Question

The production function for a firm in the business of calculator assembly is given by \[ q=2 \sqrt{l} \] where $q$ denotes finished calculator output and $l$ denotes hours of labor input. The firm is a price-taker both for calculators (which sell for $P$ ) and for workers (which can be hired at a wage rate of $w$ per hour). a. What is the total cost function for this firm? b. What is the profit function for this firm? c. What is the supply function for assembled calculators $[q(P,w)]?$ d. What is this firm's demand for labor function $[l(P,w)]?$ e. Describe intuitively why these functions have the form they do.

Short answer

#Answer# The supply function is given by q(P, w) = P/w. This indicates that the firm is willing to produce and supply more calculators as the product price (P) increases and/or labor cost (w) decreases.

Step-by-step solution

a. Total Cost Function

First, let's find the variable cost as a function of labor input l. Since the firm hires workers at a wage rate w per hour, the variable cost is given by: Variable Cost (VC) = w * l Now, we can find the total cost function by adding the fixed cost (assume it to be zero for simplicity) to the variable cost: Total Cost (TC) = VC = w * l

b. Profit Function

Profit is the difference between total revenue (TR) and total cost (TC). Total revenue is given by the product of price (P) and quantity (q) of calculators produced: TR = P * q = P * (2 * √l) Now, we can find the profit function by subtracting TC from TR: Profit = TR - TC = P * (2 * √l) - w * l

c. Supply Function

To find the supply function q(P, w), first, we should find the optimal labor input (l*) by maximizing the profit function. To maximize profit, differentiate with respect to l and set the result equal to zero: \[ \frac{d(profit)}{dl} = \frac{d(P * (2 \sqrt{l}) - w * l)}{dl} = 0 \] Solve for the optimal labor input (l*): \[ l* = \frac{P^2}{4w^2} \] Now, substitute l* into the production function (q = 2√l) to get the supply function: q(P, w) = 2√l* = 2√(P²/4w²) = P/w

d. Demand for Labor Function

We already found the optimal labor input l* in the previous step. So, the firm's demand for labor function l(P,w) is given by: l(P, w) = l* = P²/4w²

e. Intuitive Explanation

a. The total cost function is directly proportional to the labor input and the wage rate, which reflects the fact that the firm's cost increases linearly as the labor input and wage rate increase. b. The profit function shows how the difference between the revenue (determined by the price of the product, and labor input) and labor costs impacts the overall profit that the firm can make. c. The supply function q(P, w) = P/w indicates that the firm is willing to produce and supply more calculators as the product price P increases and/or labor cost w decreases. d. The demand for labor function l(P, w) = P²/4w² shows that firms will demand more labor as the price P increases relative to the wage rate w - it is a more cost-effective input when product price is high, or labor cost is low. e. These functions outline how different factors, such as product price (P), labor cost (w), and labor input (l), interact and shape a firm's decision-making process in the calculator assembly market. Understanding these relationships can help the firm optimize its operations to maximize profit.

Key concepts

Total Cost Function

The total cost function provides insight into how much a firm spends based on labor input and wage rates. In this scenario, since the firm incurs costs by hiring workers, the variable cost (VC) is calculated by multiplying the number of labor hours ($l$) by the wage rate ($w$). So, the equation is:

• Variable Cost (VC): $VC=w\times l$

Assuming no fixed costs, the total cost ($TC$) becomes entirely the variable cost. So, we can say:

• Total Cost (TC): $TC=w\times l$

This function helps businesses track their expenses, especially with rising labor needs or wage changes.

Supply Function

The supply function shows the relationship between the quantity of goods a firm is willing to make available to the market and factors like product price and cost of production. In this calculator assembly scenario, the equation for supply ($q(P,w)$) is found through optimizing labor input to maximize profits.
This is done by maximizing profit, differentiating with respect to labor, and finding the point where profit is highest. The optimal labor input is represented as:

• Optimal labor input ($l^{\ast }$): $l^{\ast }=\frac{P^2}{4w^2}$

Substituting this back into the production function gives us the supply function:

• Supply: $q(P,w)=\frac{P}{w}$

This reveals that as the product price rises, or labor cost falls, the firm is prepared to supply more calculators.

Profit Maximization

Profit maximization is the core motive of most firms. It involves determining the level of production and resource allocation where a firm's earnings outstrip costs by the greatest margin.
For our firm, profit is calculated as the difference between total revenue and total costs. Total revenue ($TR$) is simply the price per unit ($P$) multiplied by the number of units ($q$), thus:

• Total Revenue: $TR=P\times (2\sqrt{l})$

Subtracting total costs, we derive the profit ($extProfit$) function:

• Profit: $extProfit=P\times (2\sqrt{l})-w\times l$

Solving for labor where this function is maximized leads to greater profitability.

Labor Demand Function

The labor demand function delineates how much labor a firm requires to maximize profits based on external factors like wages and price of goods. For maximized profit, the firm calculates the optimal labor ($l^{\ast }$) necessary to produce the desired output efficiently.
This is obtained from the profit maximization process:

• Demand for labor function: $l(P,w)=\frac{P^2}{4w^2}$

Essentially, as the price of the product increases relative to the wage, labor becomes a more attractive input. Consequently, firms will seek more labor resources to leverage the higher pricing power of their product.
This function underscores the sensitivity of labor demand to fluctuations in market conditions and production costs.