a Lagrangian, and derive the first order conditions for ci and c2 (b) Use the first-order conditions to derive the consumer's optimality condition. (c) Solve for the demand functions of ci and c2 (d) Suppose a shock increases P. Using comparative static analysis on your demand functions from part (c), show the directional effect on aggregate quant ty demanded for each good on each demand function in (ci, Pi) space (e) Illustrate the effect of the shock in part (d) for i 1,2. Use economic logic to explain your ans wer

Answer 1

(a) Given:

C2 C1+

The household's problem is:

max C1C2 S.t. PiC+ P2C2= Y

Setting up the Lagrange:

$\mathcal{L}= \sqrt{c_1}+\sqrt{c_2}-\lambda(P_1c_1+P_2c_2-Y)$

The first order conditions are:

2 = AP and じ=2v=AP2. A(PiC1 + P2C2-Y) = 0 C1 Il

(b) Dividing the first two order conditions, we get:

CP C2P C2 P P2 Il

Substituting this value in the optimality condition:

PP2 ) Y P2

(c) Using the above condition, demands are calculated as follows:

YP YP2 1PP2P 2 PP P2

(d) After an increase in $P_1$ , change in demands can be calculated as follows:

Y(PPP2) YP(P2) dct YP2(P2+2P1) dc2 (P P2+ P dP dP + (P3+PP2Y

Aggregate quantity demanded of
Ct
will fall and aggregate quantity demanded of $c_2$ will rise.