When the price of coffee increases by $$ 15 \% $$, the quantity demanded of coffee decreases by $$ 5 \% $$ and the demand for tea increases by $$ 8 \% $$. [1.5] I) Calculate the elasticity of demand for coffee. Hint: Remember to use the absolute value. [1.5] II) Is the demand for coffee elastic or inelastic? Why? [1.5] III) If the price of tea did not change, calculate the cross elasticity of demand for tea with respect to the price of coffee. [Hint: Good X is tea, and Good Y is coffee.] [1.5] IV) Are coffee and tea substitutes or complements? Why?

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To calculate the price elasticity of demand (PED) for coffee, we use the formula:

$$
\text{PED} = \frac{\%\text{ Change in Quantity Demanded}}{\%\text{ Change in Price}}
$$

Given that the quantity demanded of coffee decreases by $$5\%$$ and the price increases by $$15\%$$, we can substitute these values into the formula:

$$
\text{PED} = \frac{-5\%}{15\%}
$$

Now, we will calculate the absolute value of the elasticity.

Let's compute this using a tool call.
In general, if the absolute value of the price elasticity of demand (PED) is greater than 1, the demand is considered elastic. If it is less than 1, the demand is inelastic.

After calculating the PED in Step I, we will analyze the result to determine whether the demand for coffee is elastic or inelastic.
To calculate the cross elasticity of demand (XED) for tea with respect to the price of coffee, we use the formula:

$$
\text{XED} = \frac{\%\text{ Change in Quantity Demanded of Tea}}{\%\text{ Change in Price of Coffee}}
$$

Given that the quantity demanded of tea increases by $$8\%$$ and the price of coffee increases by $$15\%$$, we can substitute these values into the formula:

$$
\text{XED} = \frac{8\%}{15\%}
$$

Now, we will compute this using a tool call.
Goods are considered substitutes if an increase in the price of one good leads to an increase in the quantity demanded of the other good. Conversely, goods are complements if an increase in the price of one good leads to a decrease in the quantity demanded of the other good.

Based on the results from Step III, we will analyze whether coffee and tea are substitutes or complements.Now, let's perform the calculations for Steps I and III.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{-5}{15}$$
- step1: Reduce the fraction:
  $$\frac{-1}{3}$$
- step2: Rewrite the fraction:
  $$-\frac{1}{3}$$
Calculate or simplify the expression $$ 8/15 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{8}{15}$$
The price elasticity of demand (PED) for coffee is calculated as follows:

$$
\text{PED} = \frac{-5\%}{15\%} = -\frac{1}{3} \approx -0.33
$$

Taking the absolute value, we have:

$$
|\text{PED}| = 0.33
\
Since the absolute value of the price elasticity of demand (0.33) is less than 1, the demand for coffee is inelastic. This means that the quantity demanded is not very responsive to price changes.
The cross elasticity of demand (XED) for tea with respect to the price of coffee is calculated as follows:

\[
\text{XED} = \frac{8\%}{15\%} = \frac{8}{15} \approx 0.53
$$

This indicates the relationship between the price of coffee and the quantity demanded of tea.
Since the cross elasticity of demand (0.53) is positive, this indicates that coffee and tea are substitutes. An increase in the price of coffee leads to an increase in the quantity demanded of tea.
I) The elasticity of demand for coffee is approximately $$0.33$$. II) The demand for coffee is inelastic because the absolute value of the elasticity is less than 1. III) The cross elasticity of demand for tea with respect to the price of coffee is approximately $$0.53$$. IV) Coffee and tea are substitutes because an increase in the price of coffee leads to an increase in the quantity demanded of tea.

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