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Question (1)Using the information contained in lectures, explain how you would set thesecond order conditions for an unconstrained global maximum of a function of two or morevariables, using both...

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Question (1)Using the information contained in lectures, explain how you would set thesecond order conditions for an unconstrained global maximum of a function of two or morevariables, using both algebraic and diagrammatic arguments as appropriate (15 marks).Question (2):Suppose that the function of two or more variables of question 1 above is strictly concave, andsuppose you calculate its first order condition: do you believe strict concavity of this function to bea necessary and sufficient condition for the solution to its first order condition to be a globalmaximum? Explain your answer (5 marks).No word limitation, I attached some relavant lecture notes and the cw help note
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EC3075 Coursework help ? Economic models are derived from the behaviour of individuals, firms or policy makers who are trying to maximise some measure of welfare subject to constraints on their behaviour. Optimisation is, therefore, the most important mathematical concept in economics. Optimisation encompasses three broad categories of problems: ? The first distinction is between univariate and multivariate optimisation, that is, finding extreme values of functions of one variable versus finding extreme values of functions of many variables. You are already familiar with the first and second order conditions for solving univariate optimisation problems (lectures 3-4) . ? The second distinction is between constrained and unconstrained optimisation. You should now be familiar with the first order conditions of Lagrangian multivariate constrained optimisation problems (lectures XXXXXXXXXXYou should also be familiar with first order differentiation of multivariate functions (lectures 5-6) and therefore with first order conditions of multivariate unconstrained optimisation problems (for example, profit maximisation in lectures XXXXXXXXXXThe scope of the coursework is for you to research the second order conditions for multivariate unconstrained optimisation problems; and link these to concavity and/or convexity of functions (lectures XXXXXXXXXXand 15-16). ? The final distinction is between static and dynamic optimisation, i.e. between one shot optimisation decisions and decisions in which your current choice may affect a subsequent optimisation decision. You certainly are familiar with one shot optimisation on which this module is based, and you have also certainly already studied inter-temporal consumption choice problems in other modules.Intuition and reminder in the univariate caseRemember: whether the function lies everywhere above or below its tangents at any point … or the function is cut by its tangent at an inflection point (lectures...

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Answered Same Day Dec 22, 2021

Solution

Robert answered on Dec 22 2021
124 Votes
Hello,
Before going into the solutions, I would like to make the concept
of maximum of a function clear in a simple way. For starters, I would choose a
function of one variable.
Say, “y = f(x) “. Consider the below shown figure, y is a function of x and it
increases to an extent and then decreases, maximum is defined as when the
function stops increasing i.e.”df = 0”, since this is a one variable function the
values of ‘x’ for which
df
dx
= 0 are called ‘critical points’, at which either
maximum or minimum exists and in the below shown figure it is maximum and
also the slope of the function “d(df) < 0”, and at those critical points if
d2f
dx2
0, then max. will exist at that point. Thus, equations for solving maximum
of a function of single variable are shown in the figure.
In the case of two variable function, the tangent plane is horizontal if its normal
vector points in the z direction. Hence, critical points are solutions of the
equations. Because horizontal planes have normal vector parallel to z-axis. The
two equations above must be solved simultaneously are
Solution 1)
Consider function “z = f(x, y)”, where ‘z’ is a...
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