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The automobile assembly plant you manage has a Cobb-Douglas production function given by P =...

Question:

The automobile assembly plant you manage has a Cobb-Douglas production function given by {eq}P = 20x^{0.2}y^{0.8} {/eq}

where {eq}P {/eq} is the number of automobile it produces per year, and {eq}x {/eq} is the number of employees, and {eq}y {/eq} is the daily operating budget (in dollars). Assume that you maintain a constant work force of 170 workers and wish in increase production on order to meet a demand that is increasing by 80 automobiles per year. The current demand is 800 automobiles per year.

How was should your daily operating budget be increasing? (Round your answer to the nearest cent.)

Operating Budget:

We have to find that how was should your daily operating budget be increasing. First we will find the operating budget with help of the given production function and again find the operating budget for increased demand with help of the given production function. Compare the both values to get the desired result.

Answer and Explanation:

We have given a Cobb-Douglas production function given by {eq}P = 20x^{0.2}y^{0.8}....(1) {/eq}

The current demand is 800 automobiles per year and you maintain a constant work force of 170 workers so, we have {eq}P = 800, x=170 {/eq}. Substitute the value into equation (1) and we have $$\begin{align*} P &= 20{x^{0.2}}{y^{0.8}}\\ 800 &= 20 \cdot {170^{0.2}} \cdot {y^{0.8}}\\ 800 &= 55.86 \cdot {y^{0.8}}\\ {y^{0.8}} &= \left( {\frac{{800}}{{55.86}}} \right)\\ y &= {\left( {14.32} \right)^{\frac{1}{{0.8}}}}\\ y &\approx 27.86. \end{align*} $$

When you wish in increase production on order to meet a demand that is increasing by 80 automobiles per year, then we have {eq}P = 880, x=170 {/eq}. Substitute the value into equation (1) and we have $$\begin{align*} P &= 20{x^{0.2}}{y^{0.8}}\\ 880 &= 20 \cdot {170^{0.2}} \cdot {y^{0.8}}\\ 880 &= 55.86 \cdot {y^{0.8}}\\ {y_1}^{0.8} &= \left( {\frac{{880}}{{55.86}}} \right)\\ {y_1} &= {\left( {15.75} \right)^{\frac{1}{{0.8}}}}\\ {y_1} &\approx 31.39. \end{align*} $$

Thus, your daily operating budget should be increasing $$\begin{align*} \Delta y &= {y_1} - y\\ &= 31.39 - 27.86\\ \Delta y &= 3.53/year. \end{align*} $$


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