Malachi only consumes 2 goods: DVD rentals and coffee. His utility function is
U(R,C) = R0.75C0.25
where R is the number of rentals and C is cups of coffee. Malachi has $16 in his pocket right now, and he plans to spend all of the $16 on DVD rentals and coffee today.
The price of one rental is $4 and the price of coffee is $2 per cup. Solve for Malachi’s optimal bundle.
Suppose that Malachi signs up for a membership that reduces the price of a rental to $2, while leaving his income unchanged. Find the substitution effect, the income effect, and the total effect of the decrease in the price of video rentals on Malachi’s consumption of rentals and coffee.
From your answer to part (b), are DVD rentals and coffee normal or inferior goods for Malachi? Explain.
Suppose that a consumer has utility given by U(X,Y) = XY + 10Y and income of $100 to spend on goods X and Y.
The prices of X and Y are both $1 per unit. Use a Lagrangian to solve for the optimal basket of goods.
Suppose that the price of X increases to $5 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.
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 when the price of X increases to $5 per unit?
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.
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.