Question

2 Perfect substitutes Consider an agent with perfectly substitutable utility over R The agent has total wealth w>0 1. Suppose the agent faces linear prices and that P1くPi for every i > 1, what is the agent's optimal consumption bundle? What fraction of her wealth does she spend on each good? Show that the tangency conditions for optimality are satisfed for the bundle you've found. 2. Suppose instead she faces the same linear price for every good. Describe the set of optimal consumption bundles 3. Now suppose she faces the nonlinear price schedule P(x)-E:-12, what is the agent's optimal consumption bundle? 4. Now the agent faces the price schedule P(x)-Σ=1VTi-Describe the set of optimal consumption bundles. Show that the tangency conditions for optimality are satisfied for cach optimal bundle. 5. Finally, suppose n 2 and the agent faces the price schedule P(x)-2, ritv T2. What is the agent's optimal consumption bundle? ustrate this solution with a diagram Show that the tangency condition for optimality is satisfied for this bundle.

Answer 1

Solution:

Given Utility f^n U (X_1, . . . . ., X_n) = X_1 + X_2 + . . . . . + X_n Budget constraint: P_1 X_1 + P_2 X_2 + . . . . + P_n X_n lessthanorequalto w P_1 < P_i Forall i > 1 Here, we see that goods X_1, X_2, . . . . , X_n are perfect substitute . so, consumption will depend upon the price of these goods . Here, P_1 < P_i Forall i > 1 Rightarrow price of X_1 is less than all other goods i.e, X_2, X_3, . . . . , X_n so the consumer will consume only good X_1 Rightarrow X^*_1 = (w_*/p_1) \r\n mathematically, since, P_1 < P_i Forall i > 1 Rightarrow w = P_1 X_1 at the equilibrium Rightarrow X^*_1 = w/p_1 optimal consumption bundle [(X^*_1 = w/P_1) , (X_2 = X_3 = . . . . X_n = 0)} = X^*_1 = w/P_1 Total wealth is spend on good X_1 and no wealth