a producer of light bulbs claims to have the folllowing production function: Q=10L*G.
a) what is the marginal product of labor? or glass?
b)Draw the relavant isoquant map and isocost line if the cost of labor is $4 per work-hour and the cost of glass is $4 per pound and 90 light bulds are to be produced.
What is the cost minimizing combination of glass and labor>
c) If Q were to increase to 160 light bulbs, would this firm exhibits constant, increasing or decreasing returns to scale. Explain.
a) The marginal product of labor is the change in output resulting from a one-unit increase in the quantity of labor used, holding the quantity of glass constant. Mathematically, it is the derivative of the production function with respect to labor.
To find the marginal product of labor, we take the derivative of the production function with regard to L, while assuming G is constant:
∂Q/∂L = 10G.
Similarly, the marginal product of glass is the change in output resulting from a one-unit increase in the quantity of glass used, holding the quantity of labor constant. Mathematically, it is the derivative of the production function with respect to glass:
∂Q/∂G = 10L.
b) To draw the isoquant map, we need to find different combinations of labor (L) and glass (G) that produce a certain level of output. In this case, we want to produce 90 light bulbs.
Given the production function: Q = 10LG,
we can rearrange the equation to solve for one variable in terms of the other:
L = Q/(10G).
Let's assume labor (L) is on the vertical axis and glass (G) is on the horizontal axis. We plug in different values for Q in the equation above to find the corresponding combinations of L and G that produce the desired output (90 light bulbs). For example, when Q = 90:
L = 90/(10G),
L = 9/G.
Now we can plot different combinations of L and G on a graph.
To draw the isocost line, we need to consider the cost of labor and the cost of glass. Since the cost of labor is $4 per work-hour and the cost of glass is $4 per pound, we can choose different cost values to create multiple isocost lines on the same graph.
c) To determine whether the firm exhibits constant, increasing, or decreasing returns to scale when Q increases to 160 light bulbs, we need to assess how output changes with respect to the scale of inputs (in this case, labor and glass).
If output increases proportionally to the increase in inputs, it indicates constant returns to scale. If output increases more than proportionally, it indicates increasing returns to scale. Conversely, if output increases less than proportionally, it indicates decreasing returns to scale.
To determine the effect, we compare the initial production function (Q = 10LG) with the new production function (Q = 160).
For the original production function: Q = 10LG,
let's assume L0 and G0 are the labor and glass inputs for Q = 90.
For the new production function: Q = 160,
let's assume L1 and G1 are the labor and glass inputs for Q = 160.
If (L1/L0) = (G1/G0), it indicates constant returns to scale.
If (L1/L0) > (G1/G0), it indicates increasing returns to scale.
If (L1/L0) < (G1/G0), it indicates decreasing returns to scale.
Based on the given information and the comparison of inputs for the two production levels, we can determine the type of returns to scale exhibited by the firm.
a) To find the marginal product of labor (MPL), you need to calculate the change in output (Q) resulting from a unit increase in labor (L), while keeping the other input (glass) constant. The formula for MPL is given by the derivative of the production function with respect to labor.
To find MPL, differentiate the production function Q = 10L*G with respect to L while treating G as a constant. The derivative of 10L with respect to L is simply 10, so MPL = 10G.
Similarly, to find the marginal product of glass (MPG), differentiate the production function Q = 10L*G with respect to G while treating L as a constant. The derivative of 10G with respect to G is simply 10, so MPG = 10L.
b) To draw the relevant isoquant map and isocost line, we need to first understand the concept of isoquants and isocost lines.
- Isoquants: An isoquant represents all the combinations of labor and glass that can produce a given level of output (Q). In this case, we'll draw the isoquant for producing 90 light bulbs.
- Isocost Line: An isocost line represents all the combinations of labor and glass that have the same cost. In this case, we'll draw the isocost line for a total cost of producing 90 light bulbs, considering the cost of labor ($4 per work-hour) and the cost of glass ($4 per pound).
To find the cost-minimizing combination of glass and labor, you need to determine the point where the isoquant and the isocost line are tangent. This point represents the least-cost combination to produce the required output.
c) To determine if the firm exhibits constant, increasing, or decreasing returns to scale, we need to compare the change in output to the change in inputs when scaling up production.
Given that the original output (Q) is 90 light bulbs and the new output is 160 light bulbs, we can calculate the scale factor by dividing the new output by the original output: Scale Factor = 160/90 = 1.78 (approximately).
If the scale factor is equal to one, the firm exhibits constant returns to scale, meaning that the change in output is proportional to the change in input.
If the scale factor is greater than one, the firm exhibits increasing returns to scale, meaning that the change in output is more than proportional to the change in input.
If the scale factor is less than one, the firm exhibits decreasing returns to scale, meaning that the change in output is less than proportional to the change in input.
In this case, since the scale factor is 1.78, the firm exhibits increasing returns to scale. This means that the firm experiences a greater proportionate increase in output compared to the increase in inputs when scaling up production.