Question

1 Discrete goods Consider a setting withn 2 goods, here each good must be purchased in discrete one-unit increments. However, they need not be consumed that way each good is infinitely divisible once purchased and the agent may throw away portions if she so desires. Prices are linear, with P1 = 2 and P2-1. The agent's wealth is w = 8. 1. Draw the agent's budget set. 2. Suppose the agent has utility U(x) = r r . Find the agent's optimal consurnption bundle. Draw the corresponding indifference curve on your graph from the previous part 3. Now suppose the agent has utility U(x)- minfxi, x2). Find the set of all optimal consunption bundlies. Shade in this set on your graphfrom hprts Do all optimal consumption bundles exhaust the agent's budget? If not, identify which one(s) don't exhaust the budget

Answer 1

1.

Agent's budget constraint is

u(x_1 x_2) = x_1^3 x_2^2 and 2x_1 + x_2 = 8 using languages optimization as L = x_1^3 x_2^2 + lambda(8 - 2x_1 - x_2) partial differential L/ partial differential x_1 = 0, partial differential L/ partial differential x^2 = 0 partial differential L/ partial differential n = 0 Rightarrow partial differential L/ partial differential x_1 = 3x_1^2 x_2^2 + (- 2 lambda) = 0Rightarrow1 Rightarrow partial differential l/ partial differential x_2 = 2 x_2 x_1^3 + (- lambda) = 0Rightarrow2 Rightarrow partial differential l/ partial differential lambda = 0Rightarrow2x_1 + x_2 = 8Rightarrow3 Rightarrow3x_1^2x_2^2/2x_2x_1^3 = 2lambda/lambda partial differential the optimal bundles in this case goes out of the budget constraint and the value of budget is 8.01 not 8 so bundle 2.67 doesnot exhaust the existing budget line. u(x_1,x_2) = min(x_1,x_2) and 2x_1 + x_2 = 8 now the utility function is leontiff function which is L shaped and optimum will be wherex_1 = x_2 putting in budget constraint Rightarrow3x_1 = 8 x_1 = 2.67 and x_2 = 2.67

where x is the units of good 1 consumed and y is the units of second good consumed.

2.

,

3.

the optimal bundles in this case goes out of the budget constraint and the value of budget is 8.01 not 8 so bundle 2.67 doesnot exhaust the existing budget line.