Question

The value of the marginal product of labor for Panera is VMPL=20-(1/2)L. Panera hires L workers in a competitive labor market and only sells coffee, also in a competitive market. The price of coffee is $2. First, sketch Panera's demand for labor. Clearly label each axis.

What market wage would induce Panera to hire only 4 workers?

Suppose someone provided Panera with an infinite supply of free labor. Panera would not use an infinite number of workers; instead it would use how many workers? (Any more and the boss would complain, “there are too many cooks in the kitchen!”)

If the price of coffee increases to $3, what is the new expression for VMPL? If you believe this change has no effect then explain why.

Relative to a market wage of $12, what effect will a binding minimum wage of $15 have on the average productivity (in cups per worker) at Panera? Increase, decrease, or no effect; or not enough info?

Answer 1

VMPL = 20-(1/2)L

Also, note that the perfectly competitive markets are defined as those markets where factors are paid according to their marginal product ( Wage W = VMPL) and by the same logic, Commodities are sold at their marginal costs (P = MC).

Panera's demand for labor is shown in the following image:

The labor demand curve is downward sloping meaning that at higher level of wages, less labour is demanded (point W1&L1) and at low level of wages, higher labour is is demanded (point W2&L2). Also we can call labour demand curve as Value of marginal product of the labour (VMPL) which is downward sloping as when large number of workers are hired, MP falls, lower wage is paid and when less number of workers are hired, MP is higher, Higher wage is paid, assuming a constant P.

(1) So now we know that wage = VMPL in competitive labour markets

So W = 20-(1/2)L

Put L=4 in the above equation.

we get, W = 20-2 = 18

So, market wage rate of 18. Panera will hire only 4 workers.

(2). No Panera would not use an infinite number of workers because as more and more workers are hired, The marginal product of successive worker falls (as shown in diagram above). So the number of workers hired depend on the wage rate. If the wage rate is 2 then, we have

W =  20-(1/2)L

Putting W = 2 we get, 2 = 20-1/2(L)

Solving this we get L = 11

So at wage rate 2 , Panera hire 11 workers. As the W goes above 2 number of workers will be less than 11 and as W goes below 2, number of workers will be more than 11.

(3). Value of Marginal product of labour is given by VMPL = P*MPL, where P = Price and MPL = Marginal product of the labour.

At P= 3, VMPL = 3*MPL. Note that VMPL is still a downward sloping curve with with a new constant 3, so effectively there is no change in either demand curve, MPL or VMPL . As long as Marginal product of labor is falling, there is no change in VMPL.

(4). Average Product of labor is defined as Total Product/Total labor.

As the market wage is reached at $15, we know from downward sloping demand curve that less number of people will be hired by Panera. As the number of worker is reduced, marginal product (MP) of a worker rises and hence, Total product (TP) will also rise (because MP is the rate of change in TP). So, because TP rises And Total labour falls, the ratio  Total Product/Total labor will also rise and Average product (AP) increase, keeping everything else same.