In the US , beer and meat are the only two commodities produced and consumed. A gallon of beer requires 5 minutes of labor to produce, and a pound of meat requires 8 minutes of labor to produce. Assume that labor is the only input needed to produce beer and meat. There is a total of 1 2 3 billion hours of labor per year, and a total of 6 . 3 billion gallons of beer and 4 8 billion pounds of meat are produced each year in the US . Assume that wage is the same in the production of both goods, and the markets for the goods are both perfectly competitive. Within the US under autarky, what is the ratio of the price of meat to the price of beer? Let the ratio above be k * . Suppose that on the international market, we can trade any amount of beer for meat or meat for beer, at a ratio of k gallons of beer to 1 pound of meat. For what values of k will international trade increase the welfare in the US relative to autarky? will it be equal to k * or lesser or greater than k * ?

- Beer production: \( 5 \text{ minutes/gallon} \times 6.3 \text{ billion gallons/year} \times \frac{1}{60} \text{ hours/minute} = 1.05 \text{ billion hours/year} \)

- Meat production: \( 8 \text{ minutes/pound} \times 48 \text{ billion pounds/year} \times \frac{1}{60} \text{ hours/minute} = 6.4 \text{ billion hours/year} \)

The total labor hours used in beer and meat production is \( 1.05 + 6.4 = 7.45 \text{ billion hours/year} \).

Given that there are \( 123 \text{ billion hours/year} \) of labor available, there is \( 123 - 7.45 = 115.55 \text{ billion hours/year} \) available for other goods and services.

Now, let's find the price ratio under autarky:

- Price of beer: \( \frac{123}{6.3} = 19.523 \text{ billion hours/gallon} \)

- Price of meat: \( \frac{123}{48} = 2.5625 \text{ billion hours/pound} \)

The ratio of the price of meat to the price of beer (\( k^* \)) under autarky is \( \frac{2.5625}{19.523} \approx 0.131 \).

For international trade to increase welfare in the US relative to autarky, the international trade ratio (\( k \)) should be such that it improves the terms of trade for the US. This means \( k \) should be less than \( k^* \), making meat cheaper relative to beer in the international market compared to the autarky price ratio.

Therefore, international trade will increase welfare in the US relative to autarky for values of \( k \) that are lesser than \( k^* \).