Question

1. Sweet tea (ST) requires tea bags (T) and lots of sugar (S). Suppose the perfect ratio for one serving of Sweet tea is 3 tea bags to 5 spoons of sugar.

(a) Write down the production function. (b) (2.5 points) Does the production function exhibit CRS, IRS, or CRS?

(c) What is the optimal ratio for production (between T & S)?

(d) Suppose that tea bags cost 10 cents each, five spoons of sugar cost 1 cent, and we want five servings. How much would the five servings cost?

(e) Draw two isoquant and isocost lines. One of the isoquant and iscocost lines must be the one representing five servings. Make sure to label properly.

Answer 1

1. Sweet tea (ST) requires tea bags (T) and lots of sugar (S). Suppose the perfect ratio for one serving of Sweet tea is 3 tea bags (T) to 5 spoons of sugar (S). The ratio of Sweet tea and Suger is

T : S = 3 : 5

(a) This is the Perfect Complement Production Function. The Production Function is,

Q = ST(T, S) = Min{T/3, S/5}.........(1)

where, Q = Number of servings.

The production function is

ST(T, S) = Min{T/3, S/5}.

(b) For any value of a>0, if we put (a.T) and (a.S) in the production function, we get

ST(a.T, a.S) = Min{a.T/3, a.S/5}

or, ST(a.T, a.S) = a.Min{T/3, S/5}

or, ST(a.T, a.S) = a.ST(T, S)

Hence, the production function exibits Constant Returns to Scale.

The production function exibits CRS.

(c) This is a fixed ratio production function. Hence, the Tea bags and Suger will be used at a fixed ratio. The optimal production is along the line

T/3 = S/5

or, T/S = 3/5

This is the optimal ratio for production.

The optimal ratio for production is (3/5).

(d) The cost of Tea bag cost 10 cents each. Hence, Pt = 10 cents.

And, 5 spoons of Suger costs 1 cent. Hence, 1 spoon of Suger costs, Ps = 1/5 = 0.2 cent.

Hence, Total Cost of T tea bags and S spoons of suger is

C = Pt.T + Ps.S = 10T + 0.2S........(2)

Now, from the production function, we can see that, Q = Min{T/3, S/5}

Hence, at optimum, Q = T/3 or Q = S/5

Hence, T = 3.Q

And, S = 5.Q

Putting T and S as function of Q in the Cost Function, we get

C = 10.T + 0.2.S

or, C(Q) = 10.(3Q) + 0.2.(5.Q)

or, C(Q) = 31.Q

Now, we have to find the cost of 5 servings. Hence, we put Q = 5.

Hence, Cost of five servings is

C(5) = 31×5

or, C = 155 cents

Five servings costs 155 cents.

(e) When, Q = 5, then

T = 3.Q = 3×5

or, T = 15

And, S = 5.Q = 5×5

or, S = 25

Hence, 5 servings requires 15 tea bags and 25 spoons of suger. The following diagram shows two isoquant and isocost lines. One of the isoquant and iscocost lines represents five servings.

Here, IQ1 and IQ2 represents two isoquants. Also, IC1 and IC2 represents two isocost lines. IQ2 represents 5 servings i.e. Q = 5.

Hope the solutions are clear to you my friend.