You are the owner of a winery. Your production function isq=K^1/3L^1/3=cube rootKL, where

q measures the number of bottles of wine you can produce. K measures how many "capital-hours" you rent, and L is the number of hours of labor you hire. It costs $10 to rent capital, and $5 to hire a worker. Each bottle of your wine sells for $45, regardless of how many you sell (i.e. you are a price taker in both your input and output markets).

a) First, let's look at the short run. Assume capital is fixed at K=27 units of capital, but you are free to adjust labor as necessary to maximize profits. Calculate the marginal revenue product of labor in this situation, and use it to determine the profit-maximizing amount of labor to hire. How many bottles of wine will you produce? How much profit will you earn at this level of production?First, use the marginal revenue product rule to find L*. Then plug L* and K=27 back into the production function. Lastly, remember that C = wL +rK.The answers should all come out as integers

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b)Now it's the long run, so you can alter the amount of capital you are using. Assume you want to produce the same output as in part a. Calculate the cost minimizing level of capital and labor you should use. Calculate the total cost of production using the cost minimizing combination of inputs, and the profit you would earn.

c-Draw an isoquant/isocost graph that illustrates the short-run input choice from part a, and the cost-minimizing long-run input choice in part b.You don't need to be

d-Consider the suboptimal input mix we used in the short run (2a). Use either the "buck for the bang" rule or the "equal slopes" rule to suggest a small shift in input mix that would allow the firm to produce the same amount at lower cost.