Cheap Homework Help-Economics Questions: A person with utility function u(x1, x2) = 2×1 + 3×2 wants to find the least costly consumption that provides a utility of 60 when prices are p1 = 4, p2 = 2. To do this you need to solve the following minimization problem

Cheap Homework Help-Economics Questions: A person with utility function u(x1, x2) = 2×1 + 3×2 wants to find the least costly consumption that provides a utility of 60 when prices are p1 = 4, p2 = 2. To do this you need to solve the following minimization problem

Question 4 A person with utility function u(x₁, x₂) = 2x₁ + 3x₂ wants to find the least costly consumption that provides a utility of 60 when prices are p₁ = 4, p₂ = 2. To do this, you need to solve the following minimization problem.

 

 

min 4x₁ + 2x₂ subject to

x₁,x₂

(i) 2x₁ + 3x₂ ≥ 60

• x₁ ≥ 0
• x₂ ≥

 

Solve the optimization problem graphically. How much money does the person need to afford this consumption bundle.

 

Question 5 A person with utility function u(x₁, x₂) = x^0.5 + x^0.2 wants to find the con-

1           2

sumption bundle that provides the highest utility at prices p₁ = 4, p₂ = 2 and income I. Thus, the person solves

max x^0.5 + x^0.2 subject to

x₁,x₂    1          2

(i) 4x₁ + 2x₂ ≤ I

• x₁ ≥ 0
• x₂ ≥

Using the Excel solver, compute the solution (i.e. the optimal consumption of goods 1 and 2, x₁, and x₂) for income levels I = 100 and I = 200. Determine the percentage of income that the person spends on goods 1 and 2, respectively, for the two levels of income. (Note that p_i x_i is the amount spent on good i).

Question 6 The homework 2 spreadsheet contains data, where x and y are some variables that determine z, the dependent variable. The objective is to find a function f (x, y) that describes the data. To do this, we consider the following family of functions:

 

f (x, y) =              β1

1 + exp(w₁ x + w₂y + b₁)

+              β₂

1 + exp(w₃ x + w₄y + b₂)

 

(this family of functions is a “feed-forward neural network, with one hidden layer” and used in many applications). Parameters w₁ to w₄, β₁, β₂, and b₁ and b₂ are cho- sen to minimize the square differences between the value f (x, y) and the observed value of z, i.e., we solve

 

 

min

25

.( f (x_i, y_i) − z_i)².

 

w₁,w₂,w₃,w₄,β₁,β₂,b₁,b₂ i=1

The spreadsheet already contains a column with the values of ( f (x_i, y_i) − z_i)² for each data point. Thus, to determine the objective you simply need to sum all ele- ments in that column.

1. Select three different start values for parameters w_i, β_i, and b_i, and use the non-linear solver in Excel. Do you always get the same answer?
2. Now impose the bound that −100 ≤ w_i, β_i, b_i ≤ 100 and use the genetic algo- rithm from Excel (i.e., ”evolutionary”). Redo your computation with different seeds.
3. Finally, go back to the non-linear solver (keep the bounds −100 ≤ w_i, β_i, b_i ≤

100) and choose the “multistart” option. Try your computation with larger number of start values (parameter population size) and different seeds.

4. What is the best solution you found? Make a plot of f (x, y) for these param- eters where −1 ≤ x, y ≤ For example, this can be done using Wolfram Alpha, using the comand ”plot f(x,y), x=-1..1, y=-1..1.”
5. Can you be completely sure that this solution is the true global minimum?