Microeconomics Assignment PDF

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mandatory group assignment...

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DEVELOPMENT MICROECONOMICS GROUP ASSIGNMENT Question 3: Given the utility function , derive average and marginal utility functions. Find the value of X at which total utility is maximum and the value of X at which average utility is maximum? i.

Average Utility Function AU = TU/Q =

50X + 40X2 – X3 = 50 + 40X - X2 X

ii.

Marginal Utility Function: - It is first order derivative of given utility function. MU = -3x2 + 80x + 150

iii.

Find the value of X at which total utility is maximum, - total utility reaches maximum point when MU = 0 MU = -3x2 + 80x + 150 = 0

-80

√802 – 4 *(-3)*150 2 (-3)

X = -1.758 or X = 28.425 Negative amount of a product does not make sense, X = 28.425 is where total utility is maximum

iv.

The value of X at which average utility is maximum; AU = 50 + 40X - X2

To find X when Average Utility (AU) is Maximum we need to compute its FOD, where the slope becomes zero. And second order derivative must be negative.

∂ (-X2 + 40X + 150)

∂X =

-2X + 40 = 0 2X = 40

X = 20 ∂ (-2X + 40)

∂X = -2 < 0 Therefore, X = 20 is the point where AU reaches maximum point Question 4: Illustrate using graph and explain the income and substitution effects for inferior and Giffen goods (for both price rise and fall cases) i)

The substitution effect and income effect of a price increase for an inferior good. Y C

A

sub

U1 B

inc

total

BL3 U2

x3 x2 x1

BL2

BL1

X

Good Y

Good X

The price of x increases which causes the budget line to shift from BL1 to BL2. The consumer changes his consumption from the bundle of x and y represented by point A to the bundle represented by point B. The movement from A to B represents the total effect of the price change. Consumption of x goes down from x 1 to x2 for two reasons. The substitution effect occurs because x is now more expensive relative to y (BL2 is steeper than BL1). The income effect of the price change occurs because real income (I/Px) has decreased. B is on a lower indifference curve than A. The total effect is the substitution effect plus the income effect. To separate the substitution effect from the total effect, first draw a new budget line, BL3. BL3 is parallel to BL2 because it represents the higher price for x. It must be tangent to the original indifference curve U1. In the graph above this is point C. Point C shows us how much x the consumer would buy if the price of x were increased and at the same time he was given more income so that he was no worse off than he was before the price went up. The movement from A to C is the substitution effect. The income effect is what is left when the substitution effect (A to C) is subtracted from the total effect (A to B), which is B to C in the graph above. X is an inferior good because when then the budget line shifts from BL3 to BL2 (income decreases), consumption of X increases from x3 to x 2. The demand for x is downward sloping because when price increases consumption of x goes down from x1 to x2. ii. The substitution effect and income effect of a price decrease for an inferior good.

BL3

BL1

x1

x2

BL2

x3

The price of x decreases which causes the budget line to shift from BL1 to BL2. The consumer changes his consumption from the bundle of x and y represented by point A to the bundle represented by point B. The movement from A to B represents the total effect of the price change. Consumption of x goes up from x1 to x2 for two reasons. The substitution effect occurs because x is now cheaper relative to y (BL2 is flatter than BL1). The income effect of the price change occurs because real income (I/P X) has increased. B is on a higher indifference curve than A. The total effect is the substitution effect plus the income effect. To separate the substitution effect from the total effect, first draw a new budget line, BL3. BL3 is parallel to BL2 because it represents the lower price for x. It must be tangent to the original indifference curve U1. In the graph above this is point C. Point C shows us how much x the consumer would buy if the price of x were decreased and at the same time he was given less income so that he was no better off than he was before the price went down. The movement from A to C is the substitution effect. The income effect is what is left when the substitution effect (A to C) is subtracted from the total effect (A to B), which is B to C in the graph above. When the budget line shifts from BL3 to BL2 (income increases), consumption of X decrease from x 3 to x2. The demand for x is downward sloping because when price decreases consumption of x goes up from x1 to x2.

i.

The substitution and income effect of a price increase for a giffen good

The price of x increases which causes the budget line to shift inward from BL1 to BL2. The consumer changes his consumption from the bundle of x and y represented by point A to the bundle represented by point B. The movement from A to B represents the total effect of the price change. In the Giffen good case, even though the price has increased, consumption of x has gone up. The demand curve will be upward sloping.

To separate the substitution effect from the total effect, first draw a new budget line, BL 3. BL3 is parallel to BL2 because it represents the higher price for x. It must tangent to the original indifference curve U 1. In the graph above this is point C. Point C shows us how much x the consumer would buy if the price of x were increased and at the same time he was given more income so that he was no worse off than he was before the price went up. The movement from A to C is the substitution effect. The income effect is what is left when the substitution effect (A to C) is subtracted from the total effect (A to B), which is B to C in the graph above. What makes this a Giffen good is that the size of the income effect is bigger than the size of the substitution effect.

ii.

The substitution and income effect of a price decrease for a giffen good

The price of x decreases which causes the budget line to shift outward from L1 to L2. The consumer changes his consumption from the bundle of x and y represented by point Q to the bundle represented by point R. The movement from Q to R represents the total effect of the price change. To separate the substitution effect from the total effect, first draw a new budget line, B. it is parallel to L2 because it represents the lower price for x. It must be tangent to the original indifference curve IC 1. In the graph above this is point S. Point S shows us how much x the consumer would buy if the price of x were decreased and at the same time he was given less income so that he was no better off than he was before the price went down. The movement from Q to S is the substitution effect. The income effect is what is left when the substitution effect (Q to S) is subtracted from the total effect (Q to R), which is R to S in the graph above. What makes this a Giffen good is that the size of the income effect is bigger than the size of the substitution effect.

Question 5: Suppose the production function = 50 + 5.93x 0.5 where y = corn yield in quintal per hectare, x = quintals of nitrogen applied per hectare, Find

i. 

MP and AP Y=50 + 5.93x0.5 MP = ∂Y/∂X = 5.93(0.5) X-0.5 =2.97/ √X



AP = Y/X =

50 + 5.93X0.5 X

=

50 + 5.93 X0.5

ii.

Identify the type of return to scale Y(X,L) = 50 + 5.93 X0.5 L0 Y(tX,tL) = 50 + 5.93 (tX)0.5 (tL)0

= 50+ 5.93 t0.5 X0.5 t0 L0 = 50 + 5.93 t0.5+0 X0.5+0 = 50 + 5.93 t0.5 X0.5 Since 0.5 < 1 we have decreasing returns to scale.

Question 6: Suppose that the production function is given

i.

The MPP of X1.

MPPx1 = ∂Y / ∂X1 = 2(0.5)X1(0.5-1)X20.333 X20.333

=

X10.5 ii.

MPPx2 = ∂Y / ∂X2 = 2(0.333)X10.5X20.333-1 = 0.666X10.5 X2 0.667

iii.

MRT X1 to X2 = MPPx1 MPPx2 = X20.333 X10.5 0.666X10.5 X2 0.667 =

X2 0.666X1

Find...