3.2 WELFARE ECONOMICS P

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WHAT IS WELFARE?

Welfare is the measure of living standard or utility

Welfare analysis is concerned with measuring the living standard or level of utility or in terms of productivity taking in to account the degree of efficiency in allocating resources

If welfare is concerned about issues of efficiency, how do we know whether a resource allocation is efficient or not

Pareto efficiency is used as a standard measure of efficiency

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PARETO EFFICIENCY

Assumptions economy consists of two persons (A and B); two goods (X and Y) are produced;production of each good uses two inputs (K and L) each available in a fixed quantity

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I. ECONOMIC EFFICIENCY An allocation of resources is efficient if it is not possible

to make one or more persons better off without making at least one other person worse off.

A gain by one or more persons without anyone else suffering is a Pareto improvement.

When all such gains have been made, the resulting allocation is Pareto optimal (or Pareto efficient).

Efficiency in allocation requires that three efficiency conditions are fulfilled

1. efficiency in consumption2. efficiency in production3. product-mix efficiency

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1 EFFICIENCY IN CONSUMPTION

•  Consumption efficiency requires that the marginal rates of utility substitution for the two individuals are equal:

• Consumer 1:Max U1(x1, y1)

s.t pxx1 +pyy1 m1

Form the lagrangian function L = U1(x1, y1) + (m1 - pxx1 - pyy1 )

FOC L/ x1 =0 U1x1 - px = 0 = U1x1/ px ……….1

L/ y1 =0 U1y1 - py = 0 = U1y1/ py ……….2

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L/ =0 m1 - pxx1 - pyy1 = 0

pxx1 +pyy1 = m1 …………….3

From equation 1 and 2 U1x1/ px = U1y1/ py

U1x1/ U1y1 = px/ py

MRS1x,y = px/ py ……………4

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CONSUMER TWO’S PROBLEM

Similarly for consumer two we have Max U2(x2, y2)

s.t pxx2 +pyy2 m2

Form the lagrangian function L = U2(x2, y2) + (m2 – pxx2 - pyy2)

FOC L/ x2 =0 U2x2 - px = 0 = U2x2/ px

……….1’L/ y2 =0 U2y2 - py = 0 = U2y2/ py

……….2’L/ =0 m2 – pxx2 – pyy2 = 0

pxx2 +pyy2 = m2 ……………………………………..3’ 7

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From equation 1’ and 2’U2x2/ px = U2y2/ py

U2x2/ U2y = px/ py

MRS2x,y = px/ py ……………..4’

Therefore, from equation 4 and 4’ we have MRS1x,y= MRS2x,y = px/ py

If this condition were not satisfied, it would be possible to re-arrange the allocation as between 1 and 2 of whatever is being produced so as to make one better-off without making the other worse-off

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2 EFFICIENCY IN PRODUCTION

Efficiency in production requires that the marginal rate of technical substitution be the same in the production of both commodities. That is

Problem of the producer in producing good one (x) is:

Min wL + rKs.t F(Lx, Kx) ≥ x-

L = wL + rK + (x- - F(Lx, Kx) ) from FOC and some steps we get

MRTSX = MRTSY

If this condition were not satisfied, it would be possible to re-allocate inputs to production so as to produce more of one of the commodities without producing less of the other

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3 PRODUCT-MIX EFFICIENCY

The final condition necessary for economic efficiency is product-mix efficiency

This requires that:  

MRTx(L, K) = MRTy(L, K) = MRUS1 = MRUS2

 

 

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ALL THREE CONDITIONS MUST BE SATISFIED

An economy attains a fully efficient static allocation of resources if the three condition we have discussed earlier are satisfied simultaneously.

The results readily generalise to economies with many inputs, many goods and many individuals.

The only difference will be that the three efficiency conditions will have to hold for each possible pair wise comparison that one could make.

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II. THE SOCIAL WELFARE FUNCTION AND OPTIMALITY

 In order to consider such choices we need the concept of a social welfare function, SWF.

A SWF can be used to rank alternative allocations. For the two person economy that we are examining, a

SWF will be of the general form:  

  The only assumption that we make here regarding the

form of the SWF is that welfare is non-decreasing in UA and UB.

Just as we can depict a utility function in terms of indifference curves, so we can depict a SWF in terms of social welfare indifference curves.  

)U,U W(=W BA

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SOCIAL WELFARE FUNCTION

Max W = W(UA, UB)

Subject to UA = UA(XA) and UB = UB(XB)

and X = XA + XB

gives the necessary condition

WAUAX = WBUB

X

where WA and WB are the derivatives of the social welfare function wrt UA and UB and UA

X and UBX are the derivatives of the

utility functions, marginal utilities, so that the condition is that marginal contributions to social welfare from each individual’s consumption are equal.

For W = wAUA(XA) + wBUB(XB)

where wA and wB are fixed weights the condition is

wAUAX = wBUB

X

and for wA = wB = 1 so that the fixed weights are equal

UAX = UB

X

In this case, if the individuals have the same utility functions, social welfare maximisation implies equal consumption levels.

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RAWLSIAN WELFARE FUNCTION

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One way to give utilitarianism a Rawlsian character is to use a particular form of Social Welfare Function, which for two individuals would be

W = min(UA, UB)

so that W is the smallest of UA and UB.

Raising utility for the worst off will increase welfare

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UTILITARIAN SW

W = W(UA, UB) the utilitarianism SWF is given as:

W = φ0U0 + φ1U1

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Maximised social welfare.

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shows a social welfare indifference curve WW, which has the same slope as the utility possibility frontier at b, which point identifies the combination of UA and UB that maximises the SWF.

The fact that the optimum lies on the utility possibility frontier means that all of the necessary conditions for efficiency must hold at the optimum.

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