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ECON5101 price vector
Creation date：2024-05-30 17:17:48
Belonging category：经济Economics
price vector

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ECON5101

Motivation
Elements of a decision problem:
What is feasible?
What is desirable?
We can now put them together to talk about choice. We will:
1 solve the consumer’s optimisation problem to derive their individual demand function. (now)
2 look at some comparative statics: how consumer demand changes as exogenous variables change.
(next)

Assumptions about Individuals and Preferences
Behavioural postulate: An individual always chooses their most preferred alternative among available
choices.
In a market environment where the consumption space consists of two perfectly divisible goods
(X = R2+), and the price vector p = (p1, p2) is given,
their feasible set is B(p,m) = {x ∈ R2+ : p1x1 + p2x2 ≤ m},
their preferences are described by the rational preference relation ⪰,
they will choose a bundle that is both affordable and weakly preferred to all other affordable
bundles.
Assume that preferences are continuous.

Rational Choice: Graphically
For now, also assume strictly convex and
strongly monotone preferences.
We will choose the bundle that lies on the
highest indifference curve we can ”reach.”

Rational Choice: Graphically
At the optimal bundle (x∗1 , x
∗
2 ),
the indifference curve it belongs to is tangent
to the budget constraint,
the budget is fully exhausted.
This rests on the assumptions we made on the
previous slide, and won’t always be true otherwise.

Rational Choice
At the optimal bundle (x∗1 , x
∗
2 ),
the indifference curve it belongs to is tangent to the budget constraint,
the budget is fully exhausted.
We will now see how to derive optimal choice more formally, as the solution to a constrained
optimisation problem. We will:
discuss the assumptions we need for the two above conditions to characterise the optimum.
discuss how to derive optimal choice in those circumstances.
do some examples, including those in which our assumptions don’t hold.

1 Constrained Utility Maximisation
Assumptions
Individual Demand
Cobb-Douglas Utility Functions
2 Optimisation with Other Common Utility Functions
Violation of Assumptions
Quasilinear Utility Functions
Perfect Substitutes
Perfect Complements

Constrained Utility Maximisation
We need a systematic approach to the consumer’s optimisation problem. Instead of the rational
preference relation ⪰, we will work with a utility function that represents it.
Utility Maximisation Problem (UMP)
Given price p = (p1, p2) with p1, p2 > 0, a consumer with income m > 0 solves the following problem:
max
x1,x2≥0
u(x1, x2) subject to p1x1 + p2x2 ≤ m.
Assumptions: The utility function u : R2+ → R is continuous, differentiable, strictly increasing, and
strictly concave.
Notation: For convenience, I will occasionally write (x1, x2) using the vector boldface, x.

Assumptions
u is continuous
Ensures the existence of a solution to the UMP.
Since we assumed our preferences are rational and continuous, the Utility Representation Theorem
tells us that we will be able to represent them with a continuous utility function.
u is strictly increasing
Ensures the consumer spends all their income (Walras’ Law), i.e., p1x
∗
1 + p2x
∗
2 = m.
This is guaranteed when preferences are strongly monotone.
u is strictly concave
Ensures a unique solution the UMP.
This is guaranteed when preferences are strictly convex.

Assumptions
Claim: If u is strictly concave, then there is a unique solution to the UMP.
Proof (not assessed):

Assumptions
Claim: If preferences are strictly convex, then a utility function u which represents these preferences is
strictly concave..
Proof (not assessed):

Individual Demand
Definition 1: Demand
For each price-income combination, the consumer solves the UMP to obtain an optimal consumption
bundle. The function that relates the optimal quantity of good i to prices and income is called the
demand for good i , xi (p,m).

Deriving Individual Demand
The Lagrangian for the UMP, given prices p and income m > 0 is
L = u(x)− λ(p1x1 + p2x2 −m).
Our assumptions that u is strictly concave, strictly increasing, and continuously differentiable ensure
that first order conditions are necessary and sufficient for constrained optima.
Case category
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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