Economic Analysis Economic Analysis for Businessfor Business

Session XVI: Theory of Session XVI: Theory of Consumer Choice – 2 (Utility Consumer Choice – 2 (Utility Analysis) with Production Analysis) with Production FunctionFunctionInstructorInstructorSandeep BasnyatSandeep Basnyat98418922819841892281Sandeep_basnyat@yahooSandeep_basnyat@yahoo.com.com

Utility FunctionUtility FunctionTwo ways to represent consumer

preferences:◦ Indifference Curve◦ Utility

Utility is an abstract measure of the satisfaction that a consumer receives from a bundle of goods.

Indifference Curve and Utility are closely related.◦ Because the consumer prefers points on higher

indifference curves, bundles of goods on higher indifference curves provide higher utility.

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Cones per hour

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tal U

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Utility Function Utility Function (Similar to Production (Similar to Production Function)Function)

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Total Utility

(Utility/Hr)

Cones/ Hour

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6

Marginal UtilityMarginal UtilityThe marginal utility of consumption of any

input is the increase in utility arising from an additional unit of consumption of that input, holding all other inputs constant.

Marginal Utility = ∆Tot. Utility∆Consumption

Marginal UtilityMarginal Utility

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Total Utility

(Utility/Hr)

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Marginal Utility (Utility/Cone)

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The Law of Diminishing Marginal Utility

The tendency for the additional utilitygained from consuming an additional unitof a good to diminish as consumptionincreases beyond some point

Utility MaximizationUtility MaximizationAssume:Two goods: Chocolate and vanilla ice creamPrice of chocolate = $2/pintMarginal Utility from chocolate = 16 utils/pintPrice of vanilla = $1/pintMarginal Utility from vanilla = 12 utils/pintSarah’s ice cream budget = $400/yrCurrently Sarah’s consumption of:

Vanilla ice-cream = 200 pints chocolate ice cream = 100 pints

Is Sarah maximizing her total utility? If not, should she buy more chocolate and less vanilla or more vanilla and less chocolate?

Utility MaximizationUtility MaximizationHere, Marginal Utility from chocolate = 16 utils/pintPrice of chocolate = $2/pintMarginal utility from chocolate/$ = 16/2 = 8 utils/$For every $ Sarah spends on chocolate, she derives additional utlity of 8 .Similarly, Marginal Utility from vanilla = 12 utils/pintPrice of vanilla = $1/pintMarginal utility from chocolate/$ = 12/1 = 12 util/$For every $ Sarah spends on vanilla, she derives additional utlity of 12

Sarah should spend more on vanilla than chocolate. By How much?

Utility MaximizationUtility MaximizationThe Rational Spending Rule:

Spending should be allocated across goods so that the marginal utility per dollar is the same for each good. i.e,

, Marginal utility from chocolate/$ = Marginal utility from vanilla / $

From our example, Marginal utility from chocolate/$ (= 16/2 = 8 utils/$) =Marginal Utility from chocolate / Price of chocolate = MUc /

PcSimilarly, Marginal utility from vanilla/$ (= 12/1 = 12 util/$) =Marginal Utility from vanilla / Price of vanilla = MUv / Pv

The Rational Spending Rule: MUc / Pc = MUv / Pv

Utility MaximizationUtility MaximizationThe Rational Spending Rule:

Spending should be allocated across goods so that the marginal utility per dollar is the same for each

MUc / Pc = MUv / Pv or,MUc / MUv = Pc / Pv or,

MUv / MUc = Pv / Pc orMUx / MUy = Px / Py

Here, MUv / MUc = Marginal Rate of substitution of

vanilla for chocolatePv / Pc = Relative price of vanilla and chocolateTherefore, How much should Sarah spend on

each?MRS = MUv / Muc = Pv / Pc

Utility MaximizationUtility MaximizationAt present,MRS = MUv / Muc = 12 /16 = 0.75 and,Pv / Pc = 1/2 = 0.5So,Sarah’s should reschedule her spending such

that herMRS = 1/2.

Utility Maximization-Change in PricesUtility Maximization-Change in PricesAssumeBudget = $400MUc = 20; MUv = 10PC = $2 & PV = $1QC = 75 & QV = 250 So, MUc / Pc = 20 / 2 = 10

MUv / Pv = 10 / 1 = 10 = MUc / Pc

If the price of chocolate falls to $1. Should you buy more chocolate or less?

Utility Maximization-Change in PricesUtility Maximization-Change in PricesAssumeBudget = $400MUc = 20; MUv = 10PC = $2 & PV = $1QC = 75 & QV = 250 So, MUc / Pc = 20 / 2 = 10

MUv / Pv = 10 / 1 = 10 = MUc / Pc

If the price of chocolate falls to $1. Should you buy more chocolate or less?

Now, MUc / Pc = 20 / 1 = 20. Should buy more chocolate.

Worked out examplesWorked out examplesJane receives utility from days spent traveling on vacation domestically (D) and days spent traveling on vacation in a foreign country (F), as given by the utility function

U(D,F) = 10DF. In addition, the price of a day spent traveling domestically is $100, the price of a day spent traveling in a foreign country is $400, and Jane’s annual travel budget is $4,000.Find Jane’s utility maximizing choice of days spent traveling domestically and days spent in a foreign country.Find her total utility.

Hint: Find consumer’s optimum using the rational spending rule.Marginal utility of D or F can be found by differentiating Total Utility keeping one factor constant at a time.

Worked out examplesWorked out examplesSolution:

The optimal bundle is where the slope of the indifference curve (MRS) is equal to the slope of the budget line (Relative price of the good), and Jane is spending her entire income.  So:

PD / PF = 100/400 = 1/4

MRS = MUD / MUF = F. 10D1-1/D. 10F1-1 = 10F/10D = F/D

Setting the two equal we get: F / D = 1/4 or, 4F = D ---(i)

From the above question: 100 D + 400 F = 4000 -------(ii)

Solving the above two equations gives D=20 and F=5.  Utility is 1000.

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No. of workers

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Recall Example:Recall Example: Production Function Production Function

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Q (bushels of wheat)

L(no. of

workers)

Relationship between input and output

Recall: Properties of Production Functions: Recall: Properties of Production Functions: Returns to ScaleReturns to Scale

production functions have increasing, constant or decreasing returns to scale.

(i) Q = 3L (ii) Q = L0.5 (iii) Q = L2

(Constant) (Decreasing) (Increasing)

Marginal product of labor (MPL) = ∆Q∆L

Marginal product of capital (MPk) = ∆Q∆K

Total ProductTotal ProductTotal Product of Labour:

Maximum output (Q) produced by the labour keeping the capital constant.

Total Product of Labour function:

Q = TPL = f (K, L)Total Product of Capital:

Maximum output (Q) produced by the capital keeping the labour constant.

Total Product of capital function:

Q = TPK = f (K, L)

•Average Product of Labour (APL) = TPL / L

•Average Product of Capital (APK) = TPK / K

Worked out examplesWorked out examplesFor the Production function: Q = 10K0.5L0.5

whose labour input (L) is 4, capital (K) is fixed 9 units calculate: TPL and APL

Solution: (TPL) = Q = f (K, L) = 10K0.5L0.5

= 10 x 90.5 x 40.5

= 10 x 3 x 2 = 60APL = TPL / L = 60 / 4 = 15

Relationship between Total, Average, Relationship between Total, Average, & Marginal Products: Assume K is & Marginal Products: Assume K is fixed and L is variablefixed and L is variable

Number of workers (L)

Total product (Q) Average product (AP=Q/L)

Marginal product (MP=Q/L)

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Total, Average & Marginal Total, Average & Marginal ProductsProducts

L Q AP MP

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Total, Average & Marginal Product Total, Average & Marginal Product CurvesCurves

Panel A

Panel B

Total product

Average product

Marginal product

Q1

L1

L1

L2

Q2

L2

L0

Q0

L0

Stage 1

Stage 2

Stage 3

Increasing Marginal returns

Diminishing Marginal returns

Negative Marginal returns

Marginal Product functionsMarginal Product functions

If the Production Function is Cobb-Douglas:

Q = AKαLβ

Then, Marginal Product of Labour (MPL) = dQ / dL =

βAKαLβ-1

(keeping K constant)

Marginal Product of Capital (MPK) = dQ / dK = αAKα-

1Lβ

(keeping L constant)

Value of Marginal Product of Labour (VMPL) = P x MPL

Worked out examplesWorked out examplesFor the Production function: Q = 10K0.5L0.5

whose labour input (L) is 4, selling price per unit is Rs. 5, capital (K) is fixed 9 units calculate: MPL, VMPL.

Solution:MPL = dQ / dL = βAKαLβ-1

= 10 (1/2)K0.5L0.5-1 =5(K0.5 / L0.5)= 5 x (3/2) = 7.5

VMPL = P * MPL = 5 x 7.5 = Rs. 37.5

How many Labour or Capital to How many Labour or Capital to hire?hire?

What determines how many extra labour is hired?

Suppose, MPL = 2 units;

Price of each unit sold = $20,000

Cost of hiring worker (wage rate) = $30000

Will the firm hire the worker?

Additional Revenue for the firm from hiring extra labour (Marginal Revenue Product : MRPL) = 2 x 20,000 = $40,000

Additional cost for the firm for hiring extra worker (Wage Rate -W) = $30,000

Formally,

Firms hires extra worker until, MRPL = P. MPL = w

(Note: In Perfectly competitive market, P = MR. So, MRPL = MR. MPL

Similarly, the in case of capital, capital could be employed until MRPK equals the price of the capital (r): MRPK = r

Worked out exampleWorked out exampleFor the Production function: Q = 10K0.5L0.5 whose, capital (K) is fixed 9 units, selling price per unit is Rs. 5 and wage rate is Rs. 3, calculate how many labour should be hired. Find out the profit maximizing rate of labour hired if the wage rate increased to Rs.5.

Worked out exampleWorked out exampleFor the Production function: Q = 10K0.5L0.5 whose, capital (K) is fixed 9 units, selling price per unit is Rs. 5 and wage rate is Rs. 3, calculate how many labour should be hired. Find out the profit maximizing rate of labour hired if the wage rate increased to Rs.5.

Solution:MRPL = P. MPL = P. dQ / dL = P. βAKαLβ-1

= 5.10 (1/2)K0.5L0.5-1 =25(K0.5 / L0.5)= 25 x (3/L) = 75/ L0.5

Total labour hired until:MRPL = w

75/ L0.5 = 3L = 252 = 625Therefore total no. of labour hired is 625.

If w = 5, then profit maximizing no. of labour hired is:75/ L0.5 = 5. Therefore, L = 225

Production IsoquantsProduction IsoquantsIn the long run, all inputs are

variable & isoquants are used to study production decisions◦An isoquant is a curve showing all

possible input combinations capable of producing a given level of output

◦Isoquants are downward sloping; if greater amounts of labor are used, less capital is required to produce a given output

Assume a Cobb-Douglas Production function:

Q = 100K0.5L0.5

If K = 8 and L = 2, Q = 400To obtain, Q = 400, other possible

options:K = 4 and L = 4 or,K = 2 and L = 8

Production IsoquantsProduction Isoquants

Typical Isoquants Typical Isoquants

Units of Labour

Units of Capital

0

Q1 = 400

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Q3 = 600

Q2 = 200

Marginal Rate of Technical Marginal Rate of Technical SubstitutionSubstitution

The MRTS is the slope of an isoquant & measures the rate at which the two inputs can be substituted for one another while maintaining a constant level of output

K

MRTSL

MRTS

K LThe minus sign is added to make a positivenumber since , the slope of the isoquant, isnegative

9-31

Marginal Rate of Technical Marginal Rate of Technical SubstitutionSubstitution

The MRTS can also be expressed as the ratio of two marginal products:

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K

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MP

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MPKMRTS

L MP

The Production IsocostThe Production Isocost If, No. of capital = KPer unit price of capital = rNo. of Labour = LPer unit wage rate = wTotal expenditure on capital and Labour:

C = rK + wLNow, if r= 3 and w = 2, thenCombination of 10 units of capital and 5 units

of labour will cost 40. I.e, 40 = 3(10) + 2(5).There might be others combinations which will

result the constant expenditure.

Isocost CurvesIsocost Curves

The ration w/r is the rate at which K can be traded for L.The ration r/w is the rate at which L can be traded for K.

( C ) ( w, r )

Show various combinations of inputs thatmay be purchased for given level ofexpenditure at given input prices

•

C w

K Lr r

•

C = rK + wL or

Profit MaximizationProfit MaximizationAs stated before,Firms will hire extra labour and capital until:

MRPL = w and MRPK = r

PxMPL = w and PxMPK = r

Dividing,

MPL / MPK = w / r

MPL / w = MPK / r ……………..(i)

Equation (i) is known as efficiency condition. Or

Least cost combination input.

Meaning: if a firm is maximizing in the above condition, then it is efficiently operating.

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