# Cobb-Douglas and Increasing Returns to Scale

Safeway is one of many grocery stores that offers self-checkout lines and checkout lines withcashiers. Consider output (y) as the number of customers served per hour, and humans(cashiers and baggers) and registers are respectively input 1 (X1) and input 2 (m). The old-fashioned way, one register (X2=1) with one cashier and one bagger (X1=2) can serve on average100 scanned items per hour (y=100). The new way, four self-checkout registers (X2=4) with onesupervising cashier (X1=1) can serve on average 100 scans per hour (y=100), since customers arenot as fast. In either case, you need both kinds of inputs (labor and capital) to serve customers.In addition, the more the registers the shorter the lines, so that the average number ofcustomers served increases by scaling up. w Types of registerlines X1: Employees y: Customers/hourOseIf-checkout/S old-fashionm Oself-checkout/lo old-fashion 4000 Gself-checkout/Bold-fashion 4000m Sself-checkout/O old-fashion105elf-checkout/0old-fashion —_ 4000 a)

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The table above shows the results from a sample of 5 stores with different combinationsof old-fashion registers with cashier and bagger, and new self-checkout registers. In agraph with employees (X1) on the horizontal axis, and registers (X2) on the vertical axis, plot the input combinations for each store. With a line, connect the stores that belongto the same isoquant. b) With a larger sample of stores, Safeway estimated the following production function: y =X12X2. What is the marginal product of adding additional personnel (By/OX1)? What is themarginal product of adding additional registers (By/om)? c) What is the Technical Rate of Substitution between personnel and registers? d) If you double both inputs, does the number of customers served double, more thandouble or less than double?

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