Application to economics Consumer choice Profit maximisation

Application to economics Last week we saw how to find a constrained maximum: i.e.: finding the maximum of a function within a space defined by a set of constraints This is a methodology that is often used in economics We will see 2 examples today, based on the week 9 exercises

Application to economics Consumer choice Maximising profits

Consumer choice How do we use constrained optimisation to understand the choice of agents? We need a function to maximise:  This will be the utility function  This is our ‘satisfaction function’, which tells us how satisfied we are when we consume certain goods We need a constraint:  This is simply the consumer’s budget  It depends on income and prices

Consumer choice As for last week’s example, the 3-D function is represented by contours Lines of constant utility

The budget constraint   Good 2 Good 1 The budget constraint is Dividing by p 1 and rearranging: slope intercept

The budget constraint Any bundle within the budget constraint is affordable, but not all the budget is spent (C,D). Any bundle beyond the budget constraint cannot be afforded (H,G). C H D G Any bundle on the budget constraint is affordable and ensures all the budget is spent (E,F).   F E Good 2 Good 1

The budget constraint   Budget constraint Budget set Good 2 Good 1

The optimal consumer choice   Which is the best bundle ?  F Here !  C  D  E  B  A Good 2 Good 1

The Lagrangian method Formally the Lagrangian is Differentiating

The Lagrangian method The ratio of the two conditions gives: This can be expressed as

Application to economics Consumer choice Maximising profits

Another example is the maximisation of profits by firms The nice aspect is that it illustrates how constrained maximisation is just another form of free maximisation !! We need a function to maximise:  This is the revenue function: the revenue made from selling a given quantity of output We need a constraint:  This is the cost function, which gives the cost of producing a given quantity of output

The Lagrangian method Profit maximisation can be presented Either as a constrained maximisation Or as a free maximisation This is because in this case the Lagrangian multiplier is equal to 1

The Lagrangian method Taking the first order condition Either as a constrained maximisation... Or as a free maximisation... Is therefore the same!

The Lagrangian method The implication of this result Simplification: lets assume that marginal revenue is price. Then the profit is maximised when the marginal cost of production (the cost of producing an extra unit) is equal to the price the good is sold at.