Question 5 (12 points) Darren likes to consume cups of tea (T) and sugar cubes (

Question

Question 5 (12 points) Darren likes to consume cups of tea (T) and sugar cubes (S) in a fixed proportion - for every 1 cup of tea, Darren likes exactly c cubes of sugar, where c > 0. He has an income of $10 and the price of a sugar cube is PS = 0.25.

a. (5 points) Derive the demand function for tea, T(PT , c), which depends on the price of tea, PT , and c.

b. (3 points) How does an increase in c shift Darren’s demand curve for tea? Provide intuition.

c. (4 points) Now suppose that the price of a cup of tea (without sugar) is PT = 3 and Darren’s income is a free parameter denoted by I. Determine Darren’s Engel curve for cups of tea, T(I, c).

Explanation / Answer

Utility function: U = min[(c x T), S]

(a) Budget line: 10 = T x PT + 0.25 x S

For a fixed-proportion utility function, utility is maximized when (c x T) = S

Substituting in budget line,

10 = T x PT + 0.25 x (c x T)

10 = T x PT + 0.25c x T

10 = T x (PT + 0.25c)

T = 10 / (PT + 0.25c) [Demand function for T]

(b) From abode demand function, as c increases, (PT + 0.25c) increases, therefore demand for T decreases. Intuitively, higher number of sugar cubes increase the cost of the bundle, so consumer reduces her demand for T.

(c) Budget line: I = 3 x T + 0.25 x S

Substituting (c x T) = S,

I = 3 x T + 0.25 x (c x T)

I = 3 x T + 0.25c x T

I = T x (3 + 0.25c) [Equation of Engel curve]

[Alternatively we can write:

T = I / (3 + 0.25c)]

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