Question

1.Suppose a firm produce paper using 2 inputs. When input prices are ($8,$10) the firm uses input bundle (10,10). When the prices are ($10,$6) the firm uses bundle (2,18). The amount of output is the same in both case, this behavior is consistent with Weak Axiom of Cost Minimization

True or False?

2.Jimmy's utility function is U = Apple x Banana. The price of apples used to be $4 per unit and price of bananas $1 per unit. His income is $160 daily. If the price of apples decreases to $2 and the price of banana stays the same. In order to be able to just afford his old bundle before prices change, Jimmy's would have to have a daily income of: ______, the substitution effect of the decrease in price of apple leads to an increase in the quantity of apple consumed by: _______,the total effect of the decrease in price of apple leads to an increase in the quantity of apple consumed by: _______

Answer 1

1)Ans. True.

This because weak axiom cost minimization said in a particular combination of input prices with input bundle must be less than equal to other combination of input bundle with previous input price.

More clearly let suppose first group of input price we say w1 and r1. With this input price cost minimise input bundle is L1 and K1. Now, with another input price group say, w2 and r2 the cost minimise input bundle is L2 and K2. Then to hold weak axiom cost minimization we will must have -

w1*L1 + r1*K1 <= w1*L2 + r1*K2

w2*L2 + r2*K2 <= w2*L1 + r2*K1.

Here in the input price combination ($8,$10) (say, w1, r1) cost minimise bundle is (10,10)(say, L1, K1). In case of input price combination ($10,$6)(say, w2, r2) cost minimization bundle is (2,18) (say, L2, K2).  

Therefore, foe weak axiom cost minimization to hold at input price($8,$10) the bundle(10,10) must less cost or equal to bundle (2,18)

$8*10 + $10*10 = $180

$8*2 +$10*18 =$196

As $180 < $196 , for first input price combination it is holding the condition of weak axiom cost minimization.,

Now, in the input price combination ($10,$6) cost minimise bundle is (2,18). To hold weak axiom cost minimization at input price combination ($10,6) cost of bundle (2,18) must be less than equal to cost of bundle (10,10) at this price($10,6).

Cost of bundle (2,18) at input price ($10,$6) is $10*2 +$6*18 = $138

Cost of bundle (10,10) at input price ($10,$6) is $10*10+$$6*10 = $160.

As $138< $160, it is holding the condition of weak axiom cost minimization.

Therefore this combination of Input price and input bundle consistent with weak axiom cost minimization. (Ans : True).

2) Utility function is U = A*B, A = Apple, B = Banana.

Price of apple is( say Pa)= $4 and price of banana (say Pb) =$1. Income (M) =$160.

At the utility maximisation point, MUa/MUb = Pa/Pb

MUa = B, and MUb =A , Pa =$4 , Pb =$1

So, B/A = 4/1 , or , B =4A.

From income constraint or budget constraint we must have -

M = Pa* A + Pb*B

160 = 4* A + 1* B

160 = 4A + B

Substituting the relation B = 4A , we get -

160 = 4A + 4A

160 = 8A, or A = 20

Therefore, B = 4A = 4*20 =80.

So, at the price level (Pa =$4, Pb =$1) utility maximisation bundle is ( A = 20, B = 80).

Now if the apple price decreases to $2 , then to afford the old bundle (A =20, B =80) Jimmy requires the income of $2*20 +$1*80 = $40+$80 = $120. (Ans).

Jimmy's would have to daily income of $120 to afford his old bundle.

At the income level of $120 , price of apple Pa =$2, Price of banana Pb =$1 , in utility maximisation point we must have

MUa/MUb = Pa/ Pb

B/A = 2/1 , or, B = 2A

Now with budget $120 we must have

Pa*A + Pb*B =120

or, 2*A +1*B = 120

or, 2A+B =120

or, 2A +2A = 120 , ( Putting B =2A)

or, 4A = 120 , or , A =30.

So, substitution effect of Apple due to price decrease of Apple is 30 - 20 = 10 .

The substitution effect of Apple due to decrease in the price of apple leads to an increase in the quantity consume of apple by 10. (Ans)

Total effect of will be calculated by initial income M =$160. The relation of B and A will be B =2A

Therefore , M = Pa*A + Pb*B

Or, 160 = 2*A + 1*B

Or, 160 = 2A + B

Or, 160 = 2A + 2A

Or, 160 = 4A , or, A = 40.

In the total effect quantity consume of apple will increase by 40 -20 =20

The total effect of the decrease in price of apple leads to an increase in the quantity of apple consume by 20. (Ans).