Problem 1. The utility function is (1, 2) = 12 + 1 . Suppose that the constraints are as follows: 11 + 22 , 1 0, and 2 0 where is income, 1 is the price of good 1, and 2 is the price of good 2. Let's assume that = 20 and 2 = 4. 1. Find the optimal consumption bundle when 1 = 1. 2. Find the optimal consumption bundle when 1 = 4. 3. Calculate the total effect of the change in 1 from 1 to 4 on the quantity demanded. 4. Calculate the substitution and income effects corresponding to the change in 1 from 1 to 4 using the Slutsky decomposition.

Problem 2. Suppose that the market is composed of 8 consumers, {????, ????, ????, ????, ????, ????, ????, ????}. Consumer i's individual demand is given by ????????(????) = 10 2???? for ???? {????, ????, ????} and is ????????(????) = 10 4???? for ???? {????, ????, ????, ????, ????}. 1. Find the market demand function, Q(p), and plot the inverse market demand curve. 2. Find the consumer surplus when ???? = 2. 3. Find the price elasticity of demand, ????(????).

Problem 3. Consider the production function ????(????1, ????2) = (????1 + ????2)???? where ???? > 0. 1. Calculate the marginal product of input 1. 2. For which values of the parameter ???? does this production technology exhibit increasing returns to scale (IRS), constant returns to scale (CRS), and decreasing returns to scale (DRS)?

Problem 4. Consider the production function ????(????1, ????2) = ????1 ???? ????2 ???? where ????, ???? > 0. 1. Find the marginal production functions of input 1 and input 2. 2. For which values of the parameters ???? and ???? does this production technology exhibit increasing returns to scale (IRS), constant returns to scale (CRS), and decreasing returns to scale (DRS)?