l. The quantity of sandwiches. Q. made in a small coffee shop is given by the equation Q = 2L3 + 121..2 where L is the number of labour-hours hired. (a) Write down the equations for the marginal product of labour (MP1) and average product of labour (APL). Give a verbal description ofMPL and APL at L = 1.5. (b) Calculate the units of labour at which MP1, and APL are maximised, and plot both graphs on the same diagram. Conrm algebraically that MPL and APL are equal when APL is at a maximum. (c) Find the turning points and point of inection for the production function. Plot the pro! duction function. (d) Use the graphs and any other results calculated in (a) and (b) to describe how productivity (Q = number of sandwiches) and the rate of change in productivity as the number of labour units employed (L) increase. 15 there any relationship between the turning points and points of inection in the graphs plotted in (b) and (c)? A utility function is given by the equation U = ZOxe'O'\" , where x is the number of glasses of wine consumed. (a) Show that this utility function has a maximum value and calculate the maximum utility. (b) Describe how marginal utility changes for glasses of wine consumed after the maximum utility is reached. Do you consider this reasonable? Give an explanation. (a) Show that the function Q = a/PC , where a and c are constants, has a constant elasticity of demand: Ed = c. (b) (i) Determine the elasticity of demand for train journeys on a given route when the demand function is Q = 1200/1712, where Q is the number of fares in thousands. (ii) If the fare increases by 5%, use elasticity to calculate the percentage change in demand. (iii) If the fare decreases by 5%, use elasticity to calculate the percentage change in demand