Question

Exercise 9. Cobb-Douglas and increasing Returns to Scale (Self-Checkout lines) Safeway is one of many grocery stores that offers self-checkout lines and checkout lines with cashiers. Consider output (y) as the number of customers served per hour, and humans (cashiers and baggers) and registers are respectively input 1 (xi) and input 2 (x2). The old- fashioned way, one register (X2=1) with one cashier and one bagger (X12) can serve on average 100 scanned items per hour (y=100). The new way, four self-checkout registers (x2=4) with one supervising cashier (x1=1) can serve on average 100 scans per hour (y=100), since customers are not 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 of customers served increases by scaling up. x: Registers Stores Store 1 Store 2 Store 3 Store 4 Store 5 Types of register lines O self-checkout/5 old-fashion O self-checkout/10 old-fashion 6 self-checkout/3 old-fashion 5 self-checkout/O old-fashion 10 self-checkout/0 old-fashion xi: Employees 10 20 15 ol y: Customers/hour 500 manatili 4000 mm 4000 500 4000 10 a) The table above shows the results from a sample of 5 stores with different combinations of old-fashion registers with cashier and bagger, and new self-checkout registers. In a graph with employees (X) 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 belong to the same isoquant. b) With a larger sample of stores, Safeway estimated the following production function:- X;"x2, What is the marginal product of adding additional personnel (ay/dx.)? What is the marginal product of adding additional registers (dy/ox)? 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 than double or less than double?

Answer 1

1. Please find below the graph reprenting the number of employees with respect to the number of register.

The different colours of the line represent stores with the same isoquant.

Store Combination Register 5 10 15 20 25 Employees

2.

Production Function: Y = x1 * X2 Marginal product of adding additional personnel: DY/dX = 2X* X2 dy Marginal product of adding additional personnel : Jy = x1

3.

2X1 X2 Technical Rate of Substitution: Technication disminuen = 2X2/X1 *

4.

If we double both the inputs , the average number of customer served will be more than double.

This can be understood by the fact that , more the number of serving points, shorter will the lines and hence the average number of customer served will increase due to scaling.