Question

Assume that output is given by with price of labour L = w and price of capital K = r

1.If capital in the short run is fixed at what is the short-run total cost?

2.Write the values for the derivatives of the Total cost with respect to w and r. Does Shephard’s lemma hold in this case?

Answer 1

Price of labour L = w

price of Capital K = r

1. In the short run perspective a firm's total cost can be divided in to two - fixed cost and variable cost. Fixed cost actually have no role in any economic decisions in connection with production or pricing. On the other hand marginal returns of producing each additional unit of output very much depends upon variable cost.  

Hence capital in the short run is fixed, we can't add any more additional labour, which is a variable cost, then no additional quantity can be produced. If more labour is added, more quantity can be produced, and the fixed cost remains the same. Then in the short run cost of production will be less.

But in certain situations when capital remians fixed, no additional labour, no additional unit of output, fixed cost also remains the same the cost of production will increase.  

In short total cost is the sum of fixed and variable cost of production.

The cost of production of each unit of output is closely depended upon the amount of labour and capital required. That is why the cost of production of an SUV is different from the cost of production of computer software, hare cut or fast food items.

Shephard's Lemma

This is a micro economic theory focused on the relationship between indirect utility function and demand function. According to this theory if the expenditure or cost function are convex, the cost minimising point of a given good with price will be unique. This theory is closely depended on consumer choice.

According to this theory the consumers show a tendency of purchasing a certain amount of each item as per their need, with an expectation of obtaining the product with minimum price level. This theory states that the expenditure function of the price of goods are equal to the demand function of the relevant goods.