Question

1. Suppose a consumer buys clothing for the heat it provides and that each unit of clothing provides 2 units of heat. Assume that the consumer also buys housing for the shelter it provides, and that one unit of housing provides one unit of shelter. If H is the number of housing units and C is the number of clothing units consumed, then his utility is U(H; C) = min {H; 2C}. The per-unit prices of housing and clothing are ph and pc, respectively, and his income is m. (a) Set up the maximization problem and derive the demand functions for H and C . (20 points) (b) Re-do part (a) assuming that U(H; C) = H + 2C, ph = 2, pc = 3, and m = 300: (20 points)

Answer 1

Hi

So the answer of the following question are as follows :

Ans.A) U = min{H,2C}

So now We can see from the utility function that the consumer always wants 2 unit of H for every 1 unit of C. Hence they are perfect compliments(or leontieff) function.

In this type of Consumption function a consumer is always prefers to consume at a point where the budget line intersect kinked portion of thee Indifference Curve(Like described in the below figure.

Budget Constraint : phH + pcC = m

We have to Maximize : U = min{H,2C}

Subject to : phH + pcC = m

In this type of Consumption function a consumer always prefers to consume at a point where the budget line intersect kinked portion of thee Indifference Curve(Like described in the below figure.

Kink of the Indifference Curve(IC) occurs when H = 2C.

Hence H = 2C. Putting this in Budget Constraint we get

phH + pcC = m => 2phC + pcC = m

=> C* = m/(2ph + pc) and H* = 2C = (2m)/(2ph + pc).

Ans.B) So let's solve the next problem We have to solve the following Model as

Maximize : U = H + 2C

subject to :2H + 3C = 300

We can see from the Utility Function that C and H are perfect substitutes and this consumer values C two times of what he values C. Hence

He will buy only C if Price of C < 2*Price of H

He will buy only H if Price of C > 2*Price of H

He will buy any combination of C and H that lies on the Budget constraint (2H + 3C = 300)

Hence As pc < 2ph, Hence He will buy only C with his entire income.

Hence C* = 300/3 = 100 and H* = 0.

I hope I have served the purpose well.

Thanks.