Question

Suppose that Andrew has the utility function: U(C,M)= C^0.5+M^0.5 , where C represents cupcakes and M represents muffins. Andrew has income of $150. Suppose that the initial price per cupcake is $2 and the price per muffin is $1. At these prices, Andrew spends $100 on muffins. When cupcakes are on sale, each cupcake costs $1. When cupcakes are on sale, Andrew spends $75 on muffins. The price of muffins and Andrew’s income is unchanged throughout.

1. Draw a diagram that illustrates Andrew’s (a) budget constraints, (b) indifference curves, and (c) (d) optimal bundles before and after the change in cupcake price. Place cupcakes on the horizontal axis.

2. Calculate the (a) substitution and (b) income effects resulting from the decrease in the price of cupcakes, and illustrate the substitution and income effects on your diagram in part 1.

3. Are cupcakes a normal or an inferior good for Andrew? Why?

Answer 1

Given:

Utility function:

U = C^0.5+M^0.5

where C is the number of cupcakes and M is the number of muffins

Andrew's income (I) = $150

Price of cupcakes = $2

Price of muffins = $1

a.) Diagram of budget constraint:

Budget constraint shows all the combinations of cupcakes and muffins that Andrew could purchase with his budget of $150.

Therefore this is Andrew's budget constraints, we can plot it by finding the x and y-intercepts. Plotting cupcakes on the x-axs and muffins on the y-axis.

X-axis, is when M = 0

Therefore, x-intercept is 75.

Y-axis, is when C = 0

Therefore Y-intercept is 150.

Now, Plotting Andrew's budget constraints;

b.) the indifference curve for the assume utility values. We will assume utility level to plot the indifference curves.

Lets say U = 10,12,14

Plotting three indifference curves for utility levels 10, 12 and 14

c.) The optimal bundle at initial prices:

Therefore Andrew will maximize utility subject to his budget constraints.

The utility is maximized where the marginal rate of substitution is equal to the ratio of prices. where the marginal rate of substitution is the ratio of marginal utilities of both goods.

That is,

Therefore finding the marginal utilities of both goods from the utility function. Marginal utility is the derivative of the utility with respect to the good.

Similarly, the marginal utility of muffins:

Now putting these values into the formula of the marginal rate of substitution condition:

Therefore the marginal rate of substitution is:

now the utility is maximized where:

Putting the values of the marginal rate of substitution and the prices, we get:

Now putting this condition into Andrew's budget constraints, we get the optimal quantities of cupcakes and muffins.

Now putting the utility maximization condition: M = 2C

Now putting this the equation we got above,

Therefore the optimal bundle with initial price is:

(37.5 cupcakes, 75 muffins)

d.) The optimal bundle when the price of cupcakes changes to $1.

Now, finding the new optimal bundle with the changed price of cupcakes.

The utility maximisation condition is:

Putting the values of the marginal rate of substitution and the prices, we get:

As it is given that Andrew spent $75 on the muffins after the change in price of cupcakes. therefore the new optimal bundle:

As money spent on muffins =$75 and the price of muffins = $1

Putting this into the new utility maximisation condition:

Therefore the new optimal bundle with price of cupckaes (Pc = $1) is:

(75 cupcakes, 75 muffins)

The two optimal bundles are E and E'.

The consumption of cupcakes increases from 37.5 to 75. while muffins remain same as 75.

Therefore because of the decrease in price of cupcakes to $1, the total effect is:

Total price effect = Change in number of cupcakes

Total price effect = 70-37.5

Total price effect = 32.5 cupcakes

Because of the decrease in price of cupcakes, the consumption of cupcakes increase by 32.5 and consumption of muffins doesn't change.

2.) Now, finding the substitution and income effect from the total effect:

The substitution effect is found out by finding the compensating demand curve. The compensating demand curve is where the real income of Andrew is held constant with the change in the price of cupcakes:

It is found out by keeping Andrew on the same utility as before.

Therefore the utility as the initial optimal bundle is:

Therefore keeping Andrew at the same utility and putting the new utility maximization condition into the utility function, we get the compensated demand function:

When the price of cupcakes fall to $1, we found that C = M, therefore

As

Therefore because of the substitution effect alone, the consumption of cupcakes increases from 37.5 to 54.62.

Therefore substitution effect = 54.62 - 37.5

Increase in consumption of cupcakes because of the substitution effect = 17.12

Therefore because of the substitution effect alone, the consumption of muffins falls from 75 to 54.62.

Substitution effect in muffins = 54.62-75

Substitution effect in muffin consumption = -20.38

The substitution effect on consumption of muffins falls by 20.38

Now the income effect is:

Total price effect = Substitution effect + Income effect

Income effect in consumption of cupcakes:

32.5 = 17.12+Income effect

Income effect on cupcakes = 15.38

Income effect in consumption of muffins:

0 = 20.38+Income effect

Income effect on muffins = -20.38

Income effect is from point E to e. and substitution effect is from point e to E'

Therefore conclusion:

Because of the substitution effect the consumption of:

Cupcakes increases from 37.5 to 54.62

Muffins fall from 75 to 54.62

Because of the income effect the consumption of:

cupcakes increases from 54.62 to 75

muffins increases from 54.62 to 75

3.) Cupcakes are normal goods.

As normal goods are those that have a positive relationship between income and quantity demanded. We can see that the income effect on the consumption of cupcakes is positive.

The income effect on cupcakes is 17.12 cupcakes. That is with the increase in the real income Andrew, the consumption of cupcakes increases from 37.5 to 54.62. The increase in real income increases the consumption of cupcakes. SO there exists a positive relationship between the income of Andrew and the quantity demanded of cupcakes.