Question

Write a demand and a supply curve expressed in the form of an exponential and logarithmic functions respectively. Demonstrate graphically or otherwise, that the functions reflect all the characteristics of a demand and supply curve. [4 marks] 3. James has two job offers. Massy offers James the job with an initial annual salary of $38,800.00 along with guaranteed annual increases of $3,200.00 for the first five years. Unicomer offers him an initial annual salary of $47,600.00 along with guaranteed annual increases of $1,000.00 for the first five years.

a. Compare and contrast the two offers to determine the company that offers the highest annual salary for the fifth year of employment. Show all necessary calculations.
b. Compare and contrast the two offers to determine the company that proposes the most (total) renumeration over James' first five years of employment. Show all necessary calculations.
c. Which company should James take up his employment with if both companies offer him an annual salary of $60,000.00 in his sixth year? Why?

Answer 1

A demand curve can be modeled with an exponential function, decreasing as price increases, while a supply curve is modeled with a logarithmic function, increasing as price increases. For the job offers, Unicomer offers a higher fifth-year salary and more total remuneration over five years. If both companies offer the same salary for the sixth year, Unicomer is preferable based on the initial five-year remuneration.

Step-by-step explanation:

To write demand and supply curves using mathematical functions, we can use an exponential function for demand and a logarithmic function for supply.

A simple exponential demand curve could be expressed as D(q) = a · e^{-bq}, where a and b are positive constants, e is the base of the natural logarithm, and q represents the quantity demanded. As the price increases (which corresponds to an increase in q), the quantity demanded decreases, reflecting the law of demand.

For the supply curve, a simple logarithmic function can be S(q) = c · łn(dq + 1), where c and d are positive constants, and łn represents the natural logarithm. As the price increases, the quantity supplied increases, which aligns with the law of supply.

When graphed, the exponential demand curve slopes downwards from left to right, showing that as price decreases, quantity demanded increases. The logarithmic supply curve slopes upwards, with quantity supplied increasing as the price increases. These curves reflect the typical characteristics of demand and supply in economic theory.

Comparison of Job Offers

To compare the two job offers for James:

• For the fifth year's salary: Massy offers $38,800 + 4 · $3,200 = $51,000. Unicomer offers $47,600 + 4 · $1,000 = $51,600. Unicomer offers a higher fifth-year salary.
• Total remuneration over five years: Massy's total is a sum of an arithmetic series: Σ $38,800 + (n-1) · $3,200 = $204,800. Unicomer's total is also an arithmetic series: Σ $47,600 + (n-1) · $1,000 = $217,600. Unicomer offers more total remuneration over five years.
• Choice with equal sixth-year salary: Since both companies offer the same salary for the sixth year, James should consider the total remuneration from the first five years. Therefore, Unicomer is the better option.