Utility functions are discussed in detail in appendix

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(Utility functions are discussed in detail in Appendix 3 of your text.) Economists also use the concept of a production function , which relates combined quantities of inputs to the quantity of output each combination produces. If there are just two types of input, labour-days ( L ) and machine-days ( K , for “capital”), then we can treat the output Q (in tonnes, say) as a function of the inputs of L and K : Q = f ( L, K ) (M.1.10) Equation M.1.10 is a general form of the production function; Appendix 9 in the text dis- cusses a number of speciFc production functions. (See also Module 8.) Even demand and supply functions, in their general form, are functions of several variables. We normally assume, for instance, that an individual’s demand for a good X ( X D ) is determined by (among other factors) its own price P X , the price of other goods P Y , and the individual’s income ( M ); that is, X D = f ( P X , P Y , M ). (M.1.11) It is only when we assign speciFc values to all of the other determinants of the quanti- ty demanded of a good except its own price and make the ceteris paribus (“other deter- minants remain unchanged”) assumption that we can draw the conventional demand curve, with its quantity demanded as a function of its own price alone. We shall see how this works in the Exercises for this Module. 2. Exercises 1. ±or the following equations, state whether y is or is not a function of x ; whether the equation is linear or nonlinear; whether the equation is strictly increasing, strictly decreasing, constant, or non-monotonic; and whether an inverse function exists. (a) xy = 20 (b) xy = – 20 (c) x = 4 – 2 y (d) y = 4 – 2 x (e) y = 3 x 2 + 4 x – 4 (f) y = 2 x (g) x= 2 y (h) x = 3( y 2 – 6) (i) 6 x + 12 y = 9 (j) x = 7 (k) y = 14 2. Plot the following on the same graph, and label the vertical and horizontal intercepts and the slope in each case:
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3. Plot the following 6 functions on the same graph: 4. Plug in values for x , calculate the corresponding values for y , and graph the follow- ing two functions: (a) y = x 3 and (b) y = x 1/3 . Which of these two functions has an inverse function? 5. You are given the following demand function for good X , where X is measured in kilograms, P X is in $/kg, the price of good Y ( P Y ) is in $/unit, and income M is in thousands of dollars (so that if income is $10,000, M = 10): X D = 20 – P X + 0.5 P Y + 5 M.
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