Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

EC 500 Micro > Term One > Flashcards

Term One Flashcards

1
Q

What does A>~B mean?

What happens it B>~A is also true

What if the middle statement is not true?

A

The top means the A is weakly preferred to B.

if the bottom is true it means they are different

It means that A is strictly preferred to B A>B

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

What must hold true for rationality to hold true? Define these two concepts.

A

Completeness: We must have either A>~B or B>~A or both

Transitivity: if A>~B and B>~C then A>~C

Both of these hold true for rationality.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Define the concepts of ; Monotonicity and convexity.

A

Monotonicity: Implies that people prefer more than less (max)

Convexity: Suggests that the average of two bundles is better than all of one or the other (interior solution rather than corner)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

How to map utility functions?

A

Rearrange the function to be a function of Y and X, then sub in values of K (which you choose) to give the curves.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

What does A~B mean?

A

It means that the individual is indifferent between A and B

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

What is the tangency equation in consumer optimisation

A

MRS=Px/Py

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

How to test for convexity?

A

Take the MRS and it should be decreasing in X and increasing in Y

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

How to test for an interior Solution

A

Point A: Find Y=M/Py and X=0

MUx/Px > MUy/Py

Point B: Find X=M/Px and Y=0

MUy/Py > MUx/Px

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Draw a diagram to show the effect of an income increase for goods when X and Y, when;

Both goods are normal

X is an inferior good

A

Normal both increases normally and shifts out parallel

X inferior

It will cause X to decrease and Y to increase further.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Draw a diagram to show the effect of a price increase in X.

Show what would happen with different set of preferences.

A

There will be less X consumer firstly.

Then even more less X and higher Y.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Plot indifference curves and budget constraints for perfect complements and given the utility function

A

Diagram where ax=By and there is right angles through this line. Then a budget constraint.

U(X,Y)=Min{ax,BY} Px, Py, M

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

What is shown by a demand curve?

A

A demand curve shows how much an individual will consume at a given price.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

What is shown by an Engel curve.

A

An Engel curve shows how much an individual will consume at each level of income.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

How to derive;

  • Demand Curve
  • Engel Curve
A

Take preferences, utility function and prices for goods X and y. Then plot down the level of X consumed for different levels of utility X1, X2, X3.

  • Demand, trace the levels to prices, P1 is highest to X1.
  • Engel, trace the levels to income, M1 is the lowest to X1.
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

How to derive a demand function.

What does the demand function contain?

A

Use the tangency condition and sub in the result to the budget constraint.

it shows information from both the demand curve and the Engel curve.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

What does a Straight Engel curve show?

What is an example

A

It shows the presence of Homothetic Preferences.

This is where the demand increases in line to income .

Perfect Complements are an example.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

Draw a demand curve and engel curve for quasi-linear utility

A

Income less than aPy/B will all spend on X and nothing above this.

Then the curve for Y will be the opposite, where things above aPy/B all will be spent on Y.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

Classify;

  • Inferior Good
  • Homothetic
  • Normal Goods
  • Necessary Goods
  • Luxury
A

Inferior: if income increases demand decreases.

Normal: if income increases demand increases

Homothetic: income and demand increase proportionally

Necessary: demand increases less than proportional than income.

Luxury: Demand increases greater than proportional than income

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

Outline what happens (two effects) from an increase in price of good X.

A

Substitution Effect; X becomes more expensive relative to Y. This means consumers will substitute Y in place of X.

Income Effect; Consumers have relatively less income, they will change their consumption amounts of both goods.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

What effects does the Marshallian Demand curve show?

How do you isolate the substitution effect

A

It shows both substitution and income effects.

Sub effect is isolated by finding the real income level.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

Give two measures of REAL income

A

Slutsky: The Purchasing power of the income.

Hicks: The utility that the income provides.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

What is shown by a Hicksian Demand curve?

A

it shows the change in demand for a good, as a result of a price change. Keeping the utility of the consumer constant.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

What is shown by the Slutsky demand curve?

A

It shows the change in demand of a good as a result of a price change. When purchasing power is kept constant..

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

Derive the demand curve for Slutsky and Hicks

A

Find the asnwers at Micro Lecture 7, slide 9-11

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

Is consumer surplus a good measure of welfare?

A

Consumer surplus is an inexact measure of welfare.

This is because the Marshallian Demand curve shows the consumer’s demand for a good with a fixed amount of nominal income.

Price changes will also change the value of real income.

It does not show a consumer’s true valuation of the goods.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

What is compensating variation?

A

Following a price change it is the income change that would be required for a consumer to move back to their original indifference curve that they were on before the price change.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
27
Q

What are the axis labels for a CV and EV diagram?

A

Good x on x axis

Money on Y axis.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
28
Q

Draw a CV diagram

A

Lecture 8 slide 17

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
29
Q

What is Equivalent Variation (EV)

A

It is the change in income, with constant prices, that would have the same effect on consumer welfare as a change in the prices with the income constant.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
30
Q

What effect on the budget constraint does

  • Compensating Variation
  • Equivalent Variation

have.

A
  • CV: Will cause a shift out of BC, parallel to new prices.

- EV will cause a shift inwards of BC, parallel to old prices.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
31
Q

How to graphically represent CV, EV , CS

A

Lecture 8 slide 26-30

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
32
Q

How to work out CV and EV

A

CV: old utility new prices

EV: New utility Old prices.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
33
Q

How to find Hicks and Slutsky

A

Hicks: Keep utlity constant and solve utliity function for new M, using demand functions found first.

find new demand AMOUNTS (remember demand functions cannot change).

Slutsky: Using old amounts and new prices, use budget constraint to find new income.

Use new income to find new demand amounts.

Use the change in demand amounts to calculate the substitution effect.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
34
Q

What is an individal’s budget equation with respect to labour supply?

A

pY+(T-L)w = M +Tw

Where L represents Leisure,

therefore T-L is the amount of hours worked.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
35
Q

What does w represent?

A

It is the rate at which she sells her labour hours and buys her leisure time.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
36
Q

Draw the budget constraint for labour market.

A

Straight gradient from Y intercept (M+Tw)/P. down to a point where M/P. Then straight down to x axis, where leisure is T hours.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
37
Q

If x is the number of leisure hours, express the budget constraint for this.

A

(M+(T-X)w)/P

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
38
Q

What happens if there is an increase in non-labour income?

A

It will cause the budget constraint to shift straight upward from M to M1.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
39
Q

Which effect dominates at each job wage level?

A

High wage: Income effect.

Low wage: Substitution Effect.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
40
Q

What is the effect of;

Lowering Government benefits?

Raising Income Tax?

A

GB: will be the same as reducing non-labour income. Therefore, there will be less leisure.

IT: It is the same as reducing the wage rate, this will mean less leisure.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
41
Q

What is the effect of an decrease in the interest rate?

A

Ambiguous for lenders

Lower C1 for big lenders as income effect will dominate substitution effect.

Raises C1 for borrowers

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
42
Q

What is the effect of a increase in interest rate?

A

Lowers C1 for a borrower.

increases C1 for big lenders.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
43
Q

What can be classed as uncertainty in the consumer choice model.

A

not sure of the quality of the product.

not sure of the state of the future environment.

Financial risk is embedded in the actual product, i.e; a financial asset.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
44
Q

What must be true about states of the world?

A

Only one can occur

at least one must be true in the future.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
45
Q

Give two examples of state contingent commodities.

A

Insurance,

Financial assets

What clothes you choose to wear.

Investment decisions

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
46
Q

What is a contingent consumption plan?

A

it is something which specifies what you will consume in each state of the world.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
47
Q

how do you apply consumption theory to uncertainty situations?

A

you will have your different levels of consumption as different states of the world.

mishap wealth

no mishap wealth

for example.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
48
Q

How do you find the budget constraint under uncertainty?

A

You equal the contingent consumption plans and solve

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
49
Q

how to you find the prices in uncertainty.

A

You can see on the budget constraint that they are in the form xpx+ypy=M

Slope of budget equation = -px/py.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
50
Q

What is the general form of utility functions for an uncertainty problem?

A

U(X,Y) =piex V(x) + piey V(Y)

Pie is the probability if the state of the world.

V is the expected utility function. It is an increasing function.

V is known as a Bernoulli Utility function.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
51
Q

What does it mean if a contract is actuarially fair?

A

It means the expected gain of the individual and insurance firm is zero.

where Piex/piey = px/ py and

X=Y.

52
Q

What does it mean if V (.) is;

  • Concave
  • Linear
  • Convex
A

Concave:

means they are risk adverse, they always buy actually fair insuranse, the have convex preferences .

Linear:

they are risk neutral, indifferent about actually fair insurance and have linear preferences.

Convex:

they are risk loving, they do not buy actuarial fair insurance and have concave preferences.

53
Q

How do you find the shape of the V(.) function?

A

You difference it twice with respect to x or y.

54
Q

What does the certainty line show?

A

it shows all of the points of actuarially fair insurance.

55
Q

What is the independence assumption?

A

It is that the utility of one is independent of the utility of another option.

56
Q

What is the Allias paradox?

A

it is the notion that peoples actual choices are not aligned with their predicted choices, based upon their expected utility.

57
Q

How do you find the financial portfolio diversification choice budget constraint?

A

You will find the returns from both companies from one state of the world and label it x and then do the same for the other state and label it y. Then you will form the constraint from these two curves.

Remember than you aim to remove the K from the equation, you ideally want just a numerical figure.

58
Q

What financial portfolio should an individual choose?

A

They choose the one which is tangential to the highest indifference curve possible.

59
Q

What does the fair odds line show?

A

It connects all of the consumption bundles which have the same average value.

The further out it is, the higher the return it gives.

it is always perpendicular to the certainty line.

60
Q

how much consumption will a household give up to forgo consumption risk?

A

The difference between the fair returns intersects the certainty line and where the indifference curve intersects the certainty line.

61
Q

What is the mathematicaal example of risk adversion?

A

U(X,Y)= - (piex exp(-rx) + piey exp(-ry))

Where r is a measure of how risk adverse the individual is.

-exp(-rx) is the same as bernoulli utility function V(.).

62
Q

What is the technology of a firm?

A

it is the different input-output combinations that are available to that particular firm.

They can be represented using a production set.

63
Q

What is a production function?

A

It is something which represents the maximum output that can be obtained for each input level.

Y=F(K,L)

64
Q

How to find the returns to scale of a production function?

A

F( tK , tL) > t × F( K , L)
where t > 1.

Then we say that the production function exhibits increasing returns to scale.

Similarly,
F( tK , tL) 1, then increasing returns to scale
If α + β

65
Q

What is the marginal product of an input?

A

The margina; product of an input is the extra amount of output that is gained from increasing the input by one unit .

Partial differentiation of prod.func/partial input.

66
Q

What is diminishing marginal product?

A

It is where the marginal returns decrease with every extra unit that you input.

67
Q

What is the technical rate of substitution?

A

it is the rate at which you substitute Labour for capital.

TRS= MPL/MPK

The slope of the isoquant is equal to the TRS.

As the isoquant gets flatter, it is known as diminishing technical rate of substitution.

68
Q

What is fixed proportions technology?

A

it is where K and L must be used in fixed proportions.

{ak,BL) = f(K,L)

It is optimal for K/l = B/a

Diagram are perfect right angle isoquants.

69
Q

What axis are K and L on

A

K = Y axis

L = x axis.

70
Q

What is the profit maximising condition for a firm and their technology?

A

MPl=w/p and MPK = r/p

71
Q

what to consider about the profit maximising conditions?

A

Is it short- run or long-run.

In the short-run, capital (kbar) is fixed, therefore you can only max L.

72
Q

Where does a firm maximise their profits in the short run.

A

They maximise where the production function is tangent to the iso-profit line.
i.e w/p = MPL.

73
Q

What is cost minimisation.

A

It is where a firm produces at the lowest cost for given output level.

74
Q

How can you see cost minimisation mathematically and graphically.

A

Maths;

w/r = MPl/MPk

Graphs

Where the iso cost and iso quant are tangent.

75
Q

What is an isocost curve?

A

K= TC/r - wL/r

Slope is -w/r

Further from the origin, the higher the total cost.

76
Q

What is the ratio of the prices equal to?

A

It is equal to the technical rate of substitution.

77
Q

What happens when Y appears in the optimal demand function in cost minimisation?

A

They are then called conditional factor demand functions.

78
Q

How do you mathematically find a cost function?

A

You firstly equate MPL/MPK to w/r.

You then find K in terms of w,r and L.

Sub back into prod fucn.

Use this to get ‘conditional factor demands’

in terms of Y,r and w.

Then sub into

rK+wL to give the cost function

79
Q

What is the cost minimisation form for fixed proportions technology?

A

r(y/a) + w(y/B)

When F(K,L) = {aK,BL)

80
Q

What does it mean for average cost if Returns to scale are;

  • increasing
  • constant
  • decreasing
A
  • increasing rts means AC is decreasing

Constant = constant

-decreasing rts means AC is increasing.

81
Q

Why are fixed costs not a long-run problem?

A

Because you can shut down any business in the long-run, so it is no issue.

82
Q

At what point do MC and AC cross?

A

Where AC is at its lowest,

MC above AC = AC increasing.

MC below AC = AC decreasing.

83
Q

What is the relationship between short and long run K

A

K in short run must be at least as high as K in the long run as K can adjust to optimal in the long run.

84
Q

Do the long run and short run cost curves ever cross?

A

Yes, at the level of output where K (short run) minimises costs.

85
Q

Draw the relationship between the short run and long run cost curves

A

Lecture 16, slide 20.

86
Q

how do you find a firm’s supply function?

A

You find the P=MC output for many prices. Then that Y is what the firm will supply.

Essentially the marginal cost is the firm’s supply curve (above the AC curve, so they are not making a loss).

87
Q

How to find max profits using the cost function?

A

You use pY- c(Y) and then differentiate it.

Set it to 0.

Then solve for Y.

88
Q

What conditions must hold true in a competitive market equilibrium?

A

All consumers and firms act as price takers.

Consumers maximize their utility.

Firms maximize utility.

The markets will clear.

89
Q

How do you find the industry supply curve?

A

It is the sum at each price of the supply of each firm.

90
Q

How do you find the market demand curve?

A

You just add up the individual demand curve of each consumer.

91
Q

What is partial equilibrium analysis?

A

It is when you only look at the effects in one market and not all markets. You may miss some important information.

92
Q

What are the assumptions for a competitve General Equilibrium??

A

There is a market for every good.

There are no externalities in production or consumption.

Firms and consumers are price takers.

Firms max profit, consumers max utility.

Demand equals supply in each market.

93
Q

When is an allocation feasible?

A

When the total endowment is equal to the total consumption.

In an edgeworth box it is any point within the box.

94
Q

What points do markets clear in the exchange economy?

A
  • Robinson and friday both max utility given the prices.

- Both robinson’s and friday’s markets clear.

95
Q

What is a pareto improvment?

A

Where you make some people better off without making someone else worse off.

96
Q

What is the Pareto Efficient Condition in the exchange economy?

A

When MRSf=MRSr

Assuming convex preferences.

97
Q

What is the First Fundamental Welfare Theorem?

A

All market equilibria are Pareto efficient.

Assuming;
no externalities. 
perfect info and comp.
no public goods.
complete markets
market clearing prices.
98
Q

What is the Second Fundamental Welfare Theorem.

A

If all agents have convex preferences then any pareto efficient allocation can be supported as a market equilibrium for an appropriate set of prices and initial endowments.

Essentially there is always a way to redistribute for the market to operate.

99
Q

At what point does government intervention cause inefficiency in the market?

A

it is where the governments change the incentives of the consumer.

100
Q

Imagine you have a subsidy of 40 units at 80 per unit. The full amount is 125 per unit from 100. Draw a diagram to illustrate this when this is good x. The other good is Y with price of 1 per unit. You have an income of 1000.

A

Do xpx+ypy=1000 to find that 40 units at 80 means 680 of the other will be consumed.
px is subsidy price.

px’ is highest price.

Then do 40+(m-xpx)/Px’ =Final amount.

Ensure that budget constraint is kinked for change in price

101
Q

If you choose B on day one over A and you can still afford it on day 2. What does the theory of revealed preferences sat you will do?

A

It says that you will choose something other than A, it is likely that you will choose B. However, the theory does assert that you will not definitely choose B.

102
Q

Define the Robinson-Crusoe Model?

A

Only one person: Robinson.

Two Goods; Coconuts and Leisure.

Break into two aspects;

  • The consumer
  • The producer
103
Q

What is the marginal rate of transformation?

A

It is the rate at which a firm can switch production of one good for another.

it is given by the slope of the production possibility frontier.

104
Q

What is the TRS

A

Technical Rate of Substitution.

It is the rate that you substitute one factor input for another.

105
Q

Where will robinson the producer maximise his profit?

A

Where the production function and ISO Profit are tangent.

W/p=MPL

W/P is gradient of Iso- profit

y=pie/p + (w/p) L

106
Q

Where will robinson the consumer maximise his utility?

A

his total time is T, he can buy coconut at price p.

pY+w(T-L)=pie* +wT

he will max where -MRS=W/P

The budget constraint of the consumer is the same as the producer ISO-Profit:

y=pie/p + (w/p) L,

so where the indifference curve and budget constraint ar tangent is where utility is maxed.

107
Q

What do the market equilibrium price set entail?

A
  • Each consumer maxes utility.
  • Each firm maxes profit.
  • All markets clear
108
Q

Where is the robinson crusoe economy in equilibrium?

A

Where the production function and indifference curve are tangent to each other and also to the ISO-PROFIT/BUDGET

109
Q

Explain how the First Welfare Theorem applies for Robinson Crusoe Economy?

A

-MRS=w/p and MPL=w/p

MPL= -MRT this is because their is only one good in the economy.

Therefore; MRT=MRS

110
Q

Explain the Second Welfare Theorem in terms of the Robinson Crusoe Economy

A

There is only one consumer for equitable allocation.

Any pareto efficient allocation will be a market equilibrium for a price set and initial endowments if the production function is convex.

If there are increasing returns, the second welfare theorem will not hold in the economy.

111
Q

What is a Shadow Price?

A

It is the price at which the individual will be willing to transact in the Robinson-Crusoe economy.

112
Q

Define the Crusoe, Friday economy

A

Two consumers; Crusoe and Friday.

Two goods; Bananas and Coconuts.

They do not care about leisure

Production of Bananas and Coconuts come from the labour of Crusoe and Friday

113
Q

What is the slope pf

  • PPF
  • Production Function
A
  • PPF: The marginal rate of transformation.

- PF: The marginal product of Labour/Capital

114
Q

What does it mean if the PPF is a convex feasible set?

A

It means there is diminishing marginal product to production.

115
Q

Define;

  • Absolute Advantage
  • Comparative Advantage
A

Absolute Advantage;

Where you produce more of one good per unit time than the other producer.

Comparative advantage;

Where the amount of a good that you make in the time it takes to make one unit of the other good is more than the other producer.

i.e; Your opportunity cost of production is lower.

116
Q

If one person has absolute advantage for both goods will it mean they have both comparative advantage?

A

No it does not.

people will produce at the point where they have comparative advantage.

117
Q

What three things must hold for an economy to be Pareto Efficient?

A

1) Production efficiency; You cannot change factor inputs in any way to increase production.

TRS of all firms is the same.

2) Consumption Efficiency;
There is no way for consumers to exchange goods and be better off.

MRS of consumers in the Same.

3) Top Down;

There is no way to change what is being produced and make someone better off.

MRS=MRT

118
Q

If they are employed in the Robinson Crusoe Economy, can you explain the profit max diagrams and curve

A

IN you notes topic 11 page 3,4 or lecture 20 slides 15-20.

119
Q

What must be true for the second welfare theorem to hold. With regards to the PPF?

A

The PPF must be convex.

120
Q

if the government plan to give you some money after a price rise, how will you know if they are better or worse off.

A

Calculate the budget equation with the new price and old consumption bundle. Equal to an income m +amount given.

Then solve to find this given amount. If the given amount is greater than the compensating variation then you know your utility levels will be higher and you will be better off.

121
Q

How to calculate CV?

A

Make sure your X and Y conditional demands are in terms of PX and Py, then find original utility.

Keep original utility constant.

Then sub into utility function X and Y Conditional demands with the new PX and Py.

Then solve for M’

Then M’-M is equal to the CV.

It is M- EV for EV

and M+CV for CV.

122
Q

What is implied by monotonocity?

A

That indifference curves are downward sloping more is better. It implies people prefer more to less and will consume the most that they possibly can consume.

123
Q

What is implied by convexity?

A

If you join two points on an indifference curve with a line, that line should sit above the actual indifference curve.

124
Q

Define the;

  • Income Effect

- Substitution Effect

A

Income effect is the change in consumption as a result of a change in real income.

Substitution effect is the change in consumption keeping real income constant.

125
Q

What is the Slutsky Equation? Explain it also.

A

Change in (C1) / Change in P1 = Change in (C1 sub) /Change in P1 + Change in (c1)/change in M times (M1-C1)

M1 - c1 is neg if borrower.

Sub is always neg for a normal good.

EC 500 Micro (2 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions