Question

A profit-maximizing firm uses two inputs and has the production function f(x1,x2)=x1ax2a where α+β<1 and α,β>0. It faces an output price of 1 and constant input prices, wi where i=1,2. The government imposes a tax on the firm of ti per unit of input i purchased. Suppose the taxes are proportional to the input prices such that t1=kw1 and t2=kw2 for some k>0. Show that the firm's input demands after taxes are proportional to the input demands before taxes. Problem 2: A consumer has a utility function of the form u(x1,x2)=−x11−x21. a) Compute the Marshallian demand functions. b) Show that the indirect utility function is −(p10.5+p20.5)2/m. c) Compute the expenditure function. d) Compute the Hicksian demand functions for x1 and x2. Problem 3: a) Assume that the preferences may be represented by the Cobb-Douglas utility function lnu=α1lnx1+α2lnx2. Derive the own-price, cross-price and income elasticities. b) Assume the preferences may be represented by the Linear Expenditure System (LES) utility function u=β1ln(x1−γ1)+β2ln(x2−γ2) where β1,β2>0 and β1+β2=1 and x1>γ1≥0 and x2>γ2≥0. Derive the own-price, cross-price and income elasticities. c) Show that elasticities derived from the LES utility function indicate that we have no inferior goods, only gross complements, and own-price inelastic demand. Problem 4: Outline the properties of homothetic production technology.

Answer 1

A production function illustrates how much output a firm can produce with different inputs, and these vary by product and industry.

Step-by-step explanation:

The concept of a production function is crucial in understanding the output a firm can produce given various input quantities. The production function reflects the relationship between inputs and outputs, which varies across different goods and industries. For instance, a pizza restaurant's production will differ if it makes its dough and sauce in-house versus purchasing them pre-made. The amount of labor and other resources used can significantly impact the production levels and efficiency. Inputs are categorized as either fixed or variable: Fixed inputs, like the building of a restaurant, cannot be changed quickly, while variable inputs, such as labor, can be adjusted more readily. This concept ties into the firm's decision-making process concerning resource allocation to maximize profits.

When it comes to utility maximization, consumers make choices based on their preferences, which are subject to their budget constraints. The utility-maximizing decision occurs at the point where an indifference curve, representing levels of utility, is just tangent to the budget constraint. Changes in prices lead to substitution and income effects. The substitution effect incentivizes consumers to purchase less of the relatively more expensive goods and more of the cheaper ones. Simultaneously, the income effect can make consumers buy more or less of both goods when their perceived income changes, depending on whether the goods are normal goods or inferior goods.