Assignment 1 (50 points) ECN 230 Prof. Rowan Shi Winter 2021 This assignment is take home. You may use your textbook, course material, and personal notes. Do not use the internet or work with anyone...

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Assignment 1 (50 points) ECN 230 Prof. Rowan Shi Winter 2021 This assignment is take home. You may use your textbook, course material, and personal notes. Do not use the internet or work with anyone else. 1. (20 points) Suppose social well-being is determined by the function u(x1, x2, . . . , xN) where xi is the quantity of public service i provided by the government. a) (3 points) How many first order partial derivatives does u have? Pick one of these and explain in words what it represents, in the context of social well-being and public service provision. For the rest of the problem, consider the simplified version with two ser- vices, V = u(x, y). Let x be urban green spaces and y national parks. b) (2 points) With the current level of urban green spaces and national parks provided, social well-being is W. Describe in words what the level curve u(x, y) = W represents, in the context of urban green spaces and national parks. c) (4 points) Let y = f (x) be this level curve. Using implicit differenti- ation, find f ′(x). What is its sign? Explain with economic intuition. d) (1 point) Suppose each urban green space costs p to maintain and each national park q. Write the cost function c(x, y) of maintaining x urban green spaces and y national parks. 1 Parks Canada has hired you to re-evaluate how many urban green spaces and national parks they maintain. Your goal is to streamline costs while ensuring that social well-being is kept at W, the same level as it is currently. e) (3 points) Using what you have found so far, write the first order condition of this cost minimisation problem in terms of x, p, q, and functions only. f) (2 points) You discover that, with the current number of urban green spaces and national parks, the citisen marginal rate of substitution exceeds p/q. Would you change the number of urban green spaces and national parks from their current level? g) (5 points) Let social welfare take the functional form u(x, y) = a ln ( x a ) + (1− a) ln ( y 1− a ) 0 < a="">< 1 and find the minimum cost of providing a public welfare of w = 0 in terms of p, q, and a. 2. (15 points) you sell coconuts on the beach. total sales depend on how many people buy coconuts p as well as how many coconuts each person buys b. for each coconut you sell, you make $5, so your total sales are f (p, b) = 5pb. moreover, let p = g(t, r) b = h(t, r) describe how p and b in turn are determined by the temperature t and chance of rain r. a) (3 points) let s represent your total sales. describe in words what ∂s/∂t represents, in the context of temperature, chance of rain, num- ber of buyers, coconuts per buyer, and coconut sales. b) (3 points) do the same for ∂s/∂r. c) (4 points) apply the chain rule to find ∂s/∂r. d) (5 points) what do you expect the sign of ∂s/∂r to be? explain why, discussing the contribution of each derivative or partial derivative. 2 3. (15 points) on a cruise ship, there are massage therapists and chefs. to give one hour of massage, massage therapists need to receive a 15 minute massage and eat a plate of food. to cook 10 plates of food, chefs need to receive a half hour massage and eat a plate of food. a) (4 points) overall, suppose the cruise ship provides its guests with 100 hours of massage and a huge buffet with 500 plates of food. the cruise accountant wants to know the total hours of massage given and plates of food cooked on the ship. express this system with equa- tions, specifying what each variable represents. b) (4 points) what would you tell the cruise accountant? c) (2 points) how many plates of food are eaten by massage therapists? how many hours of massage do chefs get? now suppose massage therapists no longer eat one plate of food per hour of massage they give. instead, if massage therapists give m hours total of massage, they need to eat 27 2 m− 1150 7 √ m plates of food total. they also still need to receive a 15 minute massage each hour of massage given. the requirements of chefs remain the same. d) (5 points) rewrite the system. now how many total plates of food are eaten by massage therapists? 3 multivariable functions - ch 11.1–7, excl 11.4 multivariable functions ch 11.1–7, excl 11.4 prof. rowan shi week 1 outline functions with two arguments visualising surface plots level curves partial derivatives with many arguments 2/22 outline functions with two arguments visualising surface plots level curves partial derivatives with many arguments 3/22 functions with one argument x f(x) f • reminder: a function is a rule that assigns a value to an argument • the pool of x values for which f applies is the domain • all the possible outputs given by f is the range • example: a firm’s cost function c(q) what is this function’s argument? value? domain? range? 4/22� ch 4 functions with one argument x f(x) f • reminder: a function is a rule that assigns a value to an argument • the pool of x values for which f applies is the domain • all the possible outputs given by f is the range • example: a firm’s cost function c(q) what is this function’s argument? value? domain? range? 4/22� ch 4 functions with two arguments x y f(x, y) f • a function with two arguments takes a pair (x, y) and assigns value f(x, y) • now: the rule needs two items from the domain to get a value • sometimes: • arguments are called independent or exogenous variables • value is called dependent or endogenous variable 5/22� ch 11.1 functions with two arguments • example: the demand for milk x = am 2.08 p1.5 (example 11.1.2) where a is a constant, m is income, and p is price of milk • the dependent variable: x, how much milk is demanded • independent variables: m and p • intuition: amount of milk demanded depends on both family’s income and milk’s price 6/22� ch 11.1 outline functions with two arguments visualising surface plots level curves partial derivatives with many arguments 7/22 graphs • with single-argument functions f(x), we graph them to see. . . • if f(x) increases or decreases with x • slopes and rates of change • intercepts • and more. . . • would be useful for two-argument functions, like our example • does demand increase in income? price? • what’s the maximum demand? • but: how do graph a two-argument function like f(x, y)? 8/22� ch 11.3 surface plots 1 1.5 2 2 2.5 30 5 10 p m • plotted: our milk demand function • goal: for each price p and income level m, visualise milk demand • the domain is 2d – imagine it like the horizontal surface of a table 9/22� ch 11.3 surface plots 1 1.5 2 2 2.5 30 5 10 p m • let’s represent milk demand with price p = 1 and income m = 2.5 • locate the point in the domain • represent demand as the height “above” the domain • demand is around 7 9/22� ch 11.3 surface plots 1 1.5 2 2 2.5 30 5 10 p m • let’s represent milk demand with price p = 1 and income m = 2.5 • locate the point in the domain • represent demand as the height “above” the domain • demand is around 7 9/22� ch 11.3 surface plots 1 1.5 2 2 2.5 30 5 10 p m • let’s represent milk demand with price p = 1 and income m = 2.5 • locate the point in the domain • represent demand as the height “above” the domain • demand is around 7 9/22� ch 11.3 surface plots 1 1.5 2 2 2.5 30 5 10 p m • let’s try with p = 2, m = 2.5 • demand is around 2.5 • overall: • demand is the lowest at p = 2, m = 2 • increases as p falls or m rises does that make intuitive sense? 9/22� ch 11.3 surface plots 1 1.5 2 2 2.5 30 5 10 p m • let’s try with p = 2, m = 2.5 • demand is around 2.5 • overall: • demand is the lowest at p = 2, m = 2 • increases as p falls or m rises does that make intuitive sense? 9/22� ch 11.3 surface plots 1 1.5 2 2 2.5 30 5 10 p m • let’s try with p = 2, m = 2.5 • demand is around 2.5 • overall: • demand is the lowest at p = 2, m = 2 • increases as p falls or m rises does that make intuitive sense? 9/22� ch 11.3 surface plots 1 1.5 2 2 2.5 30 5 10 p m • the surface plot is like a piece of fabric • as you move around in the domain, the fabric curves around, representing how the function changes • hard for non-machines to draw (e.g. me or you) 9/22� ch 11.3 level curves • let’s think about a simplified version of our milk demand example: x = m 2 p (numbers changed to make things easier) • we can ask: for what income and price pairs (m,p) is the demand for milk 1? • well, m = 1 and p = 1 gives us x = 1. . . so does m = 2 and p = 4 10/22� ch 11.3 level curves p m1 1 2 4 • we’ve plotted the two points that we’ve found • continuing on, we can plot all (m,p) pairs for which we get demand x = 1 • we can repeat for x = 2 • and so on 11/22� ch 11.3 level curves p m x = 1 • we’ve plotted the two points that we’ve found • continuing on, we can plot all (m,p) pairs for which we get demand x = 1 • we can repeat for x = 2 • and so on 11/22� ch 11.3 level curves p m x = 1 x = 2 • we’ve plotted the two points that we’ve found • continuing on, we can plot all (m,p) pairs for which we get demand x = 1 • we can repeat for x = 2 • and so on 11/22� ch 11.3 level curves p m x = 1 x = 2 x = 3 x = 4 • we’ve plotted the two points that we’ve 1="" and="" find="" the="" minimum="" cost="" of="" providing="" a="" public="" welfare="" of="" w="0" in="" terms="" of="" p,="" q,="" and="" a.="" 2.="" (15="" points)="" you="" sell="" coconuts="" on="" the="" beach.="" total="" sales="" depend="" on="" how="" many="" people="" buy="" coconuts="" p="" as="" well="" as="" how="" many="" coconuts="" each="" person="" buys="" b.="" for="" each="" coconut="" you="" sell,="" you="" make="" $5,="" so="" your="" total="" sales="" are="" f="" (p,="" b)="5pb." moreover,="" let="" p="g(t," r)="" b="h(t," r)="" describe="" how="" p="" and="" b="" in="" turn="" are="" determined="" by="" the="" temperature="" t="" and="" chance="" of="" rain="" r.="" a)="" (3="" points)="" let="" s="" represent="" your="" total="" sales.="" describe="" in="" words="" what="" ∂s/∂t="" represents,="" in="" the="" context="" of="" temperature,="" chance="" of="" rain,="" num-="" ber="" of="" buyers,="" coconuts="" per="" buyer,="" and="" coconut="" sales.="" b)="" (3="" points)="" do="" the="" same="" for="" ∂s/∂r.="" c)="" (4="" points)="" apply="" the="" chain="" rule="" to="" find="" ∂s/∂r.="" d)="" (5="" points)="" what="" do="" you="" expect="" the="" sign="" of="" ∂s/∂r="" to="" be?="" explain="" why,="" discussing="" the="" contribution="" of="" each="" derivative="" or="" partial="" derivative.="" 2="" 3.="" (15="" points)="" on="" a="" cruise="" ship,="" there="" are="" massage="" therapists="" and="" chefs.="" to="" give="" one="" hour="" of="" massage,="" massage="" therapists="" need="" to="" receive="" a="" 15="" minute="" massage="" and="" eat="" a="" plate="" of="" food.="" to="" cook="" 10="" plates="" of="" food,="" chefs="" need="" to="" receive="" a="" half="" hour="" massage="" and="" eat="" a="" plate="" of="" food.="" a)="" (4="" points)="" overall,="" suppose="" the="" cruise="" ship="" provides="" its="" guests="" with="" 100="" hours="" of="" massage="" and="" a="" huge="" buffet="" with="" 500="" plates="" of="" food.="" the="" cruise="" accountant="" wants="" to="" know="" the="" total="" hours="" of="" massage="" given="" and="" plates="" of="" food="" cooked="" on="" the="" ship.="" express="" this="" system="" with="" equa-="" tions,="" specifying="" what="" each="" variable="" represents.="" b)="" (4="" points)="" what="" would="" you="" tell="" the="" cruise="" accountant?="" c)="" (2="" points)="" how="" many="" plates="" of="" food="" are="" eaten="" by="" massage="" therapists?="" how="" many="" hours="" of="" massage="" do="" chefs="" get?="" now="" suppose="" massage="" therapists="" no="" longer="" eat="" one="" plate="" of="" food="" per="" hour="" of="" massage="" they="" give.="" instead,="" if="" massage="" therapists="" give="" m="" hours="" total="" of="" massage,="" they="" need="" to="" eat="" 27="" 2="" m−="" 1150="" 7="" √="" m="" plates="" of="" food="" total.="" they="" also="" still="" need="" to="" receive="" a="" 15="" minute="" massage="" each="" hour="" of="" massage="" given.="" the="" requirements="" of="" chefs="" remain="" the="" same.="" d)="" (5="" points)="" rewrite="" the="" system.="" now="" how="" many="" total="" plates="" of="" food="" are="" eaten="" by="" massage="" therapists?="" 3="" multivariable="" functions="" -="" ch="" 11.1–7,="" excl="" 11.4="" multivariable="" functions="" ch="" 11.1–7,="" excl="" 11.4="" prof.="" rowan="" shi="" week="" 1="" outline="" functions="" with="" two="" arguments="" visualising="" surface="" plots="" level="" curves="" partial="" derivatives="" with="" many="" arguments="" 2/22="" outline="" functions="" with="" two="" arguments="" visualising="" surface="" plots="" level="" curves="" partial="" derivatives="" with="" many="" arguments="" 3/22="" functions="" with="" one="" argument="" x="" f(x)="" f="" •="" reminder:="" a="" function="" is="" a="" rule="" that="" assigns="" a="" value="" to="" an="" argument="" •="" the="" pool="" of="" x="" values="" for="" which="" f="" applies="" is="" the="" domain="" •="" all="" the="" possible="" outputs="" given="" by="" f="" is="" the="" range="" •="" example:="" a="" firm’s="" cost="" function="" c(q)="" what="" is="" this="" function’s="" argument?="" value?="" domain?="" range?="" 4/22�="" ch="" 4="" functions="" with="" one="" argument="" x="" f(x)="" f="" •="" reminder:="" a="" function="" is="" a="" rule="" that="" assigns="" a="" value="" to="" an="" argument="" •="" the="" pool="" of="" x="" values="" for="" which="" f="" applies="" is="" the="" domain="" •="" all="" the="" possible="" outputs="" given="" by="" f="" is="" the="" range="" •="" example:="" a="" firm’s="" cost="" function="" c(q)="" what="" is="" this="" function’s="" argument?="" value?="" domain?="" range?="" 4/22�="" ch="" 4="" functions="" with="" two="" arguments="" x="" y="" f(x,="" y)="" f="" •="" a="" function="" with="" two="" arguments="" takes="" a="" pair="" (x,="" y)="" and="" assigns="" value="" f(x,="" y)="" •="" now:="" the="" rule="" needs="" two="" items="" from="" the="" domain="" to="" get="" a="" value="" •="" sometimes:="" •="" arguments="" are="" called="" independent="" or="" exogenous="" variables="" •="" value="" is="" called="" dependent="" or="" endogenous="" variable="" 5/22�="" ch="" 11.1="" functions="" with="" two="" arguments="" •="" example:="" the="" demand="" for="" milk="" x="Am" 2.08="" p1.5="" (example="" 11.1.2)="" where="" a="" is="" a="" constant,="" m="" is="" income,="" and="" p="" is="" price="" of="" milk="" •="" the="" dependent="" variable:="" x,="" how="" much="" milk="" is="" demanded="" •="" independent="" variables:="" m="" and="" p="" •="" intuition:="" amount="" of="" milk="" demanded="" depends="" on="" both="" family’s="" income="" and="" milk’s="" price="" 6/22�="" ch="" 11.1="" outline="" functions="" with="" two="" arguments="" visualising="" surface="" plots="" level="" curves="" partial="" derivatives="" with="" many="" arguments="" 7/22="" graphs="" •="" with="" single-argument="" functions="" f(x),="" we="" graph="" them="" to="" see.="" .="" .="" •="" if="" f(x)="" increases="" or="" decreases="" with="" x="" •="" slopes="" and="" rates="" of="" change="" •="" intercepts="" •="" and="" more.="" .="" .="" •="" would="" be="" useful="" for="" two-argument="" functions,="" like="" our="" example="" •="" does="" demand="" increase="" in="" income?="" price?="" •="" what’s="" the="" maximum="" demand?="" •="" but:="" how="" do="" graph="" a="" two-argument="" function="" like="" f(x,="" y)?="" 8/22�="" ch="" 11.3="" surface="" plots="" 1="" 1.5="" 2="" 2="" 2.5="" 30="" 5="" 10="" p="" m="" •="" plotted:="" our="" milk="" demand="" function="" •="" goal:="" for="" each="" price="" p="" and="" income="" level="" m,="" visualise="" milk="" demand="" •="" the="" domain="" is="" 2d="" –="" imagine="" it="" like="" the="" horizontal="" surface="" of="" a="" table="" 9/22�="" ch="" 11.3="" surface="" plots="" 1="" 1.5="" 2="" 2="" 2.5="" 30="" 5="" 10="" p="" m="" •="" let’s="" represent="" milk="" demand="" with="" price="" p="1" and="" income="" m="2.5" •="" locate="" the="" point="" in="" the="" domain="" •="" represent="" demand="" as="" the="" height="" “above”="" the="" domain="" •="" demand="" is="" around="" 7="" 9/22�="" ch="" 11.3="" surface="" plots="" 1="" 1.5="" 2="" 2="" 2.5="" 30="" 5="" 10="" p="" m="" •="" let’s="" represent="" milk="" demand="" with="" price="" p="1" and="" income="" m="2.5" •="" locate="" the="" point="" in="" the="" domain="" •="" represent="" demand="" as="" the="" height="" “above”="" the="" domain="" •="" demand="" is="" around="" 7="" 9/22�="" ch="" 11.3="" surface="" plots="" 1="" 1.5="" 2="" 2="" 2.5="" 30="" 5="" 10="" p="" m="" •="" let’s="" represent="" milk="" demand="" with="" price="" p="1" and="" income="" m="2.5" •="" locate="" the="" point="" in="" the="" domain="" •="" represent="" demand="" as="" the="" height="" “above”="" the="" domain="" •="" demand="" is="" around="" 7="" 9/22�="" ch="" 11.3="" surface="" plots="" 1="" 1.5="" 2="" 2="" 2.5="" 30="" 5="" 10="" p="" m="" •="" let’s="" try="" with="" p="2," m="2.5" •="" demand="" is="" around="" 2.5="" •="" overall:="" •="" demand="" is="" the="" lowest="" at="" p="2," m="2" •="" increases="" as="" p="" falls="" or="" m="" rises="" does="" that="" make="" intuitive="" sense?="" 9/22�="" ch="" 11.3="" surface="" plots="" 1="" 1.5="" 2="" 2="" 2.5="" 30="" 5="" 10="" p="" m="" •="" let’s="" try="" with="" p="2," m="2.5" •="" demand="" is="" around="" 2.5="" •="" overall:="" •="" demand="" is="" the="" lowest="" at="" p="2," m="2" •="" increases="" as="" p="" falls="" or="" m="" rises="" does="" that="" make="" intuitive="" sense?="" 9/22�="" ch="" 11.3="" surface="" plots="" 1="" 1.5="" 2="" 2="" 2.5="" 30="" 5="" 10="" p="" m="" •="" let’s="" try="" with="" p="2," m="2.5" •="" demand="" is="" around="" 2.5="" •="" overall:="" •="" demand="" is="" the="" lowest="" at="" p="2," m="2" •="" increases="" as="" p="" falls="" or="" m="" rises="" does="" that="" make="" intuitive="" sense?="" 9/22�="" ch="" 11.3="" surface="" plots="" 1="" 1.5="" 2="" 2="" 2.5="" 30="" 5="" 10="" p="" m="" •="" the="" surface="" plot="" is="" like="" a="" piece="" of="" fabric="" •="" as="" you="" move="" around="" in="" the="" domain,="" the="" fabric="" curves="" around,="" representing="" how="" the="" function="" changes="" •="" hard="" for="" non-machines="" to="" draw="" (e.g.="" me="" or="" you)="" 9/22�="" ch="" 11.3="" level="" curves="" •="" let’s="" think="" about="" a="" simplified="" version="" of="" our="" milk="" demand="" example:="" x="m" 2="" p="" (numbers="" changed="" to="" make="" things="" easier)="" •="" we="" can="" ask:="" for="" what="" income="" and="" price="" pairs="" (m,p)="" is="" the="" demand="" for="" milk="" 1?="" •="" well,="" m="1" and="" p="1" gives="" us="" x="1." .="" .="" so="" does="" m="2" and="" p="4" 10/22�="" ch="" 11.3="" level="" curves="" p="" m1="" 1="" 2="" 4="" •="" we’ve="" plotted="" the="" two="" points="" that="" we’ve="" found="" •="" continuing="" on,="" we="" can="" plot="" all="" (m,p)="" pairs="" for="" which="" we="" get="" demand="" x="1" •="" we="" can="" repeat="" for="" x="2" •="" and="" so="" on="" 11/22�="" ch="" 11.3="" level="" curves="" p="" m="" x="1" •="" we’ve="" plotted="" the="" two="" points="" that="" we’ve="" found="" •="" continuing="" on,="" we="" can="" plot="" all="" (m,p)="" pairs="" for="" which="" we="" get="" demand="" x="1" •="" we="" can="" repeat="" for="" x="2" •="" and="" so="" on="" 11/22�="" ch="" 11.3="" level="" curves="" p="" m="" x="1" x="2" •="" we’ve="" plotted="" the="" two="" points="" that="" we’ve="" found="" •="" continuing="" on,="" we="" can="" plot="" all="" (m,p)="" pairs="" for="" which="" we="" get="" demand="" x="1" •="" we="" can="" repeat="" for="" x="2" •="" and="" so="" on="" 11/22�="" ch="" 11.3="" level="" curves="" p="" m="" x="1" x="2" x="3" x="4" •="" we’ve="" plotted="" the="" two="" points="" that="">
Answered 1 days AfterFeb 08, 2021

Answer To: Assignment 1 (50 points) ECN 230 Prof. Rowan Shi Winter 2021 This assignment is take home. You may...

Komalavalli answered on Feb 10 2021
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Assignment 1 (50 points)
ECN 230
Prof. Rowan Shi Winter 2021
1)
a)
Given
    u(x1, x2, . . . , xN)
There are N first order partial derivative u have. Let us choo
se partial derivative for x1they is ∂u/∂x1, this tells us how social well being changes with a change in public service provision (x1).
b)
u(x, y) = W
Level curve describes an increase in urban green spaces and national parks whether increases or decreases the social well being.
c)
Implicit differentiation
dy/dx = f’(x)
The sign of implicit differentiation is positive .This tells us that the increase in national parks will increases the urban green spaces which in turn increases the social well being.
d)
Cost function to maintaining x urban green spaces and y national parks
c (x, y) = px + qy
e)
Cost minimization problem
Minimize cost c (x, y) = px + qy
With respect to u(x, y) = W
Lagrangian equation
L = px + qy + λ( W – u(x,y) )
First order condition of cost minimization
∂L/∂x = p – λ ∂u /∂x = 0     ------------------------ (1)
∂L/∂y = q – λ ∂u /∂y = 0    ------------------------ (2)
∂L/∂ λ = W - u(x,y) = 0    ------------------------(3)
From (1) we get ∂u /∂x
∂u /∂x = p/ λ
From (2) we get ∂u /∂y
∂u /∂y = q/ λ
MRSxy = (∂u /∂x)/( ∂u /∂y)
MRSxy = p/ λ* λ/q
MRSxy = p/q
Therefore marginal rate of substitution urban spaces for national parks is p/q
f)
If the cisten marginal rate of substitution exceeds p/q , then I would change the number of urban green spaces and national parks from their current level.
g)
Cost minimization problem
Minimize cost c (x, y) = px + qy
With respect to u(x, y) = W = a ln (x/a)+(1-a)ln(y/1-a)
Lagrangian equation
L = px + qy + λ(...
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