Suppose that a consumer has utilityU(X,Y) = 2XY+Xand income of $499 to spend on goodsXandY.

a.The prices ofXandYare $1 and $2 per unit, respectively. Use a Lagrangian to solve for the optimal basket of goods.

b.Suppose that the price ofXincreases to $2 per unit. Use a Lagrangian to solve for the new optimal basket of goods. Find the total effect of the price change on the consumption of each good.

c.Use a Lagrangian to find the substitution effect of the increase in the price of goodXon the consumption of each good. What income would the consumer need to attain the original level of utility if the price ofXincreased to $2 per unit?

d.Find the income effect of the increase in the price of goodXon the consumption of each good. Are the goods normal or inferior? Explain.

e.Show that the total effect of the increase in the price ofXis equal to the sum of the substitution effect and the income effect.