Suppose that a consumer has utility U(X,Y) = 2XY + X and income of $499 to spend on goods X and Y.
a. The prices of X and Y are $1 and $2 per unit, respectively. Use a Lagrangian to solve for the optimal basket of goods.
b. Suppose that the price of X increases 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 good X on the consumption of each good. What income would the consumer need to attain the original level of utility if the price of X increased to $2 per unit?
d. Find the income effect of the increase in the price of good X on the consumption of each good. Are the goods normal or inferior? Explain.
e. Show that the total effect of the increase in the price of X is equal to the sum of the substitution effect and the income effect.

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The Answer of Suppose that a consumer has utility U(X,Y) = 2XY + X and income of $499 to spend on goods X and Y.
a. The prices of X and Y are $1 and $2 per unit, respectively. Use a Lagrangian to solve for the optimal basket of goods.
b. Suppose that the price of X increases 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 good X on the consumption of each good. What income would the consumer need to attain the original level of utility if the price of X increased to $2 per unit?
d. Find the income effect of the increase in the price of good X on the consumption of each good. Are the goods normal or inferior? Explain.
e. Show that the total effect of the increase in the price of X is equal to the sum of the substitution effect and the income effect.

Suppose That a Consumer Has Utility U(X,Y) = 2XY