Question

Question 3: Louis got tired of living on a fixed budget and decided to take a part-time job at Smoke’s Poutinerie where he is in charge of least cost food production. As everyone knows, the production function for poutine is given by q=min[F, 2C, 2G]whereFis a cup of french fries, C is a tablespoon of cheese curds, and G is a tablespoon of gravy. The price of these inputs into production are given by (pF,pC, pG)= (2, 1, 1).

(a) Are fries, cheese curds, and gravy substitutes or complements?

(b) What is the least cost combination of inputs needed to produce one unit of poutine?

(c) What is the least cost combination of inputs needed to produce some specified amount of q units of poutine?

(d) Solve for Smoke’s Poutinerie’s food cost function, C(q).

Answer 1

The production function is $\dpi{80} q = min[F,2C,2G]$ , and the cost of production would be $\dpi{80} 2F + C + G = c$ , where I will be the cost.

(a) Fries, cheese curds and gravy are complements. The production function is given to be as Leontif production function. Without increasing in the fixed proportion of F, C and G, q would not increase, meaning that the components of production are compliments, not substitutes. These production functions have corner solution.

(b) To produce one unit of the product, the least cost combination of inputs would be $\dpi{80} F = 2C = 2G = 1$ . This is the corner of the isoquant $\dpi{80} 1 = min[F,2C,2G]$ .

(c) The least cost combination of inputs to produce q units would be $\dpi{80} F = 2C = 2G = q$ .

(d) The cost of production is $\dpi{80} c = 2F + C + G$ . We have $\dpi{80} F = 2C = 2G = q$ or $\dpi{80} C = G = 0.5F = 0.5q$ as the least cost combination of inputs. Putting $\dpi{80} C = G = 0.5q$ and $\dpi{80} 0.5F = 0.5q$ or $\dpi{80} F = q$ , we have $\dpi{80} c = 2q + 0.5q + 0.5q$ or $\dpi{80} c = 3q$ , which is the cost function.