Question

Clayton's utility function can be written as xq30x9.50 where x; is the quantity of hamburgers consumed and X2 is the quantity of milk shakes. His income is $1,600.00 1st attempt Part 1 (1 point) See Hint Hamburgers cost $8.00 each and the price of one milk shake is $4.00. If he is maximizing his utility, how much will Clayton spend on hamburgers? Part 2 (1 point) See Hint The price of hamburgers decreases to $7.37 each. Now, how much will Clayton spend on hamburgers if he is maximizing his utility?

Answer 1

Part 1

The utility maximizing condition for Clayton is

MRS = PX1 /PX2

PX1 = Price of hamburgers

PX2 = price of milk shakes

MRS = marginal rate of substitution = Marginal utility of X1(MUx1)/ Marginal utility of X2 (MUx2)

Therefore the utility maximizing condition can be also written as

(MUx1 /MUx2) = (Px1 /Px2)

The given utility function is X1^0.50X2^0.50

The MRS can be calculated as follows

Price of X1 = 8

Price of X2 = 4

Clayton's utility is maximized at

(X2 /X1) = (8/4)

Therefore

4X2 = 8X1

The budget constraint of Clayton can be written as follows

8X1 + 4X2 = 1600

Substitute 8X1 = 4X2 in the budget constraint

4X2 + 4X2 = 1600

8X2 = 1600

X2 = 1600/8 = 200

X2 = 200

Substitute X2 = 200 in the budget constraint

8X1 + 4*200 = 1600

8X1 + 800 =1600

8X1 =1600-800 =800

X1 = 800/8 = 100

X1 = 100

To maximize utility, clayton consume X1 = 100 and X2 = 200

To maximize utility, total spending on hamburgers = 8 *100 = 800

total spending on hamburgers = 800

Part 2

The new price of hamburgers = 7.37

The new utility maximizing condition is

MRS = 7.37/4

(X2 /X1) = (7.37/4)

4X2 = 7.37X1

The new budget constraint of clayton can be written as follows

(7.37 x1) + (4X2) = 1600

Substitute 4x2 = 7.37X1 in the budget constraint

7.37X1 + 7.37X1 = 1600

14.74 X1 = 1600

X1 = 1600 /14.74 = 108.54

To maximize utility, clayton has to consume 108.54 units of hamburgers

Therefore total spending on hamburgers = 108.54 * 7.37 = 799.9398

To maximize utility, Clayton has to spend 799.9398

Answer 2

Part 1

The utility maximizing condition for Clayton is

MRS = PX1 /PX2

PX1 = Price of hamburgers

PX2 = price of milk shakes

MRS = marginal rate of substitution = Marginal utility of X1(MUx1)/ Marginal utility of X2 (MUx2)

Therefore the utility maximizing condition can be also written as

(MUx1 /MUx2) = (Px1 /Px2)

The given utility function is X1^0.50X2^0.50

The MRS can be calculated as follows

0.50 0.50 U=X, X2 * MUX, 1 au axi Mux, = 0.50%, -0.50 0.50 X2 * MUX2 au =au 2x2 2 0.50 -0.50 MUX = 0.50% X; X2 MRS E MUX] MUX2 MRS - 0.50 0.50 0.50 X2 0.50 X1 0.50 -0.50 X2 NIRS = X2 X XI

Price of X1 = 8

Price of X2 = 4

Clayton's utility is maximized at

(X2 /X1) = (8/4)

Therefore

4X2 = 8X1

The budget constraint of Clayton can be written as follows

8X1 + 4X2 = 1600

Substitute 8X1 = 4X2 in the budget constraint

4X2 + 4X2 = 1600

8X2 = 1600

X2 = 1600/8 = 200

X2 = 200

Substitute X2 = 200 in the budget constraint

8X1 + 4*200 = 1600

8X1 + 800 =1600

8X1 =1600-800 =800

X1 = 800/8 = 100

X1 = 100

To maximize utility, clayton consume X1 = 100 and X2 = 200

To maximize utility, total spending on hamburgers = 8 *100 = 800

total spending on hamburgers = 800

Part 2

The new price of hamburgers = 7.37

The new utility maximizing condition is

MRS = 7.37/4

(X2 /X1) = (7.37/4)

4X2 = 7.37X1

The new budget constraint of clayton can be written as follows

(7.37 x1) + (4X2) = 1600

Substitute 4x2 = 7.37X1 in the budget constraint

7.37X1 + 7.37X1 = 1600

14.74 X1 = 1600

X1 = 1600 /14.74 = 108.54

To maximize utility, clayton has to consume 108.54 units of hamburgers

Therefore total spending on hamburgers = 108.54 * 7.37 = 799.9398

To maximize utility, Clayton has to spend 799.9398