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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@[email protected]
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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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0
50
120
150
0 1 2 3 4 5
Cones per hour
To
tal U
tilit
y in
Uti
ls
Utility Function Utility Function (Similar to Production (Similar to Production Function)Function)
1505
1404
1203
902
501
00
Total Utility
(Utility/Hr)
Cones/ Hour
1406
140
6
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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
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Marginal UtilityMarginal Utility
1505
1404
1203
902
501
00
Total Utility
(Utility/Hr)
Cones/ Hour
1406
Marginal Utility (Utility/Cone)
50
40
30
20
10
-10
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
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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?
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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?
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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
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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
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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.
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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?
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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.
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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.
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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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0
500
1,000
1,500
2,000
2,500
3,000
0 1 2 3 4 5
No. of workers
Qu
anti
ty o
f o
utp
ut
Recall Example:Recall Example: Production Function Production Function
30005
28004
24003
18002
10001
00
Q (bushels of wheat)
L(no. of
workers)
Relationship between input and output
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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
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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
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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
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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)
0 0
1 52
2 112
3 170
4 220
5 258
6 286
7 304
8 314
9 318
10 314
--
55
51.6
52
56
56.7
47.7
43.4
39.3
35.3
31.4
--
50
38
52
60
58
28
18
104
-4
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Total, Average & Marginal Total, Average & Marginal ProductsProducts
L Q AP MP
0 0
1 52
2 112
3 170
4 220
5 258
6 286
7 304
8 314
9 318
10
314
--
55
51.6
52
56
56.7
47.7
43.4
39.3
35.3
31.4
--
50
38
52
60
58
28
18
104
-4
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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
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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
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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
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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
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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.
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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
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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
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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
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Typical Isoquants Typical Isoquants
Units of Labour
Units of Capital
0
Q1 = 400
C
B
A
D
E
8
2
4
2
84
Q3 = 600
Q2 = 200
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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
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9-31
Marginal Rate of Technical Marginal Rate of Technical SubstitutionSubstitution
The MRTS can also be expressed as the ratio of two marginal products:
L
K
MPMRTS
MP
L
K
MPKMRTS
L MP
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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.
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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
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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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Thank youThank you