Assume your utility function is given as U(x,y)=5xy. Your income is equal to $1000. One unit of y costs $5 and one unit of x costs $8.
1. Determine the utility maximizing quantities of x and y by using the Lagrange method. Also, determine the Lagrange multiplier. Estimate the overall utility at this point.
2. Now assume the good y gets taxed so that its price rises from $5 to $10. The price of x stays the same. Determine the new optimal quantities consumed and the new utility level.