Solution: Mathematics for Economists MFE511S Assignment 2, Semester 1

This document provides a comprehensive solution to Assignment 2 for the Mathematics for Economists course, MFE511S. The solution covers a range of economic and mathematical concepts, starting with differentiation of various functions and simplification of the results. It then delves into the analysis of a monopolist's demand and cost functions, including the derivation of the average cost and profit functions, and the determination of the production level for maximum profit. The solution further explores production functions, calculating marginal productivity of capital and labor, and determining the marginal rate of technical substitution (MRTS). It also addresses implicit function differentiation and explores the application of the concept of consumer surplus. Finally, it provides a proof for Young's theorem through the analysis of a defined valued function. The document offers a detailed step-by-step approach to solving each question, providing clear explanations and justifications for each step.

[MATHEMATICS FOR
ECONOMISTS]
ASSIGNMENT 2
[DATE]

Question 1
1.1
1.2
1.3
1

1.4
1.5
2

1.6
Question 2
2.1 Function
At x = 1
Now,
Now,
3

It can be seen that comes out to be positive and hence, it can be said that the function is
convex.
2.2 Demand function p = 100 - 0.01 x
Cost function C(x) = 50x +10000
2.2.1 Average cost function (AC)
2.2.2 Profit function (π)
Hence,
2.2.3 Value of x for which the profit is maximum
4

First derivative of profit function =0 (Maximum profit)
2.2.4 Maximum profit would be at x = 2500
2.2.5 Price for this level of production
p = 100 - 0.01 x
x = 2500
Question 3
3.1 Production function
Where,
5

3.1.1 Marginal productivity of capital
Let
Now,
Marginal productivity of capital
Marginal productivity of labour
Let
Now,
6

Marginal productivity of labour
3.1.2 MRTS of productions of shoes
MRTS when labour work hours L = 8 hours per day
Cost of capital K = 4
Now,
3.2 Implicit function
Differentiation w.r.t.x
7

Multiply both side with 2y
Hence,
Slope of the tangent line at (1,0)
8

Question 4
4.1 Firm spends on fixed costs = $650
Total cost function =?
At
Hence,
4.2 Demand function
Marginal revenue
4.2.1 Change in TR when q has increased from 2025 to 2500
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4.2.2 The value of consumer surplus when q = 2500
Now,
4.3 Young’s theorem through
According to Young’s theorem, Z would be a valued function defined in such a way that both
the first order partial derivatives Z(x) and Z(y) would be differentiable and then,
10

Proved!!
11