Impulse Response Function. Suppose a one-time TFP shock occurs, as shown below.
As we have studied, an increase in TFP leads to an outwards shift in labor demand (recall this from our firm analysis unit), which, as long as the upward-sloping labor supply function does not shift, leads to an increase in the real wage.
Using an infinite-horizon (which, recall, is a heuristic for a "many, many, many time-period" framework) of the combined consumption-savings and consumption-labor framework (which is an extension of the brief two-period framework of Chapter 5), qualitatively plot an impulse response function for the representative consumer's optimal labor supply that lines up with the impulse response profile for TFP drawn above.
Use the lifetime utility function
in which the utility parameters ψ > 0 (the Greek lower-case letter "psi") and ν > 0 (the Greek lower-case letter "nu") are exogenous to the representative consumer.