Capital and Investment

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University of Essex Session 2011/12

Department of Economics Autumn Term

EC111: INTRODUCTION TO ECONOMICS

CAPITAL AND INVESTMENT

Like labour, capital is a factor input. But because capital takes time to install it is a

long run decision for the firm. The firms decision is whether or not (and if so how

much) to invest now in the expectation of returns (additional profits) in the future.

This is a cost-benefit calculation. If the benefits exceed the (opportunity) costs then

it is worth investing. But to make this calculation a way must be found to compare

costs incurred now with benefits that accrue in the future.

Present Values

Suppose you invest 100 today in a project that will pay you:

55 one year from now, and

70 two years from now

Should you invest?

Note that 55 one year from today is worth less than 55 today. This is because you

can put a smaller sum in the bank and accumulate interest, such that the total value

in a years time is 55.

V1invested now would be worth V1(1+R) in a years time, where R is the interest

rate. If V1(1+R) = 55, then V1= 55/(1 +R). This is the Present Value of 55 in one

years time

Similarly V2invested now would be worth V2(1+R)(1+R) in two years time. So the

present value 70 in two yearstime is V2= 70/(1+R)2.

The Net Present Value of the investment is the present value of the (future) benefits

minus the (current) cost. Suppose the interest rate is 10 percent (0.1):

85.710085.575010021.1

70

1.1

55NPV

The cost-benefit rule is that the investment is worth undertaking if the NPV is

positive.


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Note that:

Future returns (or benefits) are discounted by the interest rate.

The higher is the interest rate the more heavily they are discounted and the

less likely the NPV will be positive. If the interest rate was 20 percent overthese two years the NPV would not be positive. Q: what would it be?

The further away are the future returns the more heavily they are

discounted. Thus if the returns were nothing after one year 55 after two

years and 70 after three years the NPV would be negative even at R = 0.1.

Q: Check this calculation.

The future benefits are expected returns, they are uncertain and therefore the

discount factor used may need to take account of the riskiness of the project.

What of there is inflation? 55 in a years time has lower purchasing power if priceshave risen than if they have not. In that case we need to discount by the realinterest

rate:

Real interest rate = nominal interest raterate of inflation.

The real interest rate is uncertain even if the nominal interest rate is not since we

would need to forecast the rate of inflation over the lifetime of the investment.

The demand for capital services and the user cost of capital

A firm considers an investment opportunity as follows:

Purchase of a machine (i.e. one extra unit of capital) today at price 0.

At the end of the period the machine is sold at price 1(1is less than 0

and could be zero if the machine is scrapped).

The return from the investment is the value of the additional output for as

long as the machine is in use, This is the Marginal Value Product of Capital,

MVPK = P MPK, where P is the price of the firms output. (Q: what if the

firm is not a perfect competitor?

The one-period NPV is:

0

1

R1

MVPKNPV

If NPV is positive then the firm should invest.


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Rearranging this expression, we find that NPV > 0 is the same as:

MVPK > R0+ (01)

The left hand side is simply the Marginal Value Product of Capital. The right hand

side is the User Cost of Capital (in earlier lectures we denoted this as r; it issometimes called the Rental Price of Capital).

Note that this is an opportunity cost: interest forgone on the sum invested plus

depreciation.

Note also that even if the firm paid for the investment from its retained earnings

there is still an opportunity cost: the return that could have been obtained from

leaving the money at the bank.

More generally note that:

The user cost of capital plays the same role in the demand for capital services as the

wage does in the demand for labour.

The MVPK curve is downward sloping. MVPK declines as K increase because of

diminishing marginal returns to capital.

The firm will install capital up to the point where the MVPK is equal to the user

cost. Beyond that the NPV is negative so a further investment does not add to

profits.

The firms demand for capital decreases as the user costincreases (because the

marginal machine becomes unprofitable).

Thus, the demand for capital will be higher:

The lower is the interest rate.

The lower is the price of capital goods.

The lower the rate of depreciation

The higher the price of the firmsoutput.

The aggregate demand for capital by an industry is the (horizontal) sum of the

demand curves of all the individual firms in the industry.

Note that this is the demand for the stock of capital (just as labour demand is the

demand for the stock of workers). Investment over a given period is the change in

the stocki.e. the increase in the stock.


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The market for capital services

The supply of capital services may be upward sloping in the short run and perfectly

elastic in the long run (particularly if capital goods are produced under conditions

of perfect competition).

If the demand curve shifts out, the price, , increases in the short run, so the UC K

rises. In the long run the as the output of capital goofs expands the price falls back

to the original level. The total amount of new investment is K2K1.

MVPK,

UCK

UCK1

K1 K

MVPK

MVPK,

UCK

UCK1

K1 K2 K

DK

DK

SKLR

SKSR


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WELFARE ECONOMICS

Welfare economics assesses how well the economy works from the point of view of

welfare and if and how it can be made to work better. The two key elements are

efficiency and distribution.

Pareto Efficiency

The Pareto criterion is a key value judgement. It says that situation A is preferred to

situation B if in A at least one person is better off and no one is worse off than in B.

This is a very stringent criterion. The reason we use it is that we have no way of

weighing one persons utility against another persons utility. If we did then we

could say that A could be superior to B even if some people were worse off under A

as long as the value of the gains to the gainers outweighed the value of the losses to

the losers.

Example: shar ing a cake between two people.

At point A individual A has the whole cake and gets utility UA; at point B individual

B gets the whole cake and gets utility UB.

We cannot say whether allocation C is Pareto superior to either A or B because as

compared with C one person is worse off.

D

E

B

A

Utility Possibility Frontier

UA

UB

C


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Allocation E is Pareto superior to C, because both individuals are better off. But D

is not superior to C.

At any point inside the frontier there is a range of Pareto superior allocations. Note

that here some of the cake is lost or wasted; at least one person could be made betteroff if this inefficiency was eliminated.

Points like D and E (and A and B) are Pareto eff icient. One person cannot be made

better off without making another person worse off.

Pareto efficient points are points of maximum efficiency

Note that Pareto efficiency is of no help in resolving distributional issues. A, B, D

and E are all Pareto efficient. We have no way of choosing between them unless we

are willing to make additional value judgements.

Necessary conditions for a first-best Pareto efficient allocation

Exchange (or consumption) effi ciency

Suppose there are two individuals in the economy, A and B, and two goods, beer

and pizza.

A is willing to exchange 2 beers for one pizza.

B is willing to exchange 4 beers for one pizza.

That must mean that MRSA MRSB; the ratio of marginal utilities cannot be the

same for the two individuals.

The following exchange will make both persons better off:

B gives 3 beers to A

A gives one pizza to B

As long as MRSA MRSBthere is scope for exchanges that make one person better

off without making the other worse off. (So there is a Pareto superior allocation).

Exchange efficiency requires that MRSA= MRSB.


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Production eff iciency

This requires being on the Production Possibility Frontier

Being on the PPF implies that there is no way of reallocating inputs across

industries that would result in higher output in at least one industry without a

reduction in the output of the other industry.

Assuming two factors K and L, this means that:

MRTSKLbeer

= MRTSKLpizza

Factors of production are allocated across industries with maximum efficiency when

the marginal rate of technical substitution between factors is the same across all

industries.

Slope of PPF = Marginal

Rate of Transformation

Qbeer

Qpizza


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The Fundamental Theorems

What are the efficiency properties of market outcomes?

In particular, is a perfectly competitive general equilibrium first-best Pareto

efficient?

A perfectly competitive generalequlibrium is where there is perfect competition in

all markets. The specific conditions are:

Individuals maximise utility

Firms are price takers and maximise profits

All markets for goods and factors clear.

F ir st Fundamental Theorem of Welf are Economics

In the absence of externalities (discussed next week), and under certain conditionsthat ensure that an equilibrium exists, any perfectly competit ive equi li bri um is fi rst

best Pareto eff icient.

This can be shown by examining the three necessary conditions:

Exchange effi ciency

All individuals will have the same MRS between any two goods provided that they

maximise their utility and they all face the same set of prices. Thus for all

individuals in the economy:

Y

X

XY

P

PMRS

Production eff iciency

Each firm minimises the cost of production (a necessary condition for maximising

profits) and they all face the same set of factor prices. Thus for all firms:

r

wMRTS

LK

Output choice effi ciency

Perfectly competitive firms maximise profits so that

Y

X

Y

X

P

P

MC

MC


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The ratio of marginal costs is the marginal rate of transformation MRT, or the slope

of the Production Possibility Frontier. This is because it is the ratio of opportunity

costs.

Thus we have:XY

Y

X

Y

X

XY MRS

P

P

MC

MCMRT

ConclusionPerfect competition achieves a first best Pareto efficient allocation of

resources. There is no explicit coordination: the price mechanism provides the right

signals for an efficient allocation.

But this says nothing about distribution. As the cake-sharing example showed, there

are many Pareto efficient allocations each with a different distribution of utility

between consumers. Thus a perfectly competitive equilibrium generates a particular

Pareto efficient allocation. What determines which one?

Second fundamental theorem of welfare economics

People have different abilities and different wealth (or endowments).

Under certain conditions that have to do with the kind of preferences consumers

have and the kind of technological constraints that firms face, any fi rst-best Pareto

eff icient allocation can be achieved for some distribution of in iti al endowments across

individuals.

Implication:The issue of efficiency and the issue of distribution can be separated in

principle. To change the distribution of income in an economy, one needs to change

only the distribution of initial endowments. Under the new set of initial endowmentsthe price system will achieve a new Pareto efficient final allocation.

This seems to suggest that there is no trade-off between equity and efficiency. All ex-

post income distributions can in principle be equally efficient.

The conditions required under the first and second fundamental theorems are

rarely satisfied in practice. There are a number of sources of inefficiency in the

market economy.


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Application: Taxation

One way of redistributing income is through lump sum tax which is independent of

a persons income (or any other economiccharacteristic).

Lump sum taxes seem a good way of modifying peoples initial endowments. Buthow do you redistribute e.g. from the rich to the poor without using income as a

criterion. Example: Mrs Thatchers Poll Tax: why was it so unpopular?

So the government must impose taxes on earnings and goods, which create

distortions and deadweight loss because they affect economic incentives and

decisions of individuals and firms. As a result the conditions of the first fundamental

theorem will not be satisfied.

Recall the example of a tax on a good:

Without the tax total welfare is consumer surplus + producer surplus = AEB

With the tax we have consumer surplus (ACD) + producer surplus(GBF) + tax

revenue (CDFG). The deadweight loss is DEF.

Assuming that the supply curve represents firms marginal cost curves we have:

MC + T = P

Suppose in the situation above the tax is placed on is good X and there is no tax on

good Y.

TG

CD

E

PXA

B

F

QX

S

S


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In general equilibrium we have:

XY

Y

X

Y

X

Y

X

XY MRS

P

P

MC

TMC

MC

MCMRT

At point A before the tax the representative consumers MRS is equal to the slope of

the PPF which is the MRT and which is also the ratio of marginal costs. .

After the tax the ratio of marginal costs is lower than the price ratio, because of the

tax wedge. We have move to a point like B where the indifference curve (which is

MRSXY) is steeper than the PPF (which is MRTXY).

The tax on good X has reduced the output of X, releasing factors of production to

produce more of good Y.

One interpretation of the deadweight loss is that it is the (lump sum) payment that

consumers in situation B would require to make them just as well off as they were in

situation A.

A similar analysis could be applied to an income tax, which distorts the labour

supply decision by altering the after tax price of leisure. This imposes a distortion

such that prices fail to reflect true marginal costs (i.e. opportunity costs). Q why is

this?

QX

A

B

MRT of economyQY

MRS of consumers


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Income redistribution takes place through taxes and subsidies.

Most tax-transfer systems are progressive (they transfer from the rich to the poor)

to some degree.

Because these taxes are distortionary there is some trade-off between equity andefficiency. It may be worth some loss of efficiency to have greater equity in society.

This is a normative issue that must be decided in the political process.

The theory of the second-best

Will removing a distortion always improve economic efficiency?

Clearly, removing the tax (or other distortion) that causes the economy to settle at B

would push it back to A, where MRS = MRT.

It is important to note that there is no other distortion in the economy. If there weretwo (or more) distortions then removing one of them does not necessarily increase

efficiency.

Similarly, in the presence of one distortion introducing another one could improve

things. Take the example of the tax on good X. Putting a tax on good Y could move

the economy from point B towards point A (less Y, more X)

This is an important for policy in the real world. Policies are often evaluated as if

the conditions for Pareto efficiency were met in the rest of the economy, which

usually they are not.

Note that this does not just apply to taxesthey are not the only source of

inefficiency. There are other sources of inefficiency or market failure that make

the choice of policy even more difficult.

Capital and Investment

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