5.3 Profit Maximization in Factor Markets and Cost Minimization

5.3 Profit Max in Factor Markets

5.3 Profit Maximizing Behavior in Perfectly Competitive Factor Markets, Least-Cost Hiring Rule

  • Most factor markets, including labor, in the modern U.S. economy are perfectly competitive. This means buyers and sellers are generally wage-takers due to their small size relative to the market.
  • A local bakery, for example, can hire as many bakers as needed at the market equilibrium wage, such as $23, determined by the broader market.
Firm's Perspective
  • Firms take the market equilibrium wage as given, resulting in a perfectly elastic supply of labor to the firm.
  • This is similar to firms and customers in a perfectly competitive product market being price takers.
Market for Cookie Bakers

  • \begin{itemize}
    \item Market
    \item Perf. Comp.
    \item Prodect
    \item S
    \item P
    \item Wagis
    \item P.C. Factor Mc
    \item D=
    \item Q₁
    \end{itemize}

  • Supply of Labor to Firm Curve

  • Supply of bakers curve

  • Demand for bakers Curve

  • Firm's demand for bakers and D=MRP Curve

Cost Minimization
  • Firms aim to minimize costs while maximizing profits (MR = MC) in the product market.
  • A similar rule applies in factor markets: VMPL=MRCVMPL = MRC, where MRC is the Marginal Resource Cost.
  • For labor, this often simplifies to VMPL=wageVMPL = wage or MRP=wageMRP = wage.
Cost Minimization: the Least Cost Hiring Rule
  • Firms choose input combinations, balancing labor and capital.
  • Options range from capital-intensive (more machines, less labor) to labor-intensive (more workers, less machinery).
  • Optimal combination sought which acheives cost minimization.
Consumer Utility Maximizing Rule

  • Recall rule from Unit 1.6: MUxPx=MUyPy\frac{MUx}{Px} = \frac{MUy}{Py}. If either of those ratios is larger than the other, given some consumption level, then the consumer would be better off consuming more of the item with that larger ratio.

  • Analogy:

    • A grocery store deciding between self-checkout lanes (capital) and cashiers (labor).
    • The options are given in the table. Evaluate the best combination.
    • Option 1:
      • Capital (self-checkout stations) 20. Rental rate = $1000/month
      • Labor (cashiers) 4 Wage rate $1600/month
      • 20 \cdot 1000 + 4 \cdot 1600 = 26,400
    • Option 2:
      • Capital (self-checkout stations) 10. Rental rate = $1000/month
      • Labor (cashiers) 10 Wage rate $1600/month
      • 10 \cdot 1000 + 10 \cdot 1600 = 26.800
Marginal Analysis Approach

  • Comparing the cost-effectiveness of inputs: marginal product per dollar.

  • Example: Expanding production by:

    • Hiring a worker: $100\$100/day, 20 more units (MPL), 20$100=0.2\frac{20}{\$100} = 0.2 units per dollar.
    • Renting a machine: $300\$300/day, 50 more units (MPK), 50$300=0.17\frac{50}{\$300} = 0.17 units per dollar.

  • Decision: Choose the option with the higher ratio (in this case, the worker).

Cost-Minimization Rule
  • At the optimal combination, the marginal product per dollar should be equal:
    MPKPK=MPLPL\frac{MPK}{PK} = \frac{MPL}{PL}
Applications of the Cost-Minimization Rule
  • Evaluate costs at the total level of production.
  • Example:
    • Firm makes 2000 units.
    • Most recent worker: $32\$32/hour, 4 products/hour. 4$32=0.125\frac{4}{\$32} = 0.125
    • Most recent capital: $60\$60/hour, 6 products/hour. 6$60=0.1\frac{6}{\$60} = 0.1
    • Analysis: Not producing efficiently since 0.125≠0.10.125 \neq 0.1. Should shift towards less capital and more labor.
    • If capital produced 10 units per hour: 10$60=0.1667\frac{10}{\$60} = 0.1667 Then capital would be a better choice.
  • Start from zero production and incrementally add labor/capital based on budget/goals.
Example with Given values:

  • Cost per unit of labor: $4\$4

  • Cost per unit of capital: $2\$2

  • Total budget: $14\$14

  • Calculating marginal products (MPL, MPK) and ratios (MPL/PL, MPK/PK) to determine optimal hiring strategy. Choose the factor with the best return for each dollar spent.

  • Ultimately, choose units of labor and S units of capital, producing a total of 36 units of output with our $14 budget.

General Optimal Combination

  • In general, our optimal combination of labor and capital (and land, etc) will have: MPKPK=MPLPL\frac{MPK}{PK} = \frac{MPL}{PL}

  • If one fraction is larger than the other, then we would be better off using MORE of that factor and less of the other. Note that discrete examples like this will often end up with the ratio being unequal, but still at it's best solution.

Example

  • Producer of coffee: $5 per labor hour, $10 per capital rental hour. Marginal products given.

  • Requirement: Produce 360 coffees at least cost.

  • Calculate MPLPL\frac{MPL}{PL} and MPKPK\frac{MPK}{PK} to determine optimal combination of labor and capital hours.

  • The producer should hire units of labor and units of capital for a total cost of.

  • Note that we can use this rule to either maximize production with a given budget (first example) or minimize costs to fit a production requirement (second example).