Universal Widget produces high-quality widgets at its plant in Gulch, Nevada, for sale throughout the world. The cost function for total widget production \((q)\) is given by total cost \(=.25 q^{2}\) . Widgets are demanded only in Australia (where the demand curve is given by \(q=100-2 P\) ) and Lapland (where the demand curve is given by \(q=100-4 P\) ). If Universal Widget can control the quantities supplied to each market, how many should it sell in each location in order to maximize total profits? What price will be charged in each location? The production function for a firm in the business of calculator assembly is given by \\[ q=2 \sqrt{L} \\] where \(q\) is finished calculator output and \(L\) represents hours of labor input. The firm is a price taker for both calculators (which sell for \(P\) ) and workers (which can be hired at a wage rate of \(w \text { per hour })\) a. What is the supply function for assembled calculators \([q=f(P, w)]\) ? b. Explain both algebraically and graphically why this supply function is homogeneous of degree zero in \(P\) and \(w\) and why profits are homogeneous of degree one in these variables. c. Show explicitly how changes in \(w\) shift the supply curve for this firm.

Step by step solution

01

Identify the required equations and profit function

To start, let's identify the equations in the problem: - Total Cost is given by \(TC = 0.25 q^{2}\). - Demand curve in Australia: \(q_{A} = 100 - 2 P_{A}\). - Demand curve in Lapland: \(q_{L} = 100 - 4 P_{L}\). Next, we will find the profit function for Universal Widget. The profit function is given by \(Profit = Total \thinspace Revenue - Total \thinspace Cost\). Additionally, we know that \(TR = P \cdot q\).

02

Formulate Total Revenue

Since Universal Widget can control the quantity supplied to each market separately, we can calculate the total revenue separately for each market. So, the Total Revenue for both markets can be calculated as: \(TR = P_{A} q_{A} + P_{L} q_{L}\).

03

Find expressions for \(P_{A}\) and \(P_{L}\) using the demand curve equations

Now, we will find the expressions for \(P_{A}\) and \(P_{L}\) in terms of \(q_{A}\) and \(q_{L}\) using the respective demand curves. From the demand curve in Australia: \(q_{A} = 100 - 2 P_{A}\) and we have, \[P_{A} = \frac{100 - q_{A}}{2}\] Similarly, from the demand curve in Lapland: \(q_{L} = 100 - 4 P_{L}\) and we get, \[P_{L} = \frac{100 - q_{L}}{4}\]

04

Substitute the expressions for \(P_{A}\) and \(P_{L}\) in the total revenue equation

Now, we will substitute the expressions for \(P_{A}\) and \(P_{L}\) found in step 3 in to the total revenue equation we found in step 2. \[TR = (\frac{100 - q_{A}}{2}) q_{A} + (\frac{100 - q_{L}}{4}) q_{L}\]

05

Calculate the profit function and set up the optimization problem

First, let's calculate the total cost using the given cost function: \(TC = 0.25 (q_{A} + q_{L})^{2}\). Now, we are going to calculate the profit function by subtracting the total cost from the total revenue: \(Profit = TR - TC\). Our optimization problem now becomes: \[\max _{q_{A}, q_{L}} \left((\frac{100 - q_{A}}{2}) q_{A} + (\frac{100 - q_{L}}{4}) q_{L} - 0.25 (q_{A} + q_{L})^{2} \right)\]

06

Find the partial derivatives and solve for the optimal quantities

To maximize the profit, we will find the first-order partial derivatives with respect to \(q_{A}\) and \(q_{L}\), and set them equal to zero: \[\frac{\partial Profit}{\partial q_{A}}=0\] \[\frac{\partial Profit}{\partial q_{L}}=0\] Now, solve for \(q_{A}\) and \(q_{L}\) simultaneously to obtain the optimal quantities that maximize profits.

07

Find the optimal prices for each market using the optimal quantities

When we find the optimal quantities for both \(q_{A}\) and \(q_{L}\), we can plug these back into the expressions for \(P_{A}\) and \(P_{L}\), from step 3, to find the optimal prices that Universal Widget should charge in each market to maximize their total profits. At the end of this process, we'll have the optimal quantities (\(q_{A}\) and \(q_{L}\)) and prices (\(P_{A}\) and \(P_{L}\)) for the widget sales in Australia and Lapland that maximize Universal Widget's total profits.

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