Suppose social well-being is determined by the function u(x1, x2, . . . , xN) where xi

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Question:

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) 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 services, V = u(x, y). Let x be urban green spaces and y national parks.

b) 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) Let y = f(x) be this level curve. Using implicit differentiation, find f 0 (x). What is its sign? Explain with economic intuition.

d) 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) 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) 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) 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.