Question

Carl enjoys Coffee (q1) and smoothie (q2) and the utility function is: U=q1^2 + q2^2 Suppose that Carl has $100 spend on coffee and smoothies and the price of a pitcher of smoothie is $10 and the price of a coffee jar is $4.

e) Derive Carl’s optimal bundle. Draw the graph of the budget constraint and show the optimal bundle on the graph. Draw a free hand indifference curve. It is not necessary to use the given utility function to draw the exact indifference curve.

f) On the same graph draw a new budget line to show the effect of a gift of three jars of coffee. Show another optimal bundle on the new budget line. Again use a free hand drawn indifference curve.

g) Compare the two optimal bundles. Is Carl better off with the gift? Explain why or why not.

h) Based on the graph you have drawn, explain if the gift has the same effect as a compensation.

Answer 1

The Utility function, U = q1^2 + q2^2 depicts a concave preference means Carl prefers either coffee(q1) or smoothie(q2) but not both simultaneously.

And price of coffee Jar is less than the price of a pitcher of smoothie. So Carl will only consume coffee(q1).

e) Budget constraint of Carl: -

            4q1 + 10q2 = 100

Therefore, her optimal bundle is q1 = 100/4 = 25 and q2 = 0

The blue line is the budget constraint and the red curve is the Indifference curve.

f) The gift of 3 Jars of coffee will basically shift her budget line outwards by 3 units. So now her optimal bundle will be q1 = 28 and q2 = 0.

The green line is the new budget constraint and the violet curve is the new indifference curve.

g) Initially her optimal bundle is q1 = 25 and q2 = 0. This bundle yields her utility of U = 25^2 = 625.

Now the optimal bundle is q1 = 28, q2 = 0. This bundle yields her utility of U = 28^2 = 784.

Since her utility has increased. So she is better off with the gift.

h) As earlier mentioned, Carl has a concave preference, so she will only consume either q1 or q2. And since price of q1 is less. So she will only consume q1.

Therefore, if she would have got a compensation, she would spend it on buying 3 Jars of coffee. So, yes the gift has the same effect as a compensation.