ECON 352X Macroeconomics for Business Useful mathematical concepts Review of useful mathematical concepts 1. Functions and graphs of functions 2. Slopes of functions 3. Elasticities 4. Growth rates 5. Compounding 1. Functions and graphs of functions A `function’ is a relationship between two or more variables. Typically, we denote a function as 𝑌 = 𝐹(𝑋, 𝑍, 𝑊, … ) 𝑋, 𝑍, 𝑊, …, are the inputs in the function F and Y is the resulting output. • Example: Suppose that we own a company producing furniture. A worker can produce 5 chairs in one day. We can then express the total number of chairs produced in one day as a function of the number of employed workers as 𝑌 =5∙𝑋 Y is the number of chairs produced (output); X is the number of workers (input). Plotting a function in a graph • Consider again the production function for chairs 𝑌 = 5 ∙ 𝑋 • We would like to plot this function in the following graph Pick three different points for 𝑋 (number of workers) and compute 𝑌 (the resulting production of chairs). Number of Chairs • 25 • 15 5 • 1 3 5 Number of Workers Point 1: 𝑋=1, 𝑌 = 5∙ 𝑋 =5 Point 2: 𝑋=3, 𝑌 = 5 ∙ 𝑋 =15 Point 3: 𝑋=5, 𝑌 = 5 ∙ 𝑋 = 25 Another example • Consider again the production function for chairs. More workers produce more chairs. However, due to limited space, the increase in production tends to decline as we add more workers. • This can be represented by the following production function 𝑌 = 𝑋 0.5 Y is the number of chairs produced; 𝑋 is the number of workers. Plotting the function 𝑌 = 𝑋 0.5 Number of Chairs • 3 • 2 1 Pick three different points for 𝑋 (number of workers) and compute 𝑌 (the resulting production of chairs). • 1 4 9 Number of Workers Point 1: 𝑋=1, 𝑌 = 𝑋 0.5 =1 Point 2: 𝑋=4, 𝑌 = 𝑋 0.5=2 Point 3: 𝑋=9, 𝑌 = 𝑋 0.5 = 3 An example with two input variables • The same company produces also tables. The production of tables requires both workers and machines. The production function is 𝑌 = 𝑋 0.5 ∙ 𝑍 0.5 Y is the number of tables; 𝑋 is the number of workers; 𝑍 is the number of machines. • To plot this function we would need a three-dimensional graph, which is cumbersome. • What we can do, instead, is to plot this function keeping fixed one of the inputs. Plotting the function 𝑌 = 𝑋 0.5 ∙ 𝑍 0.5 Number of Tables Fix the number of machines to 𝑍 =1. • 2 1 Pick three different points for 𝑋 (number of workers) and compute 𝑌 (the resulting production of tables). • 3 • 1 4 9 Number of Workers Point 1: 𝑋=1, 𝑌 = 𝑋 0.5 ∙ 1 = 1 Point 2: 𝑋=4, 𝑌 = 𝑋 0.5 ∙ 1 = 2 Point 3: 𝑋=9, 𝑌 = 𝑋 0.5 ∙ 1 = 3 Shift in the curve 𝑌 = 𝑋 0.5 ∙ 𝑍 0.5 Using the production function 𝑌 = 𝑋 0.5 ∙ 𝑍 0.5 , we now change the fixed number of machines to 𝑍 = 2.25. Number of Tables • 4.5 Pick three different points for 𝑋 (number of workers) and compute 𝑌 (the resulting production of tables). • 3 2 1.5 • 1 1 4 9 Number of Workers Point 1: 𝑋=1, 𝑌 = 𝑋 0.5 ∙ 1.5 = 1.5 Point 2: 𝑋=4, 𝑌 = 𝑋 0.5 ∙ 1.5 = 3 Point 3: 𝑋=9, 𝑌 = 𝑋 0.5 ∙ 1.5 = 4.5