Dynamic Strategic Planning - Problem Sets

1. Production Functions and Optimization

Problem 1.1

Take the production function for the generation of electricity as a function of the amount of fuel (F), the amount of labor (L) and the size of the facility (S):

Q = F^{0.4} L^{0.3} S^{0.2}

Additionally, assume that the input cost function is:

C = F + 0.5 L^{1.5} + 0.25 S ^{2}

- Calculate the marginal rate of substitution of fuel for amount of labor at (F=2, L=3, S=4).
- State the conditions for optimal design as they apply to this case.
- Define in words the expansion path. Find the equation of the expansion path for the generation of electricity, stating the values of L and S in terms of F.
- Find the cost function for the generation of electricity.
- Define in words returns to scale (RTS). What can you say about the returns to scale of the production of electricity?
- Does this production process exhibit economies of scale?

Problem 1.2

Take the production function for the manufacture of a particular car as a function of the amount of raw material (R), the amount of labor (L) and the size of the facility (S):

Q = R^{0.2} L^{0.4} S^{0.3}

Additionally, assume that the input cost function is:

C = R + 0.2 L ^{2} + S ^{1.5}

- Calculate the marginal rate of substitution of raw material for amount of labor at the point (R=2, L=3, S=2).
- State the conditions for optimal design as they apply to this case.
- Define in words the expansion path. Find the equation of the expansion path for the manufacture of cars, stating the values of L and S in terms of R.
- Find the cost function for manufacturing cars.
- Define in words returns to scale (RTS). What can you say about the returns to scale of manufacturing cars?
- Does this production process exhibit economies of scale?

Problem 1.3

The shipping industry now uses larger oil tankers than it did a generation ago, that is the ratio of shipping machinery and labor inputs into process of moving oil has shifted toward a higher amount of shipping machinery.

- How would you describe this shift in terms of the production function for shipping crude oil?
- How would you describe technological progress in terms of the production function?

Problem 1.4

For the sake of simplicity, take the production function for the generation and distribution of electric power to be a function of the amount of Fuel (F), the number of customers to whom the power must be distributed (D), and the size of the production facility (S):

Q = F^{0.3} D^{0.3} S^{0.2}

Additionally, assume that in some specific context the cost for the inputs to this production function is:

Cost of Inputs = F + D^{1.5 }+ S^{2}

- What can you say about the returns to scale of this production function?
- State the conditions for optimal design as they apply to this case.
- Define the expansion path, stating the values for D and S in terms of F.
- State and justify your conclusions about whether this production process exhibits economies of scale in the context defined.

Problem 1.5

Given the production function:

Z = 8 X^{0.4} Y^{1.6}

Given the input cost function:

C = 3 X^{2} + 20 Y^{1.6}

Calculate:

- Marginal products. Are they increasing, decreasing or ?
- Marginal rates of substitution.
- Returns to scale. What kind are they?
- Expansion path.
- Cost-effectiveness function.
- Economies of scale. Are they increasing, decreasing or ?

Problem 1.6

Given the production function:

Z = X^{0.2} Y^{0.3}

Given the input cost function:

C = X + 0.2 Y^{0.5}

- What are the marginal products?
- Are returns to scale increasing?
- What is the marginal rate of substitution at (X=1; Y=1)?
- State the conditions for optimal design as they apply to this case.
- Write the equation defining the expansion path. Explain the meaning of the expansion path.
- Find the cost function and cost-effectiveness function associated with this technology. Are there economies of scale?

Problem 1.7

Given the production function:

Z = 5 X^{0.6} Y^{0.2}

And given the input cost function:

C = 3 X^{2} + 2 Y

- Calculate the marginal products and the marginal costs.
- State the conditions for optimal design as they apply to this case.
- Define in words the expansion path. Write the equation for the expansion path.
- Find the cost function associated with this technology.
- State and justify your conclusions about whether this production process exhibits economies of scale.
- Calculate the marginal rate of substitution at (X = 6; Y = 2).
- Is the feasible region convex? Please explain.

Problem 1.8

An automotive assembly plant is currently operating near design capacity. More vehicles can be produced in the plant, but not very efficiently.

The production function is:

Z = 2 F^{0.4} L^{0.1}

The input cost function is:

C = 24 F + 6 L

With:

Z = extra vehicles / hour

C = extra cost in thousands of dollars / vehicle

F = extra materials

L = extra operating and maintenance employees

- Find the marginal products and the marginal costs.
- Determine the expansion path.
- Write an equation describing the cost-effectiveness curve for manufacturing additional vehicles from the current plant.
- Over the near term, the plant can supply additional vehicles (above the present level) to its customers from 3 sources:

- extra capacity can be wrung from the current plant, as previously described,
- vehicles can be imported from a foreign subsidiary at $15,000 / vehicle,
- there is one antiquated assembly plant which can be put back in service, to produce up to 30 vehicles / hour; the cost of having the facility in service is $12,000 / vehicle, regardless of the amount of vehicles generated.

Construct the cost-effectiveness curve for supplying additional vehicles.